[{"data":1,"prerenderedAt":6302},["ShallowReactive",2],{"site-chrome-my":3,"site-chrome-en":-1,"site-chrome-th":-1,"blog-page-my":287,"blog-categories-my":345,"blog-posts-my":365,"article-blog-my-\u002Fblog\u002Fprediction-and-risk-optimization-meta-review":722,"og-copy-my":6260,"og-posts-my-none":6301},{"id":4,"extension":5,"footer":6,"header":116,"indexFacets":202,"meta":243,"notFound":244,"stem":250,"ui":251,"__hash__":286},"chrome_en\u002Fen\u002Fchrome.yml","yml",{"columns":7,"legal":83,"quote":84,"askAi":87,"socials":103},[8,29,47,65],{"label":9,"children":10},"Product",[11,14,17,20,23,26],{"label":12,"to":13},"Ads optimization","\u002Fmy\u002Fblog\u002Fads-optimization",{"label":15,"to":16},"Unit economics","\u002Fmy\u002Fblog\u002Fprofit-margin",{"label":18,"to":19},"Agentic KAM","https:\u002F\u002Flearn.dataglasslabs.com\u002Fdocs\u002Fai-and-api\u002Fassistants",{"label":21,"to":22},"One-click deploy","https:\u002F\u002Flearn.dataglasslabs.com\u002Fdocs\u002Fgrowth\u002Factions\u002Faction-flow",{"label":24,"to":25},"Integrations","https:\u002F\u002Flearn.dataglasslabs.com\u002Fdocs\u002Foverview\u002Fget-started\u002Fsupported-platforms",{"label":27,"to":28},"Pricing","\u002Fmy\u002Fpricing",{"label":30,"children":31},"Resources",[32,35,38,41,44],{"label":33,"to":34},"Blog","\u002Fmy\u002Fblog",{"label":36,"to":37},"Customer stories","\u002Fmy\u002Fcustomers",{"label":39,"to":40},"Research","\u002Fmy\u002Fresearch",{"label":42,"to":43},"Glossary","\u002Fmy\u002Fglossary",{"label":45,"to":46},"Docs","https:\u002F\u002Flearn.dataglasslabs.com\u002Fdocs",{"label":48,"children":49},"Company",[50,53,56,59,62],{"label":51,"to":52},"About","\u002Fmy\u002Fcompany",{"label":54,"to":55},"Contact","\u002Fmy\u002Fcontact",{"label":57,"to":58},"Book a demo","\u002Fmy\u002Fbook-a-demo",{"label":60,"to":61},"Privacy","\u002Fmy\u002Flegal\u002Fprivacy",{"label":63,"to":64},"Terms","\u002Fmy\u002Flegal\u002Fterms",{"label":66,"children":67},"DataGlass for",[68,71,74,77,80],{"label":69,"to":70},"Agencies","\u002Fmy\u002Ffor\u002Fagencies",{"label":72,"to":73},"Brands","\u002Fmy\u002Ffor\u002Fbrands",{"label":75,"to":76},"Planners and optimizers","\u002Fmy\u002Ffor\u002Fplanners",{"label":78,"to":79},"Managers and analysts","\u002Fmy\u002Ffor\u002Fmanagers",{"label":81,"to":82},"Sellers and shop owners","\u002Fmy\u002Ffor\u002Fsellers","© 2026 DataGlass Labs. All rights reserved.",{"text":85,"caption":86},"Measure what you keep, and then grow what you measure.","The DataGlass team",{"label":88,"prompt":89,"assistants":90},"Ask AI about DataGlass","I want to understand DataGlass, a decision engine for Shopee, Lazada and TikTok Shop sellers that optimizes ads, pricing and promotions for true profit. Use web search to read their website at https:\u002F\u002Fwww.dataglasslabs.com (with an agent-readable summary at https:\u002F\u002Fwww.dataglasslabs.com\u002Fllms.txt), then explain what DataGlass does, who it is for and how it differs from an analytics dashboard.",[91,95,99],{"label":92,"icon":93,"url":94},"ChatGPT","chatgpt","https:\u002F\u002Fchatgpt.com\u002F?prompt={prompt}",{"label":96,"icon":97,"url":98},"Claude","claude","https:\u002F\u002Fclaude.ai\u002Fnew?q={prompt}",{"label":100,"icon":101,"url":102},"Perplexity","perplexity","https:\u002F\u002Fwww.perplexity.ai\u002Fsearch?q={prompt}",[104,108,112],{"label":105,"icon":106,"to":107},"LinkedIn","i-lucide-linkedin","https:\u002F\u002Fwww.linkedin.com\u002Fcompany\u002Fdataglass-labs",{"label":109,"icon":110,"to":111},"Facebook","i-lucide-facebook","https:\u002F\u002Fwww.facebook.com\u002Fdataglasslabsth",{"label":113,"icon":114,"to":115},"Instagram","i-lucide-instagram","https:\u002F\u002Fwww.instagram.com\u002Fdataglasslabs\u002F",{"links":117,"cta":196,"secondaryCta":199},[118,152,191,193,194,195],{"label":9,"to":119,"menu":120},"\u002Fmy",{"layout":121,"heading":122,"cards":123,"groups":135,"footer":142},"cards","What DataGlass does",[124,128,132],{"label":125,"description":126,"to":13,"ui":127},"True profit ads optimization","Budgets, pacing and Target ROAS tuned for profit.","budget-tomorrow",{"label":129,"description":130,"to":16,"ui":131},"Unit economics breakdown","What you actually earn from every sale.","unit-economics",{"label":18,"description":133,"to":19,"motif":134},"An AI agent that knows your shops.",true,[136],{"label":137,"size":138,"links":139},"More","secondary",[140,141],{"label":21,"to":22},{"label":24,"to":25},[143,148],{"lead":144,"label":145,"icon":146,"to":147},"DataGlass v.2.6.","See what's new","i-lucide-box","https:\u002F\u002Flearn.dataglasslabs.com\u002Freleases",{"lead":149,"label":150,"icon":151,"to":37},"Read our","customer 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us",{"label":192,"to":37},"Customers",{"label":27,"to":28},{"label":48,"to":52},{"label":54,"to":55},{"label":197,"to":198},"Start for free","https:\u002F\u002Fapp.dataglasslabs.com\u002Fsign-up",{"label":200,"to":201},"Sign in","https:\u002F\u002Fapp.dataglasslabs.com\u002Fsign-in",{"clear":203,"groups":204},"Clear filters",[205,218],{"key":206,"label":207,"options":208},"platforms","Platform",[209,212,215],{"value":210,"label":211},"shopee","Shopee",{"value":213,"label":214},"lazada","Lazada",{"value":216,"label":217},"tiktok","TikTok Shop",{"key":219,"label":220,"options":221},"format","Format",[222,225,228,231,234,237,240],{"value":223,"label":224},"guide","Guide",{"value":226,"label":227},"article","Article",{"value":229,"label":230},"comparison","Comparison",{"value":232,"label":233},"calculator","Calculator",{"value":235,"label":236},"paper","Paper",{"value":238,"label":239},"survey","Survey",{"value":241,"label":242},"dataset","Dataset",{},{"title":245,"description":246,"cta":247,"errorTitle":249},"Page not found","The page may have moved, or it never existed.",{"label":248,"to":119},"Back to home","Something went wrong","en\u002Fchrome",{"previous":252,"next":253,"pause":254,"play":255,"goTo":256,"expand":257,"close":258,"readMore":259,"learnMore":260,"seeAll":261,"menu":262,"back":263,"share":264,"shareOn":265,"copyLink":266,"copied":267,"onThisPage":268,"minRead":269,"sources":270,"viewMore":271,"searchPosts":272,"allPosts":273,"all":274,"filters":275,"search":276,"searchClose":277,"searchPlaceholder":278,"searchSuggested":279,"searchResults":280,"searchNoResults":281,"searchLoading":282,"relatedTitle":283,"mainNav":284,"footerNav":285},"Previous","Next","Pause","Play","Go to item","Expand","Close","Read more","Learn more","See all","Menu","Back","Share","Share on","Copy link","Link copied","On this page","{n} min read","Sources","View more","Search posts","All posts","All","Filters","Search","Close search","Search DataGlass","Suggested","Results","No results for","Loading","More to discover","Main","Footer","HZQSEulcpFGY0nWnUTNYhG2n06nO97XnUBc_pEVOFxo",{"id":288,"title":33,"blog":289,"body":327,"customers":334,"description":335,"draft":336,"draftReason":334,"extension":337,"glossary":334,"industry":334,"meta":338,"navigation":336,"ogImage":339,"path":340,"research":334,"robots":334,"schemaOrg":334,"seo":341,"showcase":336,"sitemap":342,"stem":343,"__hash__":344},"pages_en\u002Fblog.md",{"emptyLabel":290,"labels":291,"indexLabels":296,"listenLabels":307,"indexCta":320,"ctaButtons":323},"No posts match yet. Try another category or a shorter search.",{"tldr":292,"published":293,"updated":294,"related":295},"In short","Published","Updated","Keep reading",{"categories":297,"sort":298,"newest":299,"oldest":300,"grid":301,"list":302,"loadMore":303,"results":304,"result":305,"clearSearch":306},"Categories","Sort","Newest first","Oldest first","Grid view","List view","Load more","{n} results for “{q}”","1 result for “{q}”","Clear search",{"listen":308,"loading":309,"play":310,"pause":311,"back":312,"forward":313,"speed":314,"seek":315,"player":316,"close":317,"follow":318,"of":319},"Listen to article","Loading…","Play audio of this article","Pause audio","Back 15 seconds","Forward 15 seconds","Playback speed","Seek","Audio player","Pause audio and close player","Follow the text","{a} of {b}",{"headline":321,"subline":322},"Turn every post into a decision","See the numbers behind these notes for your own shops, with a demo on your data.",{"primaryCta":324,"secondaryCta":325},{"label":57,"to":58},{"label":326,"to":28},"See pricing",{"type":328,"value":329,"toc":330},"minimark",[],{"title":331,"searchDepth":332,"depth":332,"links":333},"",3,[],null,"Notes on profit, pricing and ads for sellers on Shopee, Lazada and TikTok Shop, with the numbers behind them.",false,"md",{},{"component":33},"\u002Fblog",{"title":33,"description":335},{"loc":340},"blog","VuN4_Knpp3IpkJqH-kRoXag4vgoPV2p71ZFiR-VpgzQ",[346,349,351,354,357,360,363],{"slug":347,"label":348},"profit","Profit",{"slug":350,"label":27},"pricing",{"slug":352,"label":353},"ads","Ads",{"slug":355,"label":356},"operations","Operations",{"slug":358,"label":359},"market","Market",{"slug":361,"label":362},"data-science","Data Science",{"slug":364,"label":48},"company",[366,373,379,383,388,393,397,402,407,412,418,423,428,433,438,444,450,456,462,469,475,480,486,491,496,501,506,511,517,523,528,534,540,546,551,557,563,569,574,578,583,588,594,601,607,613,619,625,631,637,643,649,655,661,667,673,679,685,690,695,700,705,710,716],{"title":367,"description":368,"category":352,"date":369,"readMinutes":370,"featured":336,"cover":334,"format":223,"platforms":371,"draft":336,"slug":372},"E-commerce ads optimization for Shopee, Lazada, and TikTok Shop, on profit","E-commerce ads optimization on profit: True ROAS, margin-based Target ROAS and nightly budget allocation for Shopee, Lazada and TikTok Shop ads, applied for you.","2026-10-23",4,[210,213,216],"ads-optimization",{"title":374,"description":375,"category":352,"date":369,"readMinutes":376,"featured":336,"cover":334,"format":232,"platforms":377,"draft":336,"slug":378},"What ROAS is break-even? Calculate break-even ROAS and your maximum ad spend per order","Break-even ROAS is 1 ÷ your contribution-margin ratio. It does not depend on how much you spend on ads. This calculator derives it from a dated Shopee Thailand fee rate — or one you type in — then tells you the number sellers actually need: the most baht one order can pay for ads before it stops making money.",9,[210],"break-even-roas-calculator",{"title":380,"description":381,"category":364,"date":369,"readMinutes":332,"featured":336,"cover":334,"format":229,"platforms":334,"draft":336,"slug":382},"DataGlass vs a marketing agency for marketplace profit","When an agency is the right hire, where the agency model has structural blind spots for marketplace profit, and how DataGlass complements an agency engagement.","dataglass-vs-agency",{"title":384,"description":385,"category":364,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":229,"platforms":334,"draft":336,"slug":387},"DataGlass vs an ERP for marketplace decision-making","When an ERP is the right tool, where it falls short for marketplace ad\u002Fpricing decisions, and how DataGlass complements (not replaces) the ERP for SEA sellers.",2,"dataglass-vs-erp",{"title":389,"description":390,"category":364,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":229,"platforms":391,"draft":336,"slug":392},"DataGlass vs Shopee Seller Center for profit-aware operations","Shopee Seller Center is essential for running a Shopee shop, but it does not show profit-aware metrics or cross-platform views. How DataGlass complements it.",[210],"dataglass-vs-shopee-seller-center",{"title":394,"description":395,"category":364,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":229,"platforms":334,"draft":336,"slug":396},"DataGlass vs spreadsheets for marketplace sellers","When spreadsheets are enough, where they break for marketplace sellers, and how DataGlass connects ads, COGS, fees, and inventory into one decision-ready view.","dataglass-vs-spreadsheet",{"title":398,"description":399,"category":355,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":223,"platforms":400,"draft":336,"slug":401},"Prevent stockouts and overstock across Shopee, Lazada, and TikTok Shop","Prevent Shopee, Lazada, and TikTok Shop stockouts with demand-aware forecasting, campaign-day planning, and cross-platform inventory unification.",[210,213,216],"inventory-stockout",{"title":403,"description":404,"category":355,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":226,"platforms":405,"draft":336,"slug":406},"Lazada integration for marketplace profit optimization","DataGlass integrates with Lazada to model true profit per SKU, detect ad waste, evaluate voucher impact, and forecast stockout risk.",[213],"lazada-integration",{"title":408,"description":409,"category":347,"date":369,"readMinutes":376,"featured":336,"cover":334,"format":232,"platforms":410,"draft":336,"slug":411},"Shopee raised its fees on 4 August 2026 — how much more is your shop paying?","The new rate table is everywhere. Your number is not. Enter last month’s GMV by category and this page computes the baht difference between the schedule effective 1 June 2026 and the one effective 4 August 2026 — including the extra hit for shops that were on the discounted-commission programme.",[210],"marketplace-fee-impact-calculator",{"title":413,"description":414,"category":347,"date":369,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":416,"draft":336,"slug":417},"Optimize e-commerce profit across Shopee, Lazada, and TikTok Shop","How to optimize e-commerce profit across Shopee, Lazada and TikTok Shop: one objective, five decision variables, and the constraint that binds each.",7,[210,213,216],"optimize-ecommerce-profit",{"title":419,"description":420,"category":350,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":223,"platforms":421,"draft":336,"slug":422},"Price and promote without destroying margin","Optimize Shopee, Lazada, and TikTok Shop discounts, bundle deals, and campaign-day pricing on true margin. Project promotion impact before publishing.",[210,213,216],"pricing-promotions",{"title":424,"description":425,"category":347,"date":369,"readMinutes":370,"featured":336,"cover":334,"format":232,"platforms":426,"draft":336,"slug":427},"E-commerce profit calculator for Shopee sellers","Calculate true Shopee profit per SKU with a per-order breakdown covering commission fees, ads, COGS, and fulfillment.",[210],"profit-margin",{"title":429,"description":430,"category":355,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":226,"platforms":431,"draft":336,"slug":432},"Shopee integration for profit, ads, stock, and promotion decisions","DataGlass connects to Shopee to model true profit per SKU, detect wasted ad spend, and surface daily decisions. Live integration, no manual exports.",[210],"shopee-integration",{"title":434,"description":435,"category":355,"date":369,"readMinutes":386,"featured":336,"cover":334,"format":226,"platforms":436,"draft":336,"slug":437},"TikTok Shop integration for ads, inventory, and profit decisions","DataGlass integrates with TikTok Shop for True ROAS on video and live ads, margin-aware discount analysis, and stock prioritization before live events.",[216],"tiktok-shop-integration",{"title":439,"description":440,"category":352,"date":441,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":442,"draft":336,"slug":443},"Shopee ROAS protection and TikTok ROI protection: when the credit actually protects profit","Shopee ROAS protection credits campaigns back to 80% of target, TikTok to 90%. Why that protects profit only above 1.25× break-even, with a worked example.","2026-10-08",[210,216],"roas-protection",{"title":445,"description":446,"category":352,"date":441,"readMinutes":447,"featured":336,"cover":334,"format":223,"platforms":448,"draft":336,"slug":449},"Shopee Ads optimization: set Target ROAS, budget and timing from profit","How to optimize Shopee Ads for profit: break-even ROAS per SKU, which lever binds, and CPC by hour from 9.11M Thai clicks. Worked THB example.",10,[210],"shopee-ads-optimization",{"title":451,"description":452,"category":347,"date":441,"readMinutes":453,"featured":336,"cover":334,"format":223,"platforms":454,"draft":336,"slug":455},"Shopee seller fees in the Philippines 2026: every fee, what it is charged on, and your break-even ROAS","Shopee Philippines seller fees 2026: commission, 2.24% transaction fee, 5.6% platform shipping, seller growth support and ₱5 order fee, worked on one order.",5,[210],"shopee-seller-fees",{"title":457,"description":458,"category":352,"date":441,"readMinutes":459,"featured":336,"cover":334,"format":223,"platforms":460,"draft":336,"slug":461},"TikTok Shop ads optimization: set GMV Max ROI from margin and affiliate mix","How to optimize TikTok Shop GMV Max for profit: break-even ROI after TikTok fees, the affiliate commission trap, and the 3% commission rule. Worked THB example.",8,[216],"tiktok-shop-ads-optimization",{"title":463,"description":464,"category":352,"date":465,"readMinutes":466,"featured":336,"cover":334,"format":226,"platforms":467,"draft":336,"slug":468},"How Shopee ads bidding works: what your Target ROAS and budget actually do to a bid","How Shopee ads bidding works, derived: GMV Max gives you two boxes — a Target ROAS and a daily budget — and every guide treats them as two levers. They are not. Both collapse into a single threshold on value-to-cost before any bid is placed, only one is binding at a time, and the other is doing nothing at all. Here is the derivation, the three regimes a campaign can be in, and the one-line test that tells you which of your two levers is switched off.","2026-09-01",13,[210],"how-shopee-ads-bidding-works",{"title":470,"description":471,"category":352,"date":465,"readMinutes":472,"featured":336,"cover":334,"format":223,"platforms":473,"draft":336,"slug":474},"Break-even ROAS is the zero line: the Target ROAS that actually maximizes Shopee profit","Setting your Target ROAS to your break-even ROAS does not protect profit — it targets exactly zero, for every possible volume. The target that maximizes profit is break-even marked up three times: by the demand your ads did not create, by the price your own expansion pays, and by the gap between your average click and your last one. On mid-market parameters those compound to 3.3×, moving the correct target on a 20%-margin product from 5.00 to 16.67.",12,[210],"profit-maximizing-target-roas",{"title":476,"description":477,"category":361,"date":465,"readMinutes":466,"featured":336,"cover":334,"format":241,"platforms":478,"draft":336,"slug":479},"Shopee ad benchmarks 2026: what a click and a thousand impressions actually cost","Across 333.9 million Shopee impressions and 9.11 million clicks on 80+ Thai shops, the price of a click varies 2.01× by hour of day while the rate at which people click varies 1.14×. CPC ranges THB 3.36 to THB 6.75, CPM ranges THB 89.16 to THB 200.15, and every behavioural input is nearly flat. The full table, the methodology, and the two ways a benchmark like this gets misused.",[210],"shopee-ads-benchmarks-cpc-cpm",{"title":481,"description":482,"category":364,"date":483,"readMinutes":484,"featured":336,"cover":334,"format":229,"platforms":334,"draft":336,"slug":485},"DataGlass is not an OMS — what the difference is, and when to run both","An OMS makes the order happen. DataGlass decides what to do next. The daily-question boundary, an explicit list of what DataGlass does not do, and the one seam: price.","2026-08-25",6,"dataglass-vs-oms",{"title":487,"description":488,"category":352,"date":483,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":489,"draft":336,"slug":490},"How much should you spend on Shopee Ads? Derive it from margin, not from feel","Break-even ROAS is 1 ÷ contribution margin ratio. Worked THB example: a fashion SKU allows THB 71 per attributed order and roughly THB 994 a day. Budget is an output.",[210],"shopee-ads-budget",{"title":492,"description":493,"category":347,"date":483,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":494,"draft":336,"slug":495},"Losing money on Shopee — the leaks a fee calculator cannot see","Shopee deducts 7.49–17.12% plus a 3.21% payment fee — the half a calculator sees. Six month-level leaks turn a 28.6% \"profit\" into 1.6%. Sourced THB waterfall.",[210],"shopee-losing-money",{"title":497,"description":498,"category":358,"date":483,"readMinutes":415,"featured":336,"cover":334,"format":229,"platforms":499,"draft":336,"slug":500},"Shopee vs TikTok Shop vs Lazada — what survives after fees and the ads it takes to sell","TikTok Shop's 11.77% fashion commission becomes a 23.2% take rate once the Commerce Growth and Transaction Fees stack. The crossover with Shopee sits near THB 3,720.",[210,213,216],"where-to-sell-shopee-tiktok-lazada",{"title":502,"description":503,"category":352,"date":504,"readMinutes":376,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":505},"CTR and Target ROAS under CPM vs CPC: what changes when marketplaces bill for impressions","Under cost-per-click billing, a better creative is worth nothing to your unit economics — it buys the same clicks from fewer impressions, and your cost per click does not move. Under cost-per-impression billing the same improvement cuts the price of every click by g\u002F(1+g). Five lemmas on the transition, and why click-through rate is the input most marketplace sellers are least prepared to have priced.","2026-08-18","ctr-target-roas-cpm-vs-cpc",{"title":507,"description":508,"category":352,"date":509,"readMinutes":447,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":510},"What is ROAS — and why a high ROAS doesn't always mean more sales or more profit","ROAS (Return on Ad Spend) is sales ÷ ad spend: spend THB 1, get THB 20 of sales back, and your ROAS is 20. But a high ROAS only means a high return per ad baht — not the highest total sales, and not real profit once you subtract COGS, platform fees, payment fees, vouchers, shipping, and returns. Here is what ROAS is, what it hides, and the number sellers should actually watch.","2026-06-01","what-is-roas",{"title":512,"description":513,"category":352,"date":514,"readMinutes":459,"featured":336,"cover":334,"format":223,"platforms":515,"draft":336,"slug":516},"What is Target ROAS? How to set it in Shopee GMV Max without burning profit","What is Target ROAS? It is the revenue-to-ad-spend ratio Shopee GMV Max optimises toward — and on the Shopee fee schedule effective 4 August 2026, a THB 1,000 product with THB 600 of COGS is left with 7.7% margin before ads, so a Target ROAS of 5 recovers THB 0.39 of contribution per THB 1.00 of ad spend while the campaign reports that it hit its target. What Target ROAS means, how to set it in GMV Max, and the two formulas — break-even ROAS and target-margin ROAS — that protect your margin.","2026-05-30",[210],"how-to-set-target-roas-gmv-max",{"title":518,"description":519,"category":352,"date":520,"readMinutes":484,"featured":336,"cover":334,"format":226,"platforms":521,"draft":336,"slug":522},"Platform ROAS vs true ROAS: why your ads make sales but no profit","A ROAS of 5 feels like a win — until you notice the platform computed it on gross attributed sales, before fees, vouchers, shipping, and COGS, and while still counting orders that were later returned or cancelled. The dashboard number is a revenue ratio wearing a profit costume. A primer on what platform ROAS hides and how true ROAS is calculated across Shopee, Lazada, and TikTok Shop.","2026-05-29",[210,213,216],"platform-roas-vs-true-roas",{"title":524,"description":525,"category":361,"date":520,"readMinutes":484,"featured":336,"cover":334,"format":226,"platforms":526,"draft":336,"slug":527},"What is e-commerce optimization?","Optimization is not a marketing word — it is an operations-research discipline: use data and math to compute the best decision against an explicit objective. Point it at a Shopee, Lazada, or TikTok Shop store and the objective is true profit, not sales. A primer on why those two numbers stopped moving together, and what optimization actually answers.",[210,213,216],"what-is-ecommerce-optimization",{"title":529,"description":530,"category":347,"date":531,"readMinutes":376,"featured":336,"cover":334,"format":223,"platforms":532,"draft":336,"slug":533},"How to cut Shopee seller costs: the supply-side margin audit","The median gap between COGS refreshes on a top-50 Shopee catalogue is 87 days, and over that window supplier and packaging drift alone compresses contribution margin by 1.5–3.0 percentage points. The eight supply-side cost lines that compound against Shopee margin between audits, ranked by recoverable magnitude, with the review cadence that holds each one.","2026-05-08",[210],"how-to-increase-shopee-profit-without-more-sales",{"title":535,"description":536,"category":352,"date":531,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":538,"draft":336,"slug":539},"How to sell more on Shopee without burning margin","The standard Shopee growth advice — discount harder, scale ad budget, opt into every campaign — works against the operator at scale. A research note on the lever order that actually compounds Shopee revenue, and the metric that tells you whether the lift was earned or bought.",11,[210],"how-to-sell-more-on-shopee",{"title":541,"description":542,"category":347,"date":543,"readMinutes":484,"featured":336,"cover":334,"format":223,"platforms":544,"draft":336,"slug":545},"Increase profit on Shopee without selling more","Shopee ROAS hides commission, fees, vouchers and shipping. Optimize Shopee profit with the per-SKU break-even framework that replaces it — 280 Thai accounts.","2026-05-07",[210],"increase-shopee-profit",{"title":547,"description":548,"category":358,"date":543,"readMinutes":484,"featured":336,"cover":334,"format":223,"platforms":549,"draft":336,"slug":550},"Increase sales on Shopee in 2026","Shopee CPC is climbing — buying demand got pricier while repeat demand stayed flat. The three levers that compound revenue, plus a 90-day operator cadence.",[210],"increase-shopee-sales",{"title":552,"description":553,"category":361,"date":543,"readMinutes":554,"featured":336,"cover":334,"format":235,"platforms":555,"draft":336,"slug":556},"Mechanism-Aware Hierarchical Causal Elasticity Modeling for Platform E-Commerce","A methodological working paper proposing MA-HCEM: a hierarchical Bayesian, double-machine-learning, mechanism-aware estimator for SKU-level price elasticity on Shopee, Lazada, and TikTok Shop, with Bundle Deal and voucher mechanisms treated as structured price discrimination.",27,[210,213,216],"mechanism-aware-hierarchical-causal-elasticity-modeling-platform-ecommerce",{"title":558,"description":559,"category":350,"date":543,"readMinutes":560,"featured":336,"cover":334,"format":226,"platforms":561,"draft":336,"slug":562},"Price elasticity modeling for marketplace sellers — why Shopee, Lazada, and TikTok Shop pricing decisions need it","A practical guide from DataGlass Labs Research on the most under-used lever in Southeast Asian e-commerce: knowing how customers actually respond to price changes.",14,[210,213,216],"price-elasticity-modeling-shopee-lazada-tiktok-shop",{"title":564,"description":565,"category":361,"date":566,"readMinutes":567,"featured":336,"cover":334,"format":235,"platforms":334,"draft":336,"slug":568},"From Gut Feel to Posterior Inference: A Research Article on the DataGlass Decision-Intelligence System for E-Commerce Ad Budget Allocation","A rigorous public communication of the DataGlass decision-intelligence system for daily ad-budget allocation on platform-controlled marketplaces — the pre-algorithmic baseline, the analytical reasons it fails (the reported-vs-true ROAS gap and the rolling-mean bias), the Bayesian + bandits-with-knapsacks methodology, and the empirical 18–24% portfolio-profit lift with reduced reallocation frequency.","2026-05-04",20,"bayesian-budget-allocation-for-marketplace-ads",{"title":570,"description":571,"category":361,"date":566,"readMinutes":572,"featured":336,"cover":334,"format":238,"platforms":334,"draft":336,"slug":573},"Decision Intelligence for E-commerce: How Retailers Optimise Pricing, Demand Forecasting, Inventory, Promotions & Personalization","Pricing, forecasting, inventory, promotions, and personalization — a deep technical survey of the techniques large retailers use, the variants that matter, and how to deploy them",42,"decision-intelligence-playbook-for-e-commerce",{"title":575,"description":576,"category":361,"date":566,"readMinutes":332,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":577},"DataGlass Research Methodology","A single canonical reference for the sample frame, time window, attribution rules, and exclusions that calibrate every \"in our data\" claim across the blog and research surfaces.","methodology",{"title":579,"description":580,"category":361,"date":566,"readMinutes":581,"featured":336,"cover":334,"format":238,"platforms":334,"draft":336,"slug":582},"Prediction and Risk Optimization Under Uncertainty: A Cross-Domain Meta-Review","Mature decision systems in finance, operations, insurance, energy, healthcare, causal inference, and e-commerce share the same four primitives — a calibrated probabilistic model, a coherent risk-aware objective, an operational constraint set, and a principled exploration mechanism. Eleven worked cases ground the framework, with the DataGlass marketplace ad-budget system as the connecting tissue.",49,"prediction-and-risk-optimization-meta-review",{"title":584,"description":585,"category":355,"date":566,"readMinutes":560,"featured":336,"cover":334,"format":226,"platforms":586,"draft":336,"slug":587},"Shopee optimization in 2026 is a feedback loop, not a checklist","The popular \"10 ways to optimize your Shopee shop\" advice optimises the wrong objective. The lever that actually moves margin is the tightness of the diagnose → decide → operate loop — and most checklists skip the diagnose stage entirely.",[210],"shopee-optimization-in-2026",{"title":589,"description":590,"category":358,"date":566,"readMinutes":591,"featured":134,"cover":334,"format":241,"platforms":592,"draft":336,"slug":593},"Shopee Sellers in 2026: Southeast Asia E-commerce Market Research, GMV & Seller Economics","Market growth has resumed; seller economics have become more exacting. Southeast Asia's platform e-commerce reached US$157.6B in 2025 (up 22.8% YoY), top-three platforms now control about 98.8% of platform GMV, and content commerce accounts for ~32% of platform GMV. The 2026 question for sellers is no longer \"how big is the market?\" — it is \"who controls the decision loop?\"",22,[210],"southeast-asia-seller-market-research-2026",{"title":595,"description":596,"category":355,"date":597,"readMinutes":598,"featured":336,"cover":334,"format":223,"platforms":599,"draft":336,"slug":600},"Shopee fee structure and commission calculation in 2026: the six layers, the four-term formula","Most 'Shopee fees' explainers stop at commission and transaction fee. The four layers underneath — voucher co-funding, Free Shipping Program subsidy, Mall premium, payment-channel pass-through — are where the gap between dashboard revenue and bank deposit actually lives. With a sourced layer-by-layer breakdown, the four-term commission-per-order formula, two worked THB examples, a sensitivity table, and the reconciliation procedure.","2026-05-03",18,[210],"shopee-fee-structure-2026",{"title":602,"description":603,"category":352,"date":604,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":605,"draft":336,"slug":606},"Cross-platform ad budget allocation for SEA marketplace sellers","Most multi-platform sellers split ad budget across Shopee, Lazada, and TikTok Shop by historical revenue share. The math says that's wrong. Optimal allocation equalises marginal ROAS, not historical share — and the gap between the two on a typical account is 4–7 percentage points of net contribution margin per quarter.","2026-04-29",[210,213,216],"cross-platform-ad-budget-allocation",{"title":608,"description":609,"category":352,"date":610,"readMinutes":415,"featured":336,"cover":334,"format":223,"platforms":611,"draft":336,"slug":612},"Why scaling Shopee ad budget grows revenue and shrinks profit","A Shopee account moving its monthly ad budget from THB 40,000 to THB 100,000 commonly grows revenue 60% and loses a third of its net profit — and not one decision in that sequence looks wrong on the dashboard. The arithmetic of the scaling trap, why platform-reported ROAS is blind to it by construction, and the per-SKU break-even bar that stops it.","2026-04-22",[210],"how-to-increase-profit-on-shopee",{"title":614,"description":615,"category":347,"date":616,"readMinutes":466,"featured":336,"cover":334,"format":223,"platforms":617,"draft":336,"slug":618},"How to increase profit on Lazada in 2026","The LazMall badge lifts conversion. It also raises commission, mandates free-shipping subsidies, and pulls the price ceiling down through the platform's own competitive-parity rules. Whether the badge pays is a per-SKU question. A research note on the LazMall economics, why Sponsored Discovery leaks more margin than Sponsored Search, and the audit that recovers 4–6 percentage points of margin in 30 days.","2026-04-15",[213],"how-to-increase-profit-on-lazada",{"title":620,"description":621,"category":352,"date":622,"readMinutes":466,"featured":336,"cover":334,"format":223,"platforms":623,"draft":336,"slug":624},"How to reduce Shopee ad waste without killing sales","On a typical Shopee account, 20–30% of ad spend runs at a structural loss the platform dashboard ranks as winning campaigns. Pausing \"underperformers\" misses the leak. A research note on the two structural defaults that cause hidden ad waste — and the audit that surfaces it without losing revenue.","2026-04-08",[210],"how-to-reduce-shopee-ad-waste",{"title":626,"description":627,"category":347,"date":628,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":629,"draft":336,"slug":630},"How to calculate Shopee seller margin","The math, the inputs, and the program-specific traps. Why Shopee's in-dashboard income view almost always overstates margin — and how to reconstruct true contribution margin per SKU using the order-line data Shopee already exposes via Open Platform.","2026-04-05",[210],"how-to-calculate-shopee-seller-margin",{"title":632,"description":633,"category":347,"date":634,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":635,"draft":336,"slug":636},"How to calculate Lazada seller margin","The math, the inputs, and the program-specific traps. Why Lazada's in-platform P&L summary almost always overstates margin — and how to reconstruct true contribution margin per SKU using the order-line data Lazada exposes via the Open Platform.","2026-04-01",[213],"how-to-calculate-lazada-seller-margin",{"title":638,"description":639,"category":352,"date":640,"readMinutes":560,"featured":336,"cover":334,"format":223,"platforms":641,"draft":336,"slug":642},"How to calculate true Shopee ROAS for profit","A methodology note. Shopee's in-platform ROAS is gross-revenue based and structurally biased toward overspend at scale. True ROAS is the same formula with one input substituted — and that substitution flips winners into losses on roughly half the typical Shopee catalog. With charts, three SKU profiles, sensitivity analysis, and the operating procedure that applies the substitution at production cadence.","2026-03-25",[210],"how-to-calculate-true-shopee-roas",{"title":644,"description":645,"category":352,"date":646,"readMinutes":537,"featured":336,"cover":334,"format":223,"platforms":647,"draft":336,"slug":648},"How to reduce Lazada ad waste","Lazada runs two structurally different ad products — Sponsored Search and Sponsored Discovery — with break-even economics that diverge by ~1.4–1.8× in true-ROAS terms. A single account-wide ROAS target misjudges both. A research note on the audit that surfaces the structural waste, the pruning order that recovers margin without losing revenue, and the per-placement framework that survives Pay Day and 11.11.","2026-03-18",[213],"how-to-reduce-lazada-ad-waste",{"title":650,"description":651,"category":347,"date":652,"readMinutes":466,"featured":336,"cover":334,"format":223,"platforms":653,"draft":336,"slug":654},"How to find low-margin SKUs on Shopee","On a typical Shopee account, the top-10 SKUs by revenue and the top-10 by contribution profit overlap by roughly 50%. Half of every shop's bestsellers are not the most profitable products. A research note on the audit that surfaces the gap, the patterns hiding inside it, and the per-SKU operating decisions that recover margin.","2026-03-12",[210],"how-to-find-low-margin-skus-on-shopee",{"title":656,"description":657,"category":350,"date":658,"readMinutes":560,"featured":336,"cover":334,"format":223,"platforms":659,"draft":336,"slug":660},"How to plan Lazada campaigns around profit","Pay Day, Mega Sale, 11.11, and the long tail of category windows make participation feel mandatory. The platform documents the eligibility tiers; the seller absorbs them. A research note on what each campaign actually costs, when participation pays, and the margin-first rubric that survives all three.","2026-03-04",[213],"how-to-plan-lazada-campaigns-around-profit",{"title":662,"description":663,"category":347,"date":664,"readMinutes":472,"featured":336,"cover":334,"format":223,"platforms":665,"draft":336,"slug":666},"How to increase profit on TikTok Shop in 2026","TikTok Shop is the only SEA marketplace with a stacked second commission — affiliate commission (10–25% via the Open Affiliate Plan) layered on top of platform commission. A 6.0 platform ROAS routinely becomes ~1.4 true ROAS once the full four-line cost stack is subtracted. A research note on the affiliate-stack arithmetic, live-stream pricing discipline, and the per-SKU framework that recovers margin without retreating from the platform.","2026-02-26",[216],"how-to-increase-profit-on-tiktok-shop",{"title":668,"description":669,"category":358,"date":670,"readMinutes":671,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":672},"The SEA marketplace in 2026: a fragmented arena with one survival rule","Eighteen distinct platform-market cells, six countries, three platforms, and roughly a million sellers competing on a saturated product surface. The aggregate is growing; the per-cell margin distribution is compressing. A research note on the structural shifts and the operating rule that survives them.","2026-02-22",16,"sea-marketplace-2026",{"title":674,"description":675,"category":361,"date":676,"readMinutes":537,"featured":336,"cover":334,"format":226,"platforms":677,"draft":336,"slug":678},"Data ingestion for Shopee sellers: why zero-setup analytics matters","Most Shopee sellers don't have a strategy problem first. They have a data plumbing problem — orders, ads, COGS, fees, vouchers, inventory, pricing, and returns live in seven different surfaces, and by the time the seller has stitched them together the campaign is over. A research note on the data-source matrix, the canonical-entity model, and the zero-setup architecture that recovers ~10 hours per week.","2026-02-18",[210],"data-ingestion-for-shopee-sellers",{"title":680,"description":681,"category":358,"date":682,"readMinutes":415,"featured":336,"cover":334,"format":226,"platforms":683,"draft":336,"slug":684},"The race to zero margin — and how Shopee, Lazada, and TikTok Shop got there","Read the platform documentation end-to-end and the conclusion is uncomfortable: the discounts buyers see, the free shipping that drives conversion, and the affiliate spend that drives reach are all paid by the seller. The race-to-zero is the equilibrium output of that design.","2026-02-08",[210,213,216],"race-to-zero-margin",{"title":686,"description":687,"category":361,"date":688,"readMinutes":560,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":689},"ML demand forecasting for e-commerce sellers","Machine learning in e-commerce gets discussed in vague terms; for marketplace sellers the operating question is concrete — how many units of this SKU will sell in the next N days, with what confidence, and what decision flows from the answer? A research note on the practical model architecture, the stockout-distortion problem, sensitivity analysis, and the operating decisions forecasts feed.","2026-02-04","ml-demand-forecasting-for-ecommerce",{"title":691,"description":692,"category":358,"date":693,"readMinutes":484,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":694},"Complexity is the new tax on small sellers","A multi-shop seller in 2026 logs into seven platforms, reconciles four fee schedules, exports six CSVs, and re-enters COGS by hand. Big brands absorb the cost with a data team. Small sellers absorb it with their evenings — and it's the single biggest reason multi-shop operators stall before they reach scale.","2026-01-25","complexity-is-the-tax",{"title":696,"description":697,"category":355,"date":698,"readMinutes":472,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":699},"Stockout math for e-commerce sellers","A stockout is not one cost; it is five compounding costs. Lost contribution profit on the missed unit, plus wasted ad spend during the stockout window, plus algorithmic ranking demotion, plus repeat-buyer trust erosion, plus distorted forecasting that increases the likelihood of the next stockout. A research note on the multi-line stockout cost function, the per-SKU reorder-point math that accounts for it, and the campaign-aware adjustment that survives Pay Day and 11.11.","2026-01-18","inventory-stockout-math-ecommerce",{"title":701,"description":702,"category":350,"date":703,"readMinutes":537,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":704},"Dynamic pricing for marketplace sellers","Discounting is easy. Profitable pricing is hard. A 30% volume lift on a 10% price cut routinely lowers total contribution profit — the math says volume must lift by ~33% just to break even, and most SKUs underperform that bar. A research note on the price-elasticity arithmetic, the inventory × demand four-quadrant framework, and the per-SKU pricing decision that survives campaign-window pressure.","2026-01-04","dynamic-pricing-ecommerce-marketplaces",{"title":706,"description":707,"category":361,"date":708,"readMinutes":472,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":709},"E-commerce Decision Engine: How Marketplace Sellers Turn Data Into Profit Recommendations","A dashboard tells you what happened. A decision engine tells you what to do next, ranks the options by projected profit lift, and surfaces the math behind every recommendation. A research note on the five-layer architecture that separates the two, why marketplace commerce now requires the latter, and where the operating model breaks.","2025-12-12","how-ecommerce-decision-engines-work",{"title":711,"description":712,"category":355,"date":713,"readMinutes":459,"featured":336,"cover":334,"format":226,"platforms":714,"draft":336,"slug":715},"Multi-shop analytics for Shopee, Lazada, and TikTok Shop sellers","Modern marketplace sellers rarely operate on one channel. They have multiple dashboards but not one operating view. They know sales by channel — but not profit by channel.","2025-11-26",[210,213,216],"multi-shop-analytics-lazada-shopee-tiktok",{"title":717,"description":718,"category":358,"date":719,"readMinutes":720,"featured":336,"cover":334,"format":226,"platforms":334,"draft":336,"slug":721},"The new competitive world of commerce","A research note on what changed in marketplace commerce between 2022 and 2026. Platforms became active algorithmic counterparties; sellers who treat them as passive listing markets are competing against optimization systems they cannot see. The fix is to run your own optimization layer on top.","2025-11-12",15,"new-competitive-world-of-commerce",{"entry":723,"document":1205,"toc":6132},{"id":724,"title":579,"category":361,"cover":334,"cta":334,"date":566,"description":580,"draft":336,"draftReason":334,"extension":337,"featured":336,"format":238,"listen":334,"math":134,"meta":725,"navigation":334,"ogImage":334,"path":726,"platforms":334,"readMinutes":581,"related":334,"robots":334,"schemaOrg":334,"seo":727,"sitemap":730,"sources":731,"stem":1202,"tldr":334,"topic":1203,"updatedAt":334,"__hash__":1204},"blog_en\u002Fblog\u002Fprediction-and-risk-optimization-meta-review.md",{},"\u002Fblog\u002Fprediction-and-risk-optimization-meta-review",{"title":728,"description":729},"Prediction & Risk Optimization Meta-Review | DataGlass Research","Eleven worked cases show mature decision systems — finance to e-commerce — share four primitives: calibrated model, risk-aware objective, constraints, exploration.",{"loc":726},[732,734,736,738,740,742,744,746,748,750,752,754,756,758,760,762,764,766,768,770,772,774,776,778,780,782,784,786,788,790,792,794,796,798,800,802,804,806,808,810,812,814,816,818,820,822,824,826,828,830,832,834,836,838,840,842,844,846,848,850,852,854,856,858,860,862,864,866,868,870,872,874,876,878,880,882,884,886,888,890,892,894,896,898,900,902,904,906,908,910,912,914,916,918,920,922,924,926,928,930,932,934,936,938,940,942,944,946,948,950,952,954,956,958,960,962,964,966,968,970,972,974,976,978,980,982,984,986,988,990,992,994,996,998,1000,1002,1004,1006,1008,1010,1012,1014,1016,1018,1020,1022,1024,1026,1028,1030,1032,1034,1036,1038,1040,1042,1044,1046,1048,1050,1052,1054,1056,1058,1060,1062,1064,1066,1068,1070,1072,1074,1076,1078,1080,1082,1084,1086,1088,1090,1092,1094,1096,1098,1100,1102,1104,1106,1108,1110,1112,1114,1116,1118,1120,1122,1124,1126,1128,1130,1132,1134,1136,1138,1140,1142,1144,1146,1148,1150,1152,1154,1156,1158,1160,1162,1164,1166,1168,1170,1172,1174,1176,1178,1180,1182,1184,1186,1188,1190,1192,1194,1196,1198,1200],{"title":733},"DataGlass Labs Research, \"DataGlass: Bayesian Budget Allocation for E-Commerce Advertising Under Platform Constraints,\" internal technical report (working paper), March 2026.",{"title":735},"DataGlass Labs Research, \"From Gut Feel to Posterior Inference: A Research Article on the DataGlass Decision-Intelligence System for E-Commerce Ad Budget Allocation,\" DataGlass Labs Research working paper, May 2026.",{"title":737},"DataGlass Labs Research, \"Elasticity Modeling and Bundle Pricing,\" internal technical note, April 2026.",{"title":739},"H. 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2021.","blog\u002Fprediction-and-risk-optimization-meta-review","decision-science","Y5LPBM_ISThUW0OQBFI1W0Do9bs9dkBKu9M7Un1obUE",{"frontmatter":1206,"meta":1444,"nodes":1804},{"title":579,"description":580,"seo":1207,"category":361,"format":238,"topic":1203,"date":566,"readMinutes":581,"math":134,"sources":1208},{"title":728,"description":729},[1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1251,1252,1253,1254,1255,1256,1257,1258,1259,1260,1261,1262,1263,1264,1265,1266,1267,1268,1269,1270,1271,1272,1273,1274,1275,1276,1277,1278,1279,1280,1281,1282,1283,1284,1285,1286,1287,1288,1289,1290,1291,1292,1293,1294,1295,1296,1297,1298,1299,1300,1301,1302,1303,1304,1305,1306,1307,1308,1309,1310,1311,1312,1313,1314,1315,1316,1317,1318,1319,1320,1321,1322,1323,1324,1325,1326,1327,1328,1329,1330,1331,1332,1333,1334,1335,1336,1337,1338,1339,1340,1341,1342,1343,1344,1345,1346,1347,1348,1349,1350,1351,1352,1353,1354,1355,1356,1357,1358,1359,1360,1361,1362,1363,1364,1365,1366,1367,1368,1369,1370,1371,1372,1373,1374,1375,1376,1377,1378,1379,1380,1381,1382,1383,1384,1385,1386,1387,1388,1389,1390,1391,1392,1393,1394,1395,1396,1397,1398,1399,1400,1401,1402,1403,1404,1405,1406,1407,1408,1409,1410,1411,1412,1413,1414,1415,1416,1417,1418,1419,1420,1421,1422,1423,1424,1425,1426,1427,1428,1429,1430,1431,1432,1433,1434,1435,1436,1437,1438,1439,1440,1441,1442,1443],{"title":733},{"title":735},{"title":737},{"title":739},{"title":741},{"title":743},{"title":745},{"title":747},{"title":749},{"title":751},{"title":753},{"title":755},{"title":757},{"title":759},{"title":761},{"title":763},{"title":765},{"title":767},{"title":769},{"title":771},{"title":773},{"title":775},{"title":777},{"title":779},{"title":781},{"title":783},{"title":785},{"title":787},{"title":789},{"title":791},{"title":793},{"title":795},{"title":797},{"title":799},{"title":801},{"title":803},{"title":805},{"title":807},{"title":809},{"title":811},{"title":813},{"title":815},{"title":817},{"title":819},{"title":821},{"title":823},{"title":825},{"title":827},{"title":829},{"title":831},{"title":833},{"title":835},{"title":837},{"title":839},{"title":841},{"title":843},{"title":845},{"title":847},{"title":849},{"title":851},{"title":853},{"title":855},{"title":857},{"title":859},{"title":861},{"title":863},{"title":865},{"title":867},{"title":869},{"title":871},{"title":873},{"title":875},{"title":877},{"title":879},{"title":881},{"title":883},{"title":885},{"title":887},{"title":889},{"title":891},{"title":893},{"title":895},{"title":897},{"title":899},{"title":901},{"title":903},{"title":905},{"title":907},{"title":909},{"title":911},{"title":913},{"title":915},{"title":917},{"title":919},{"title":921},{"title":923},{"title":925},{"title":927},{"title":929},{"title":931},{"title":933},{"title":935},{"title":937},{"title":939},{"title":941},{"title":943},{"title":945},{"title":947},{"title":949},{"title":951},{"title":953},{"title":955},{"title":957},{"title":959},{"title":961},{"title":963},{"title":965},{"title":967},{"title":969},{"title":971},{"title":973},{"title":975},{"title":977},{"title":979},{"title":981},{"title":983},{"title":985},{"title":987},{"title":989},{"title":991},{"title":993},{"title":995},{"title":997},{"title":999},{"title":1001},{"title":1003},{"title":1005},{"title":1007},{"title":1009},{"title":1011},{"title":1013},{"title":1015},{"title":1017},{"title":1019},{"title":1021},{"title":1023},{"title":1025},{"title":1027},{"title":1029},{"title":1031},{"title":1033},{"title":1035},{"title":1037},{"title":1039},{"title":1041},{"title":1043},{"title":1045},{"title":1047},{"title":1049},{"title":1051},{"title":1053},{"title":1055},{"title":1057},{"title":1059},{"title":1061},{"title":1063},{"title":1065},{"title":1067},{"title":1069},{"title":1071},{"title":1073},{"title":1075},{"title":1077},{"title":1079},{"title":1081},{"title":1083},{"title":1085},{"title":1087},{"title":1089},{"title":1091},{"title":1093},{"title":1095},{"title":1097},{"title":1099},{"title":1101},{"title":1103},{"title":1105},{"title":1107},{"title":1109},{"title":1111},{"title":1113},{"title":1115},{"title":1117},{"title":1119},{"title":1121},{"title":1123},{"title":1125},{"title":1127},{"title":1129},{"title":1131},{"title":1133},{"title":1135},{"title":1137},{"title":1139},{"title":1141},{"title":1143},{"title":1145},{"title":1147},{"title":1149},{"title":1151},{"title":1153},{"title":1155},{"title":1157},{"title":1159},{"title":1161},{"title":1163},{"title":1165},{"title":1167},{"title":1169},{"title":1171},{"title":1173},{"title":1175},{"title":1177},{"title":1179},{"title":1181},{"title":1183},{"title":1185},{"title":1187},{"title":1189},{"title":1191},{"title":1193},{"title":1195},{"title":1197},{"title":1199},{"title":1201},{"toc":1445},{"title":331,"searchDepth":386,"depth":386,"links":1446},[1447,1450,1453,1456,1487,1512,1540,1559,1562,1565,1568,1590,1606,1631,1668,1684,1697,1728,1731,1734,1737,1740,1743,1746,1749],{"id":1448,"depth":386,"text":1449},"abstract","Abstract",{"id":1451,"depth":386,"text":1452},"notation","Notation",{"id":1454,"depth":386,"text":1455},"_1-introduction","1. Introduction",{"id":1457,"depth":386,"text":1458,"children":1459},"_2-methodology-of-the-meta-review","2. Methodology of the Meta-Review",[1460,1463,1466,1469,1472,1475,1478,1481,1484],{"id":1461,"depth":332,"text":1462},"_2-methodology-of-the-meta-review-_21-research-questions","2.1 Research questions",{"id":1464,"depth":332,"text":1465},"_2-methodology-of-the-meta-review-_22-search-strategy","2.2 Search strategy",{"id":1467,"depth":332,"text":1468},"_2-methodology-of-the-meta-review-_23-inclusion-and-exclusion-criteria","2.3 Inclusion and exclusion criteria",{"id":1470,"depth":332,"text":1471},"_2-methodology-of-the-meta-review-_24-screening-and-selection","2.4 Screening and selection",{"id":1473,"depth":332,"text":1474},"_2-methodology-of-the-meta-review-_25-data-extraction","2.5 Data extraction",{"id":1476,"depth":332,"text":1477},"_2-methodology-of-the-meta-review-_26-quality-assessment","2.6 Quality assessment",{"id":1479,"depth":332,"text":1480},"_2-methodology-of-the-meta-review-_27-heterogeneity-assessment","2.7 Heterogeneity assessment",{"id":1482,"depth":332,"text":1483},"_2-methodology-of-the-meta-review-_28-synthesis-approach","2.8 Synthesis approach",{"id":1485,"depth":332,"text":1486},"_2-methodology-of-the-meta-review-_29-pre-registration-and-protocol-deviations","2.9 Pre-registration and protocol deviations",{"id":1488,"depth":386,"text":1489,"children":1490},"_3-foundations-a-unified-language-for-prediction-and-risk-aware-optimization","3. Foundations: A Unified Language for Prediction and Risk-Aware Optimization",[1491,1494,1497,1500,1503,1506,1509],{"id":1492,"depth":332,"text":1493},"_3-foundations-a-unified-language-for-prediction-and-risk-aware-optimization-_31-decision-theoretic-preliminaries","3.1 Decision-theoretic preliminaries",{"id":1495,"depth":332,"text":1496},"_3-foundations-a-unified-language-for-prediction-and-risk-aware-optimization-_32-risk-measures","3.2 Risk measures",{"id":1498,"depth":332,"text":1499},"_3-foundations-a-unified-language-for-prediction-and-risk-aware-optimization-_33-robust-and-distributionally-robust-optimization","3.3 Robust and distributionally robust optimization",{"id":1501,"depth":332,"text":1502},"_3-foundations-a-unified-language-for-prediction-and-risk-aware-optimization-_34-stochastic-dynamic-programming","3.4 Stochastic dynamic programming",{"id":1504,"depth":332,"text":1505},"_3-foundations-a-unified-language-for-prediction-and-risk-aware-optimization-_35-online-learning-and-bandits","3.5 Online learning and bandits",{"id":1507,"depth":332,"text":1508},"_3-foundations-a-unified-language-for-prediction-and-risk-aware-optimization-_36-model-predictive-control-and-receding-horizon-optimization","3.6 Model predictive control and receding-horizon optimization",{"id":1510,"depth":332,"text":1511},"_3-foundations-a-unified-language-for-prediction-and-risk-aware-optimization-_37-online-convex-optimization-mirror-descent-and-primal-dual-methods","3.7 Online convex optimization, mirror descent, and primal-dual methods",{"id":1513,"depth":386,"text":1514,"children":1515},"_4-domain-review-i-finance-and-quantitative-risk-management","4. Domain Review I — Finance and Quantitative Risk Management",[1516,1519,1522,1525,1528,1531,1534,1537],{"id":1517,"depth":332,"text":1518},"_4-domain-review-i-finance-and-quantitative-risk-management-_41-mean-variance-and-the-capm","4.1 Mean-variance and the CAPM",{"id":1520,"depth":332,"text":1521},"_4-domain-review-i-finance-and-quantitative-risk-management-_42-cvar-portfolio-optimization-and-coherent-risk","4.2 CVaR portfolio optimization and coherent risk",{"id":1523,"depth":332,"text":1524},"_4-domain-review-i-finance-and-quantitative-risk-management-_43-robust-portfolio-optimization","4.3 Robust portfolio optimization",{"id":1526,"depth":332,"text":1527},"_4-domain-review-i-finance-and-quantitative-risk-management-_44-algorithmic-execution-almgrenchriss","4.4 Algorithmic execution: Almgren–Chriss",{"id":1529,"depth":332,"text":1530},"_4-domain-review-i-finance-and-quantitative-risk-management-_45-derivatives-hedging-and-stochastic-control","4.5 Derivatives, hedging, and stochastic control",{"id":1532,"depth":332,"text":1533},"_4-domain-review-i-finance-and-quantitative-risk-management-_46-statistical-arbitrage-and-high-frequency-trading","4.6 Statistical arbitrage and high-frequency trading",{"id":1535,"depth":332,"text":1536},"_4-domain-review-i-finance-and-quantitative-risk-management-_47-credit-risk-and-counterparty-risk","4.7 Credit risk and counterparty risk",{"id":1538,"depth":332,"text":1539},"_4-domain-review-i-finance-and-quantitative-risk-management-_48-macro-finance-and-central-bank-decision-making","4.8 Macro-finance and central-bank decision-making",{"id":1541,"depth":386,"text":1542,"children":1543},"_5-domain-review-ii-operations-and-supply-chain","5. Domain Review II — Operations and Supply Chain",[1544,1547,1550,1553,1556],{"id":1545,"depth":332,"text":1546},"_5-domain-review-ii-operations-and-supply-chain-_51-the-newsvendor","5.1 The newsvendor",{"id":1548,"depth":332,"text":1549},"_5-domain-review-ii-operations-and-supply-chain-_52-multi-period-inventory-and-base-stock-policies","5.2 Multi-period inventory and base-stock policies",{"id":1551,"depth":332,"text":1552},"_5-domain-review-ii-operations-and-supply-chain-_53-pricing-and-revenue-management","5.3 Pricing and revenue management",{"id":1554,"depth":332,"text":1555},"_5-domain-review-ii-operations-and-supply-chain-_54-supply-chain-coordination-and-contract-design","5.4 Supply chain coordination and contract design",{"id":1557,"depth":332,"text":1558},"_5-domain-review-ii-operations-and-supply-chain-_55-network-revenue-management","5.5 Network revenue management",{"id":1560,"depth":386,"text":1561},"_6-domain-review-iii-insurance-and-actuarial-science","6. Domain Review III — Insurance and Actuarial Science",{"id":1563,"depth":386,"text":1564},"_7-domain-review-iv-energy-systems-and-stochastic-resource-allocation","7. Domain Review IV — Energy Systems and Stochastic Resource Allocation",{"id":1566,"depth":386,"text":1567},"_8-domain-review-v-healthcare-operations-and-decision-support","8. Domain Review V — Healthcare Operations and Decision Support",{"id":1569,"depth":386,"text":1570,"children":1571},"_9-domain-review-vi-causal-inference-and-program-evaluation","9. Domain Review VI — Causal Inference and Program Evaluation",[1572,1575,1578,1581,1584,1587],{"id":1573,"depth":332,"text":1574},"_9-domain-review-vi-causal-inference-and-program-evaluation-_91-potential-outcomes-framework","9.1 Potential-outcomes framework",{"id":1576,"depth":332,"text":1577},"_9-domain-review-vi-causal-inference-and-program-evaluation-_92-randomized-experiments","9.2 Randomized experiments",{"id":1579,"depth":332,"text":1580},"_9-domain-review-vi-causal-inference-and-program-evaluation-_93-quasi-experimental-methods","9.3 Quasi-experimental methods",{"id":1582,"depth":332,"text":1583},"_9-domain-review-vi-causal-inference-and-program-evaluation-_94-machine-learning-for-causal-inference","9.4 Machine learning for causal inference",{"id":1585,"depth":332,"text":1586},"_9-domain-review-vi-causal-inference-and-program-evaluation-_95-sensitivity-analysis-and-unobserved-confounding","9.5 Sensitivity analysis and unobserved confounding",{"id":1588,"depth":332,"text":1589},"_9-domain-review-vi-causal-inference-and-program-evaluation-_96-off-policy-evaluation","9.6 Off-policy evaluation",{"id":1591,"depth":386,"text":1592,"children":1593},"_10-domain-review-vii-online-learning-online-convex-optimization-and-conformal-prediction","10. Domain Review VII — Online Learning, Online Convex Optimization, and Conformal Prediction",[1594,1597,1600,1603],{"id":1595,"depth":332,"text":1596},"_10-domain-review-vii-online-learning-online-convex-optimization-and-conformal-prediction-_101-online-convex-optimization","10.1 Online convex optimization",{"id":1598,"depth":332,"text":1599},"_10-domain-review-vii-online-learning-online-convex-optimization-and-conformal-prediction-_102-no-regret-learning-and-game-theoretic-implications","10.2 No-regret learning and game-theoretic implications",{"id":1601,"depth":332,"text":1602},"_10-domain-review-vii-online-learning-online-convex-optimization-and-conformal-prediction-_103-conformal-prediction","10.3 Conformal prediction",{"id":1604,"depth":332,"text":1605},"_10-domain-review-vii-online-learning-online-convex-optimization-and-conformal-prediction-_104-distribution-shift-and-ood-generalization","10.4 Distribution shift and OOD generalization",{"id":1607,"depth":386,"text":1608,"children":1609},"_11-domain-review-viii-e-commerce-prediction-and-optimization","11. Domain Review VIII — E-Commerce Prediction and Optimization",[1610,1613,1616,1619,1622,1625,1628],{"id":1611,"depth":332,"text":1612},"_11-domain-review-viii-e-commerce-prediction-and-optimization-_111-demand-forecasting","11.1 Demand forecasting",{"id":1614,"depth":332,"text":1615},"_11-domain-review-viii-e-commerce-prediction-and-optimization-_112-pricing-and-elasticity","11.2 Pricing and elasticity",{"id":1617,"depth":332,"text":1618},"_11-domain-review-viii-e-commerce-prediction-and-optimization-_113-recommendation-and-personalization","11.3 Recommendation and personalization",{"id":1620,"depth":332,"text":1621},"_11-domain-review-viii-e-commerce-prediction-and-optimization-_114-promotion-and-assortment-optimization","11.4 Promotion and assortment optimization",{"id":1623,"depth":332,"text":1624},"_11-domain-review-viii-e-commerce-prediction-and-optimization-_115-real-time-bidding-and-display-advertising","11.5 Real-time bidding and display advertising",{"id":1626,"depth":332,"text":1627},"_11-domain-review-viii-e-commerce-prediction-and-optimization-_116-auction-theory-and-platform-mechanism-design","11.6 Auction theory and platform mechanism design",{"id":1629,"depth":332,"text":1630},"_11-domain-review-viii-e-commerce-prediction-and-optimization-_117-marketplace-ad-budget-allocation","11.7 Marketplace ad budget allocation",{"id":1632,"depth":386,"text":1633,"children":1634},"_12-detailed-case-studies","12. Detailed Case Studies",[1635,1638,1641,1644,1647,1650,1653,1656,1659,1662,1665],{"id":1636,"depth":332,"text":1637},"_12-detailed-case-studies-_121-case-1-markowitz-mean-variance-portfolio","12.1 Case 1 — Markowitz mean-variance portfolio",{"id":1639,"depth":332,"text":1640},"_12-detailed-case-studies-_122-case-2-rockafellaruryasev-cvar","12.2 Case 2 — Rockafellar–Uryasev CVaR",{"id":1642,"depth":332,"text":1643},"_12-detailed-case-studies-_123-case-3-almgrenchriss-optimal-execution","12.3 Case 3 — Almgren–Chriss optimal execution",{"id":1645,"depth":332,"text":1646},"_12-detailed-case-studies-_124-case-4-the-data-driven-newsvendor","12.4 Case 4 — The data-driven newsvendor",{"id":1648,"depth":332,"text":1649},"_12-detailed-case-studies-_125-case-5-cramrlundberg-ruin-theory","12.5 Case 5 — Cramér–Lundberg ruin theory",{"id":1651,"depth":332,"text":1652},"_12-detailed-case-studies-_126-case-6-contextual-bandit-news-recommendation-linucb","12.6 Case 6 — Contextual-bandit news recommendation (LinUCB)",{"id":1654,"depth":332,"text":1655},"_12-detailed-case-studies-_127-case-7-real-time-bidding-as-a-constrained-mdp","12.7 Case 7 — Real-time bidding as a constrained MDP",{"id":1657,"depth":332,"text":1658},"_12-detailed-case-studies-_128-case-8-the-m5-forecasting-competition","12.8 Case 8 — The M5 forecasting competition",{"id":1660,"depth":332,"text":1661},"_12-detailed-case-studies-_129-case-9-double-machine-learning-for-treatment-effect-estimation","12.9 Case 9 — Double machine learning for treatment-effect estimation",{"id":1663,"depth":332,"text":1664},"_12-detailed-case-studies-_1210-case-10-wasserstein-dro-and-adversarial-robustness","12.10 Case 10 — Wasserstein DRO and adversarial robustness",{"id":1666,"depth":332,"text":1667},"_12-detailed-case-studies-_1211-case-11-dataglass-marketplace-ad-budget-allocation","12.11 Case 11 — DataGlass marketplace ad budget allocation",{"id":1669,"depth":386,"text":1670,"children":1671},"_13-cross-domain-quantitative-synthesis","13. Cross-Domain Quantitative Synthesis",[1672,1675,1678,1681],{"id":1673,"depth":332,"text":1674},"_13-cross-domain-quantitative-synthesis-_131-the-four-primitive-taxonomy","13.1 The four-primitive taxonomy",{"id":1676,"depth":332,"text":1677},"_13-cross-domain-quantitative-synthesis-_132-effect-size-summary-across-deployed-systems","13.2 Effect-size summary across deployed systems",{"id":1679,"depth":332,"text":1680},"_13-cross-domain-quantitative-synthesis-_133-cross-domain-transfer-assessment","13.3 Cross-domain transfer assessment",{"id":1682,"depth":332,"text":1683},"_13-cross-domain-quantitative-synthesis-_134-quality-rubric-distribution","13.4 Quality-rubric distribution",{"id":1685,"depth":386,"text":1686,"children":1687},"_14-heterogeneity-risk-of-bias-and-publication-bias","14. Heterogeneity, Risk of Bias, and Publication Bias",[1688,1691,1694],{"id":1689,"depth":332,"text":1690},"_14-heterogeneity-risk-of-bias-and-publication-bias-_141-heterogeneity","14.1 Heterogeneity",{"id":1692,"depth":332,"text":1693},"_14-heterogeneity-risk-of-bias-and-publication-bias-_142-risk-of-bias","14.2 Risk of bias",{"id":1695,"depth":332,"text":1696},"_14-heterogeneity-risk-of-bias-and-publication-bias-_143-publication-bias","14.3 Publication bias",{"id":1698,"depth":386,"text":1699,"children":1700},"_15-open-problems-and-future-directions","15. Open Problems and Future Directions",[1701,1704,1707,1710,1713,1716,1719,1722,1725],{"id":1702,"depth":332,"text":1703},"_15-open-problems-and-future-directions-p1-time-consistent-risk-averse-bellman-for-marketplace-allocation","P1 — Time-consistent risk-averse Bellman for marketplace allocation",{"id":1705,"depth":332,"text":1706},"_15-open-problems-and-future-directions-p2-wasserstein-dro-with-bandit-regret-guarantees","P2 — Wasserstein DRO with bandit regret guarantees",{"id":1708,"depth":332,"text":1709},"_15-open-problems-and-future-directions-p3-joint-optimization-of-budget-and-target-roas","P3 — Joint optimization of budget and target ROAS",{"id":1711,"depth":332,"text":1712},"_15-open-problems-and-future-directions-p4-causal-identification-under-attribution-mixing","P4 — Causal identification under attribution mixing",{"id":1714,"depth":332,"text":1715},"_15-open-problems-and-future-directions-p5-cross-marketplace-transfer-learning","P5 — Cross-marketplace transfer learning",{"id":1717,"depth":332,"text":1718},"_15-open-problems-and-future-directions-p6-conformal-prediction-integration-into-operational-decision-systems","P6 — Conformal-prediction integration into operational decision systems",{"id":1720,"depth":332,"text":1721},"_15-open-problems-and-future-directions-p7-algorithmic-fairness-constraints-in-e-commerce-decision-systems","P7 — Algorithmic-fairness constraints in e-commerce decision systems",{"id":1723,"depth":332,"text":1724},"_15-open-problems-and-future-directions-p8-mechanism-design-counter-strategies-for-opaque-platform-auctions","P8 — Mechanism-design counter-strategies for opaque platform auctions",{"id":1726,"depth":332,"text":1727},"_15-open-problems-and-future-directions-p9-foundation-model-exploitation-of-unstructured-signals","P9 — Foundation-model exploitation of unstructured signals",{"id":1729,"depth":386,"text":1730},"_16-limitations-of-this-meta-review","16. Limitations of This Meta-Review",{"id":1732,"depth":386,"text":1733},"_17-reproducibility-statement","17. Reproducibility Statement",{"id":1735,"depth":386,"text":1736},"_18-conflict-of-interest-funding-and-ethics","18. Conflict of Interest, Funding, and Ethics",{"id":1738,"depth":386,"text":1739},"_19-conclusion","19. Conclusion",{"id":1741,"depth":386,"text":1742},"appendix-a-glossary","Appendix A — Glossary",{"id":1744,"depth":386,"text":1745},"appendix-b-search-strings","Appendix B — Search Strings",{"id":1747,"depth":386,"text":1748},"appendix-c-additional-summary-tables","Appendix C — Additional Summary Tables",{"id":1750,"depth":386,"text":1751,"children":1752},"references","References",[1753,1756,1759,1762,1765,1768,1771,1774,1777,1780,1783,1786,1789,1792,1795,1798,1801],{"id":1754,"depth":332,"text":1755},"references-internal-dataglass-references","Internal DataGlass references",{"id":1757,"depth":332,"text":1758},"references-foundations-of-decision-and-risk-theory","Foundations of decision and risk theory",{"id":1760,"depth":332,"text":1761},"references-risk-measures-and-robustdistributionally-robust-optimization","Risk measures and robust\u002Fdistributionally robust optimization",{"id":1763,"depth":332,"text":1764},"references-stochastic-dynamic-programming-mdps-and-reinforcement-learning","Stochastic dynamic programming, MDPs, and reinforcement learning",{"id":1766,"depth":332,"text":1767},"references-online-learning-bandits-and-oco","Online learning, bandits, and OCO",{"id":1769,"depth":332,"text":1770},"references-methodology-of-systematic-meta-reviews","Methodology of systematic \u002F meta-reviews",{"id":1772,"depth":332,"text":1773},"references-model-predictive-control","Model predictive control",{"id":1775,"depth":332,"text":1776},"references-quantitative-finance","Quantitative finance",{"id":1778,"depth":332,"text":1779},"references-operations-and-supply-chain","Operations and supply chain",{"id":1781,"depth":332,"text":1782},"references-insurance-and-actuarial","Insurance and actuarial",{"id":1784,"depth":332,"text":1785},"references-healthcare-energy-and-other-cross-domain","Healthcare, energy, and other cross-domain",{"id":1787,"depth":332,"text":1788},"references-causal-inference","Causal inference",{"id":1790,"depth":332,"text":1791},"references-conformal-prediction-and-distribution-shift","Conformal prediction and distribution shift",{"id":1793,"depth":332,"text":1794},"references-e-commerce-forecasting-pricing-recommendation-advertising","E-commerce: forecasting, pricing, recommendation, advertising",{"id":1796,"depth":332,"text":1797},"references-real-time-bidding-and-auctions","Real-time bidding and auctions",{"id":1799,"depth":332,"text":1800},"references-e-commerce-response-curves-and-changepoint-detection","E-commerce response curves and changepoint detection",{"id":1802,"depth":332,"text":1803},"references-open-problem-references","Open-problem references",[1805,1821,1828,1831,1834,1851,1857,1863,1865,1867,1872,2140,2142,2144,2167,2178,2188,2191,2207,2209,2211,2230,2233,2236,2242,2248,2254,2260,2262,2281,2283,2301,2307,2309,2312,2314,2317,2319,2326,2333,2340,2347,2354,2357,2359,2374,2376,2383,2385,2392,2394,2396,2398,2427,2431,2445,2474,2503,2506,2508,2522,2531,2534,2541,2559,2583,2586,2592,2606,2609,2624,2637,2639,2652,2655,2669,2672,2682,2699,2716,2719,2734,2760,2762,2767,2770,2793,2811,2814,2823,2841,2862,2864,2882,2885,2917,2920,2951,2966,2970,2972,2986,2989,3011,3013,3056,3058,3060,3062,3109,3111,3117,3120,3128,3130,3159,3161,3178,3181,3210,3212,3253,3255,3266,3268,3294,3296,3309,3311,3313,3315,3325,3328,3358,3360,3381,3383,3406,3408,3419,3421,3440,3442,3444,3451,3454,3494,3501,3503,3505,3529,3531,3533,3563,3565,3567,3573,3575,3612,3614,3637,3639,3672,3674,3724,3726,3741,3743,3773,3775,3777,3779,3809,3811,3840,3842,3861,3863,3888,3890,3892,3895,3897,3955,3957,3987,3989,4029,4031,4051,4053,4060,4063,4093,4095,4126,4128,4164,4174,4177,4180,4182,4184,4187,4189,4207,4222,4243,4257,4259,4267,4272,4275,4296,4305,4307,4317,4341,4344,4351,4356,4358,4371,4387,4416,4421,4423,4428,4454,4472,4480,4482,4495,4507,4522,4530,4532,4537,4563,4577,4588,4590,4595,4600,4608,4620,4622,4633,4648,4670,4675,4677,4682,4687,4690,4695,4717,4725,4727,4732,4743,4746,4756,4768,4778,4781,4783,4785,4787,4790,4795,5017,5019,5022,5027,5191,5193,5196,5201,5413,5415,5422,5424,5426,5428,5431,5433,5436,5445,5451,5457,5459,5465,5467,5469,5472,5474,5480,5482,5488,5490,5493,5495,5504,5506,5509,5511,5517,5519,5532,5534,5542,5544,5554,5556,5558,5567,5569,5571,5574,5576,5578,5584,5590,5592,5594,5600,5607,5610,5612,5614,5620,5626,5632,5638,5644,5649,5655,5660,5665,5671,5677,5682,5687,5693,5699,5705,5711,5717,5723,5729,5735,5737,5739,5742,5751,5759,5767,5775,5783,5791,5799,5807,5815,5823,5825,5827,5832,5943,5948,6084,6086,6088,6091,6093,6095,6097,6099,6101,6103,6105,6107,6109,6111,6113,6115,6117,6119,6121,6123,6125,6127],[1806,1807,1808],"p",{},[1809,1810,1811,1812,1816,1817,1820],"em",{},"Working paper, May 2026. Companion to the DataGlass technical paper ",[1813,1814],"blog-cite",{"n":1815},"1"," and the seller-pain meta-paper ",[1813,1818],{"n":1819},"2",". Cite as: DataGlass Labs Research, \"Prediction and Risk Optimization Under Uncertainty: A Cross-Domain Meta-Review of Methods in Finance, Operations, Causal Inference, and E-Commerce Decision Intelligence,\" DataGlass Labs Research working paper, May 2026.",[1806,1822,1823,1827],{},[1824,1825,1826],"strong",{},"Keywords."," Decision under uncertainty, risk-aware optimization, distributionally robust optimization, stochastic dynamic programming, mean-variance portfolio theory, conditional value-at-risk, multi-armed bandits, online convex optimization, model predictive control, causal inference, double machine learning, demand forecasting, real-time bidding, e-commerce decision intelligence, marketplace advertising, meta-review, systematic synthesis.",[1829,1830],"hr",{},[1832,1833,1449],"h2",{"id":1448},[1806,1835,1836,1839,1840,1843,1844,1847,1848,1850],{},[1824,1837,1838],{},"Background and objectives."," The state of the art in ",[1809,1841,1842],{},"prediction"," and ",[1809,1845,1846],{},"risk-aware optimization"," under uncertainty has been developed independently in finance, operations research, insurance, energy, healthcare, causal inference, and — most recently — e-commerce decision intelligence. The literatures share a mathematical core but cite each other only weakly. We map the landscape, identify the small set of formal primitives that underlies every mature production system, and bridge from canonical finance and operations results to the decision-intelligence problems of marketplace e-commerce — with the DataGlass system ",[1813,1849],{"n":1815}," as the connecting case.",[1806,1852,1853,1856],{},[1824,1854,1855],{},"Methods and results."," Structured narrative meta-review across Scopus, Web of Science, Google Scholar, arXiv, and SSRN, 1952–2026. The corpus is 213 primary works and 41 textbook\u002Fhandbook references, screened against an AMSTAR-2-inspired rubric (formal rigor, empirical validation, reproducibility, deployment evidence). Every mature decision system in the surveyed domains is a composition of four primitives: (i) a calibrated probabilistic model, (ii) a coherent risk-aware objective, (iii) an explicit operational constraint set, and (iv) a principled exploration mechanism. Eleven worked cases — Markowitz mean-variance, Rockafellar–Uryasev CVaR, Almgren–Chriss execution, the data-driven newsvendor, Cramér–Lundberg ruin theory, contextual-bandit news recommendation, real-time bidding as constrained MDP, the M5 forecasting competition, double machine learning, Wasserstein DRO, and DataGlass marketplace ad budget allocation — instantiate the framework. Cross-domain transfer is strong in the formal core (shadow prices, Bellman recursion, no-regret guarantees) and weaker in the calibration layer (risk-measure choice, ambiguity-set design, exploration schedule).",[1806,1858,1859,1862],{},[1824,1860,1861],{},"Conclusions and plain-language summary."," Cross-domain reading is structurally under-weighted in current practice; the next wave of e-commerce decision-intelligence systems will be built by engineers fluent in the prior generation's finance and operations literatures. We close with nine open problems. The non-technical version: the same math runs stock portfolios, inventory orders, insurance pricing, and Shopee or Amazon ad budgets — once you see the shared structure, many \"novel\" e-commerce algorithms turn out to be re-derivations of finance and operations classics. The paper provides a citable map for researchers and a checklist for practitioners.",[1829,1864],{},[1832,1866,1452],{"id":1451},[1806,1868,1869],{},[1809,1870,1871],{},"Symbols recurring across the meta-review. Domain-specific specialisations are introduced where used.",[1873,1874,1875,1888],"table",{},[1876,1877,1878],"thead",{},[1879,1880,1881,1885],"tr",{},[1882,1883,1884],"th",{},"Symbol",[1882,1886,1887],{},"Meaning",[1889,1890,1891,1906,1916,1926,1936,1946,1956,1966,1976,1986,1996,2009,2019,2031,2041,2054,2064,2077,2089,2106,2122,2130],"tbody",{},[1879,1892,1893,1900],{},[1894,1895,1896],"td",{},[1897,1898],"blog-math",{"html":1899,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>θ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\theta\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1901,1902,1903],{},"Unknown parameter; element of parameter space ",[1897,1904],{"html":1905,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">Θ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\Theta\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">Θ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1879,1907,1908,1913],{},[1894,1909,1910],{},[1897,1911],{"html":1912,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>X\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">X\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1914,1915],{},"Random outcome \u002F state",[1879,1917,1918,1923],{},[1894,1919,1920],{},[1897,1921],{"html":1922,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>π\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a, \\pi\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1924,1925],{},"Action; policy mapping states to actions",[1879,1927,1928,1933],{},[1894,1929,1930],{},[1897,1931],{"html":1932,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"script\">A\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathcal{A}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1934,1935],{},"Action \u002F decision set",[1879,1937,1938,1943],{},[1894,1939,1940],{},[1897,1941],{"html":1942,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L(a, X)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1944,1945],{},"Loss functional",[1879,1947,1948,1953],{},[1894,1949,1950],{},[1897,1951],{"html":1952,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ρ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\rho(\\cdot)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ρ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">⋅\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1954,1955],{},"Risk measure (variance, VaR, CVaR, generic coherent \u002F convex)",[1879,1957,1958,1963],{},[1894,1959,1960],{},[1897,1961],{"html":1962,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"script\">P\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">Q\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathcal{P}, \\mathcal{Q}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0822em;\">P\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\">Q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1964,1965],{},"Ambiguity sets over distributions",[1879,1967,1968,1973],{},[1894,1969,1970],{},[1897,1971],{"html":1972,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>V\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">V^{\\star}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6887em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1974,1975],{},"Value function in dynamic programming",[1879,1977,1978,1983],{},[1894,1979,1980],{},[1897,1981],{"html":1982,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">T\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1984,1985],{},"Horizon length (or sample size where context indicates)",[1879,1987,1988,1993],{},[1894,1989,1990],{},[1897,1991],{"html":1992,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>K\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>N\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">K, N\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">K\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,1994,1995],{},"Number of arms \u002F campaigns \u002F assets",[1879,1997,1998,2003],{},[1894,1999,2000],{},[1897,2001],{"html":2002,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>B\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">B_t\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0502em;\">B\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,2004,2005,2006],{},"Budget at time ",[1897,2007],{"html":2008,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">t\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6151em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1879,2010,2011,2016],{},[1894,2012,2013],{},[1897,2014],{"html":2015,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsup>\u003Cmi>λ\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mu^{\\star}, \\lambda^{\\star}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,2017,2018],{},"Lagrange multiplier \u002F shadow price on a budget or resource constraint",[1879,2020,2021,2026],{},[1894,2022,2023],{},[1897,2024],{"html":2025,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>R\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">R_T\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,2027,2028,2029],{},"Cumulative regret over horizon 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class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>α\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\alpha\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1879,2055,2056,2061],{},[1894,2057,2058],{},[1897,2059],{"html":2060,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>η\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>κ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\eta, \\kappa\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">κ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,2062,2063],{},"Auxiliary scalars (context-dependent: temporary impact, decay rate, Lagrangian)",[1879,2065,2066,2071],{},[1894,2067,2068],{},[1897,2069],{"html":2070,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath 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style=\"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">P\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">Q\u003C\u002Fspan>\u003Cspan 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mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,2085,2086,2087],{},"Feasible \u002F filtration set at time ",[1897,2088],{"html":2008,"display":-1},[1879,2090,2091,2096],{},[1894,2092,2093],{},[1897,2094],{"html":2095,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo 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mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,2097,2098,2099,2102,2103,2105],{},"Daily budget for campaign ",[1897,2100],{"html":2101,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">i\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan 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style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,2114,2115,2116,2118,2119],{},"Expected profit of campaign ",[1897,2117],{"html":2101,"display":-1}," at budget ",[1897,2120],{"html":2121,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>b\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1879,2123,2124,2127],{},[1894,2125,2126],{},"AOV, CVR, ROAS",[1894,2128,2129],{},"Average order value; conversion rate; return on ad spend",[1879,2131,2132,2137],{},[1894,2133,2134],{},[1897,2135],{"html":2136,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>m\u003C\u002Fmi>\u003Cmi>g\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>r\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>f\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>ρ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">m_g, r, f, \\rho\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ρ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,2138,2139],{},"Gross margin, return rate, platform-fee rate, fulfilment overhead (Section 11.1)",[1829,2141],{},[1832,2143,1455],{"id":1454},[1806,2145,2146,2147,2150,2151,2154,2155,2158,2159,2162,2163,2166],{},"Decision-making under uncertainty is the unifying problem of quantitative finance, operations research, insurance, energy economics, healthcare operations, causal inference for program evaluation, and — increasingly — e-commerce decision intelligence. Across these domains, the working stack is structurally similar: a probabilistic model is fit to historical data, a loss or utility functional is specified, an optimization problem is solved subject to operational constraints, and the resulting policy is deployed in an environment that will eventually deviate from the model's assumptions. The methodological vocabulary is shared. Markowitz ",[1813,2148],{"n":2149},"3",", Bellman ",[1813,2152],{"n":2153},"4",", Robbins ",[1813,2156],{"n":2157},"5",", and Knight ",[1813,2160],{"n":2161},"6"," still set the agenda. What differs across domains is the ",[1809,2164,2165],{},"calibration"," of the primitives — what is observable, what is controllable, what is the appropriate notion of \"risk,\" and what is the cost of being wrong.",[1806,2168,2169,2170,2173,2174,2177],{},"This paper is a meta-review of that shared landscape. The motivating gap is twofold. ",[1824,2171,2172],{},"First, the academic literature on prediction and risk-aware optimization is fragmented along domain lines."," A paper on conditional value-at-risk in portfolio theory rarely cites the closely related work on robust newsvendor inventory, and almost never cites the production e-commerce literature on real-time bidding under budget constraints, despite the three problems being structurally isomorphic. The same Lagrange multiplier — call it a shadow price in operations, a budget multiplier in advertising, a tangency-portfolio scalar in finance — recurs verbatim across domains, but the citation graphs barely intersect. ",[1824,2175,2176],{},"Second, the e-commerce decision-intelligence stack — pricing, demand forecasting, ad budget allocation, recommendation, promotion optimization — is rapidly adopting techniques developed for finance and operations decades earlier",", often without an explicit articulation of which translation steps are required and which are not. The result is a class of \"novel\" e-commerce algorithms that, on inspection, are unwitting re-derivations of finance and operations classics — and a class of finance and operations results that fail to land in the e-commerce literature simply because no one has written the bridge. A consolidated meta-review serves both audiences.",[1806,2179,2180,2181,2184,2185,2187],{},"The contribution of this paper is ",[1809,2182,2183],{},"not"," a new methodological result. It is (i) a unified taxonomy under which finance, operations, causal inference, and e-commerce optimization can be read as instances of the same problem class; (ii) a critical, citable synthesis of the canonical results in each domain; (iii) eleven detailed case studies that ground the abstract framework in concrete formulations; (iv) an explicit bridge from the finance and operations literature to the decision-intelligence problems of marketplace e-commerce, with the DataGlass system ",[1813,2186],{"n":1815}," as one worked example; and (v) the most complete cross-domain bibliography we are aware of in this area, with 254 references organized thematically.",[1806,2189,2190],{},"The paper is structured as follows. Section 2 documents the meta-review methodology with explicit search strings, screening protocol, and quality-assessment rubric. Section 3 develops the unified theoretical foundations. Sections 4–11 are the domain reviews. Section 12 contains eleven detailed case studies. Section 13 is the cross-domain quantitative synthesis with comparison tables. Section 14 documents heterogeneity, risk of bias, and publication-bias considerations. Section 15 lists nine open problems. Sections 16–18 contain limitations, reproducibility, and conflict-of-interest statements. Three appendices provide a glossary, the search strings, and additional summary tables. Citations follow IEEE numerical style.",[2192,2193,2195,2198,2199,2202,2203,2206],"blog-callout",{"title":2194},"The gap, and why this paper exists",[1824,2196,2197],{},"The gap."," Mature decision systems in quantitative finance, operations research, insurance, energy economics, healthcare, causal inference, and e-commerce solve structurally identical problems — yet the literatures cite each other only weakly. A CVaR-portfolio paper rarely cites the closely related distributionally-robust newsvendor work, and almost never cites the marketplace ad-budget literature, even though the three are isomorphic up to renaming. The seller-side e-commerce stack, in particular, is rapidly re-deriving results that finance and operations settled decades earlier. ",[1824,2200,2201],{},"The motivation."," A citable cross-domain map. We argue that ",[1809,2204,2205],{},"every"," production decision system in this space composes the same four primitives — a calibrated probabilistic model, a coherent risk-aware objective, an operational constraint set, and a principled exploration mechanism — and we trace those primitives through eleven worked cases, from Markowitz mean-variance to the DataGlass marketplace ad-budget allocation system, so engineers in newer fields can inherit the prior generation's results instead of reinventing them.",[1829,2208],{},[1832,2210,1458],{"id":1457},[1806,2212,2213,2214,2217,2218,2221,2222,2225,2226,2229],{},"We follow the recommendations of Petticrew and Roberts ",[1813,2215],{"n":2216},"7"," for systematic reviews in the social sciences and operations research, adapted to the structural-review form because the unit of analysis is ",[1809,2219,2220],{},"method"," rather than ",[1809,2223,2224],{},"clinical effect size",". The review is not PRISMA-compliant in the strict sense ",[1813,2227],{"n":2228},"8"," because pooled effect sizes are not meaningfully comparable across domains; we therefore adopt a structured narrative synthesis with explicit inclusion criteria, quality assessment, and heterogeneity discussion.",[2231,2232,1462],"h3",{"id":1461},[1806,2234,2235],{},"The review addresses four research questions.",[1806,2237,2238,2241],{},[1824,2239,2240],{},"RQ1."," What is the formal common structure, if any, of mature decision systems in finance, operations, insurance, energy, healthcare, causal inference, and e-commerce?",[1806,2243,2244,2247],{},[1824,2245,2246],{},"RQ2."," Which mathematical primitives recur across domains and which are domain-specific?",[1806,2249,2250,2253],{},[1824,2251,2252],{},"RQ3."," Which results from finance and operations transfer cleanly to e-commerce ad budget allocation and which do not?",[1806,2255,2256,2259],{},[1824,2257,2258],{},"RQ4."," What are the binding open research problems at the intersection?",[2231,2261,1465],{"id":1464},[1806,2263,2264,2265,2270,2271,2275,2276,2280],{},"Searches were conducted between January and April 2026 in the following databases: Scopus, Web of Science Core Collection, Google Scholar, the ACM Digital Library, IEEE Xplore, INFORMS PubsOnLine, JSTOR, arXiv (cs.LG, ",[2266,2267,2269],"a",{"href":2268},"http:\u002F\u002Fstat.ML","stat.ML",", math.OC, ",[2266,2272,2274],{"href":2273},"http:\u002F\u002Fq-fin.PM","q-fin.PM",", ",[2266,2277,2279],{"href":2278},"http:\u002F\u002Fq-fin.TR","q-fin.TR",", q-fin.RM), SSRN, and RePEc. Search strings combined a methodological term (\"risk-aware optimization\", \"conditional value-at-risk\", \"distributionally robust\", \"multi-armed bandit\", \"Bayesian decision\", \"constrained Markov decision process\", \"online convex optimization\", \"model predictive control\") with a domain term (\"portfolio\", \"execution\", \"inventory\", \"newsvendor\", \"insurance\", \"ruin\", \"advertising\", \"real-time bidding\", \"demand forecasting\", \"ad budget\", \"marketplace\"). The full search-string inventory is given in Appendix B.",[2231,2282,1468],{"id":1467},[1806,2284,2285,2288,2289,2292,2293,2296,2297,2300],{},[1824,2286,2287],{},"Inclusion."," A method (or method family) qualifies if all four of the following hold: (i) it produces a ",[1809,2290,2291],{},"decision"," under non-trivial uncertainty about a state variable; (ii) it falls within the seven specified domains or in a clearly cross-cutting methodology stratum (causal inference, online learning); (iii) it is ",[1809,2294,2295],{},"either"," canonically cited in subsequent literature ",[1809,2298,2299],{},"or"," (post-2018) demonstrates production-scale deployment with a quantified empirical lift; (iv) the paper or chapter specifies its loss functional, constraint set, and uncertainty model with enough precision for reproduction or critique.",[1806,2302,2303,2306],{},[1824,2304,2305],{},"Exclusion."," Pure prediction work without an explicit decision objective; qualitative case studies without analytical content; vendor whitepapers without independent validation; methods superseded by demonstrably better successors that are themselves included; papers behind paywalls without accessible preprints.",[2231,2308,1471],{"id":1470},[1806,2310,2311],{},"Initial searches returned approximately 6,400 candidate records. After deduplication and abstract-level screening against Section 2.3 criteria, 487 records were retained for full-text screening. Of these, 213 primary research papers and 41 textbook\u002Fhandbook chapters were retained for the final corpus. Records were screened against a written checklist; in cases of ambiguity, two reviewers independently rated the record on the Section 2.6 quality rubric and disagreements (12 records) were resolved by discussion.",[2231,2313,1474],{"id":1473},[1806,2315,2316],{},"For each retained record we extracted: (i) domain; (ii) problem class (single-period vs. sequential, finite vs. infinite horizon, full information vs. bandit); (iii) probabilistic model class; (iv) loss\u002Frisk functional; (v) constraint set; (vi) exploration mechanism if any; (vii) theoretical result(s); (viii) empirical validation if any; (ix) deployment evidence if any. Extraction was performed into a structured database; summary tables in Section 13 are derived from this database.",[2231,2318,1477],{"id":1476},[1806,2320,2321,2322,2325],{},"Each record was scored on a four-dimensional rubric inspired by the AMSTAR-2 framework ",[1813,2323],{"n":2324},"9",", adapted for methodological review:",[1806,2327,2328,2329,2332],{},"1. ",[1809,2330,2331],{},"Formal rigor"," — does the paper state assumptions and prove its claims, or is the result heuristic?",[1806,2334,2335,2336,2339],{},"2. ",[1809,2337,2338],{},"Empirical validation"," — is there at least one numerical study with held-out evaluation?",[1806,2341,2342,2343,2346],{},"3. ",[1809,2344,2345],{},"Reproducibility"," — is the algorithm specified to a level that an independent implementation could reproduce headline results?",[1806,2348,2349,2350,2353],{},"4. ",[1809,2351,2352],{},"Deployment evidence"," — has the method been deployed at production scale, with an effect size and confidence interval reported?",[1806,2355,2356],{},"Scores are 0 \u002F 1 \u002F 2 on each dimension (0 = absent, 1 = partial, 2 = strong). The summary tables in Section 13 indicate the median rubric score for each method family.",[2231,2358,1480],{"id":1479},[1806,2360,2361,2362,2365,2366,2369,2370,2373],{},"Quantitative heterogeneity (in the meta-analytic sense of ",[1897,2363],{"html":2364,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>I\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">I^2\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8141em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">I\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," statistic) is not meaningful across domains. We instead document ",[1809,2367,2368],{},"structural"," heterogeneity along the four-primitive taxonomy of Section 3: probabilistic model, risk objective, constraint set, exploration mechanism. Cross-domain transfer is graded ",[1809,2371,2372],{},"strong \u002F partial \u002F weak"," based on whether the formal object, the calibration, and the empirical performance characteristics carry over.",[2231,2375,1483],{"id":1482},[1806,2377,2378,2379,2382],{},"We adopt a ",[1809,2380,2381],{},"vote-counting"," synthesis at the level of methodological primitives — not at the level of effect sizes — supplemented by detailed worked cases that exhibit the primitives concretely. This approach is appropriate when the units of synthesis are methods rather than studies and when effect sizes are not commensurable.",[2231,2384,1486],{"id":1485},[1806,2386,2387,2388,2391],{},"The protocol was not pre-registered on PROSPERO because PROSPERO's scope is restricted to health-related reviews. The protocol is recorded internally at DataGlass Labs Research and is available on request. One deviation from the protocol: the post-screening corpus expanded the ",[1809,2389,2390],{},"causal inference"," domain (Section 9) beyond the original scope after pilot extraction made clear that the primitives were structurally critical to the e-commerce sections.",[1829,2393],{},[1832,2395,1489],{"id":1488},[2231,2397,1493],{"id":1492},[1806,2399,2400,2401,2403,2404,2406,2407,2410,2411,2414,2415,2418,2419,2422,2423,2426],{},"Let ",[1897,2402],{"html":1905,"display":-1}," denote the (unknown) parameter, ",[1897,2405],{"html":1912,"display":-1}," the (random) outcome, and ",[1897,2408],{"html":2409,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">A\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a \\in \\mathcal{A}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," the action. The agent observes data ",[1897,2412],{"html":2413,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"script\">D\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathcal{D}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", forms a posterior ",[1897,2416],{"html":2417,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">D\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p(\\theta \\mid \\mathcal{D})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", and chooses ",[1897,2420],{"html":2421,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," to minimize an expected loss ",[1897,2424],{"html":2425,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>L\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". The Bayes action is",[1897,2428],{"html":2429,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>a\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>arg\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">A\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsub>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmrow>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo>∼\u003C\u002Fmo>\u003Cmi>p\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">D\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmtext> \u003C\u002Fmtext>\u003Cmsub>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmrow>\u003Cmi>X\u003C\u002Fmi>\u003Cmo>∼\u003C\u002Fmo>\u003Cmi>p\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmtext> \u003C\u002Fmtext>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmtext> \u003C\u002Fmtext>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">.\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a^{\\star} \\;=\\; \\arg\\min_{a \\in \\mathcal{A}} \\; \\mathbb{E}_{\\theta \\sim p(\\theta\\mid\\mathcal{D})}\\, \\mathbb{E}_{X\\sim p(X\\mid\\theta)}\\,[\\,L(a, X)\\,].\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7387em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7387em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.5217em;vertical-align:-0.7717em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">ar\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.3557em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">a\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∈\u003C\u002Fspan>\u003Cspan class=\"mord mathcal mtight\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7717em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3448em;\">\u003Cspan style=\"top:-2.5198em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∼\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">p\u003C\u002Fspan>\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathcal mtight\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3552em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3448em;\">\u003Cspan style=\"top:-2.5198em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∼\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">p\u003C\u002Fspan>\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3552em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003Cspan class=\"mord\">.\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","true",[1806,2432,2433,2434,2275,2437,2440,2441,2444],{},"This is the canonical Savage–Berger Bayesian decision-theoretic setup ",[1813,2435],{"n":2436},"10",[1813,2438],{"n":2439},"11",". Frequentist alternatives — minimax, empirical risk minimization, statistical learning theory — replace the outer expectation with worst-case or finite-sample analogues ",[1813,2442],{"n":2443},"12",".",[1806,2446,2447,2448,2451,2452,2455,2456,2459,2460,2463,2464,2466,2467,2470,2471,2444],{},"The four-layer decomposition that organizes the rest of this paper emerges immediately. The probabilistic model is ",[1897,2449],{"html":2450,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmi>p\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p(X \\mid \\theta) p(\\theta)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","; the ",[1809,2453,2454],{},"risk-aware"," objective is some functional ",[1897,2457],{"html":2458,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ρ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\rho[L(a, X)]\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ρ\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," generalizing ",[1897,2461],{"html":2462,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathbb{E}[L]\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","; constraints restrict ",[1897,2465],{"html":1932,"display":-1}," to feasible actions; and exploration ensures that ",[1897,2468],{"html":2469,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">D\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p(\\theta\\mid\\mathcal{D})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," becomes concentrated where it matters most for ",[1897,2472],{"html":2473,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>a\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a^{\\star}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6887em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[2192,2475,2477,2478,2481,2482,2485,2486,2489,2490,2493,2494,2497,2498,2275,2500,2444],{"title":2476},"Theorem 3.1 — Bayes-rule optimality","Under squared-error loss ",[1897,2479],{"html":2480,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L(a, X) = (a - X)^2\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", the unique Bayes action is ",[1897,2483],{"html":2484,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>a\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">D\u003C\u002Fmi>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a^{\\star} = \\mathbb{E}[X \\mid \\mathcal{D}]\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6887em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". Under absolute loss ",[1897,2487],{"html":2488,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L(a, X) = |a - X|\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", the Bayes action is the posterior median. Under pinball loss ",[1897,2491],{"html":2492,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>τ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>+\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>τ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L(a, X) = \\tau(X - a)^+ + (1 - \\tau)(a - X)^+\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0213em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0213em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", the Bayes action is the ",[1897,2495],{"html":2496,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>τ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tau\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","-quantile of the posterior predictive distribution ",[1813,2499],{"n":2439},[1813,2501],{"n":2502},"13",[1806,2504,2505],{},"The choice of loss therefore implicitly chooses the summary statistic of the posterior that the system optimizes — a point that becomes operationally critical in Section 11.4 (forecasting) and Section 11.6 (advertising).",[2231,2507,1496],{"id":1495},[1806,2509,2510,2511,2514,2515,2518,2519,2521],{},"Replacing ",[1897,2512],{"html":2513,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathbb{E}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6889em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," with a risk measure ",[1897,2516],{"html":2517,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ρ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\rho\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ρ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," produces a ",[1809,2520,2454],{}," objective. We catalogue the four classes that recur most frequently.",[1806,2523,2524,2527,2528,2530],{},[1824,2525,2526],{},"Variance and mean-variance."," Markowitz ",[1813,2529],{"n":2149}," defines portfolio risk as variance and gives the canonical mean-variance program",[1897,2532],{"html":2533,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>w\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi mathvariant=\"normal\">Σ\u003C\u002Fmi>\u003Cmi>w\u003C\u002Fmi>\u003Cmspace width=\"1em\"\u002F>\u003Cmtext>s.t.\u003C\u002Fmtext>\u003Cmspace width=\"1em\"\u002F>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>r\u003C\u002Fmi>\u003Cmo>ˉ\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmn mathvariant=\"bold\">1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\min_w \\; w^\\top \\Sigma w \\quad \\text{s.t.} \\quad \\mu^\\top w = \\bar r,\\; \\mathbf{1}^\\top w = 1,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.5991em;vertical-align:-0.7em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.4em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">Σ\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">s.t.\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0935em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5678em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1944em;\">\u003Cspan class=\"mord\">ˉ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbf\">1\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2535,2536,2537,2540],{},"which produces the Markowitz frontier and, with a risk-free asset, the Capital Market Line of Sharpe ",[1813,2538],{"n":2539},"14",". The objection that variance penalizes upside as well as downside motivates downside-risk alternatives.",[1806,2542,2543,2546,2547,2550,2551,2554,2555,2558],{},[1824,2544,2545],{},"Value-at-Risk."," ",[1897,2548],{"html":2549,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmrow>\u003Cmi mathvariant=\"normal\">V\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">a\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">R\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmi>α\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>inf\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">{\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">ℓ\u003C\u002Fmi>\u003Cmo>:\u003C\u002Fmo>\u003Cmi>Pr\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo>≤\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">ℓ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>≥\u003C\u002Fmo>\u003Cmi>α\u003C\u002Fmi>\u003Cmo stretchy=\"false\">}\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathrm{VaR}_\\alpha(L) = \\inf\\{\\ell : \\Pr(L \\le \\ell) \\ge \\alpha\\}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">VaR\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">in\u003Cspan style=\"margin-right:0.0778em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">{\u003C\u002Fspan>\u003Cspan class=\"mord\">ℓ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">:\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">Pr\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≤\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">ℓ\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"mclose\">}\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". VaR is the regulatory lingua franca of banking ",[1813,2552],{"n":2553},"15"," but is non-coherent: it fails sub-additivity, so portfolio diversification can ",[1809,2556,2557],{},"increase"," VaR.",[1806,2560,2561,2546,2564,2567,2568,2275,2571,2574,2575,2578,2579,2582],{},[1824,2562,2563],{},"Conditional Value-at-Risk (Expected Shortfall).",[1897,2565],{"html":2566,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmrow>\u003Cmi mathvariant=\"normal\">C\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">V\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">a\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">R\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmi>α\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmsub>\u003Cmrow>\u003Cmi mathvariant=\"normal\">V\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">a\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">R\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmi>α\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathrm{CVaR}_\\alpha(L) = \\mathbb{E}[L \\mid L \\ge \\mathrm{VaR}_\\alpha(L)]\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">CVaR\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8193em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">VaR\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". Rockafellar and Uryasev ",[1813,2569],{"n":2570},"16",[1813,2572],{"n":2573},"17"," showed that CVaR is ",[1809,2576,2577],{},"coherent"," in the sense of Artzner et al. ",[1813,2580],{"n":2581},"18"," and admits the Linear-Program reformulation",[1897,2584],{"html":2585,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmrow>\u003Cmi mathvariant=\"normal\">C\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">V\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">a\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">R\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmi>α\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>η\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>η\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmrow>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>α\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">[\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>η\u003C\u002Fmi>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">]\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">.\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathrm{CVaR}_\\alpha(L) \\;=\\; \\min_{\\eta} \\;\\eta + \\frac{1}{1-\\alpha}\\,\\mathbb{E}\\big[(L - \\eta)^+\\big].\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">CVaR\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.504em;vertical-align:-0.8361em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.4em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8361em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:2.0908em;vertical-align:-0.7693em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3214em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7693em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">[\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8213em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">]\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">.\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2587,2588,2589,2591],{},"When ",[1897,2590],{"html":2425,"display":-1}," is linear in decisions, this becomes a tractable LP — a watershed result that made CVaR the dominant practical risk measure in modern portfolio optimization, capital allocation, and increasingly in fairness-aware machine learning.",[1806,2593,2594,2597,2598,2601,2602,2605],{},[1824,2595,2596],{},"Convex and distortion risk measures."," Föllmer and Schied ",[1813,2599],{"n":2600},"19"," and Frittelli and Rosazza Gianin ",[1813,2603],{"n":2604},"20"," generalize coherence to convex risk measures, which preserve the dual representation",[1897,2607],{"html":2608,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>ρ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmrow>\u003Cmi>sup\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>Q\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">Q\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmtext> \u003C\u002Fmtext>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">{\u003C\u002Fmo>\u003Cmsub>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmi>Q\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>−\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>α\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>Q\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">}\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\rho(L) \\;=\\; \\sup_{Q \\in \\mathcal{Q}}\\,\\big\\{\\mathbb{E}_Q[L] \\;-\\; \\alpha(Q)\\big\\},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ρ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.9249em;vertical-align:-1.0749em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4306em;\">\u003Cspan style=\"top:-2.1612em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">Q\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∈\u003C\u002Fspan>\u003Cspan class=\"mord mathcal mtight\">Q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">sup\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.0749em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">{\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">Q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">Q\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">}\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2610,2611,2612,2615,2616,2619,2620,2623],{},"with ",[1897,2613],{"html":2614,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>α\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>Q\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\alpha(Q)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">Q\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," a penalty function on test measures. Distortion risk measures of Wang ",[1813,2617],{"n":2618},"21"," cover the actuarial premium-principle literature and recover CVaR as a special case of a piecewise-linear distortion. Spectral risk measures ",[1813,2621],{"n":2622},"22"," sit between CVaR and the broader convex class.",[2192,2625,2627,2628,2630,2631,2634,2635,2444],{"title":2626},"Theorem 3.2 — Coherent risk axioms (Artzner et al.)","A functional ",[1897,2629],{"html":2517,"display":-1}," on ",[1897,2632],{"html":2633,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>L\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L^\\infty\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6644em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">∞\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is coherent iff it satisfies (i) monotonicity, (ii) sub-additivity, (iii) positive homogeneity, and (iv) translation invariance. CVaR satisfies all four; VaR fails (ii) in general ",[1813,2636],{"n":2581},[2231,2638,1499],{"id":1498},[1806,2640,2641,2642,2644,2645,2546,2648,2651],{},"When the distribution of ",[1897,2643],{"html":1912,"display":-1}," is itself uncertain, the agent can hedge against the worst case in an ",[1809,2646,2647],{},"ambiguity set",[1897,2649],{"html":2650,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"script\">P\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathcal{P}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0822em;\">P\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",":",[1897,2653],{"html":2654,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmrow>\u003Cmi>sup\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>P\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">P\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsub>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmrow>\u003Cmi>X\u003C\u002Fmi>\u003Cmo>∼\u003C\u002Fmo>\u003Cmi>P\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">.\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\min_a \\; \\sup_{P \\in \\mathcal{P}} \\; \\mathbb{E}_{X\\sim P}[L(a, X)].\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.7161em;vertical-align:-0.9661em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.4em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4306em;\">\u003Cspan style=\"top:-2.1612em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">P\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∈\u003C\u002Fspan>\u003Cspan class=\"mord mathcal mtight\" style=\"margin-right:0.0822em;\">P\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">sup\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9661em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∼\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">P\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003Cspan class=\"mord\">.\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2656,2657,2658,2661,2662,2665,2666,2444],{},"Ben-Tal, El Ghaoui, and Nemirovski ",[1813,2659],{"n":2660},"23"," systematize the deterministic robust counterpart for ellipsoidal uncertainty in linear and conic programs, achieving tractable second-order-cone reformulations. Bertsimas and Sim ",[1813,2663],{"n":2664},"24"," give the \"price of robustness\" budgeted-uncertainty framework that interpolates between the nominal and worst-case problems with a tunable conservatism parameter ",[1897,2667],{"html":2668,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">Γ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\Gamma\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">Γ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2670,2671],{},"Distributionally Robust Optimization (DRO) replaces the parametric ambiguity set with a divergence ball.",[1806,2673,2674,2677,2678,2681],{},[1824,2675,2676],{},"Moment ambiguity."," Delage and Ye ",[1813,2679],{"n":2680},"25"," characterize the worst-case expectation under known mean and second-moment intervals and prove SDP reformulability.",[1806,2683,2684,2690,2691,2694,2695,2698],{},[1824,2685,2686,2689],{},[1897,2687],{"html":2688,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ϕ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\phi\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ϕ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","-divergence ambiguity."," Ben-Tal et al. ",[1813,2692],{"n":2693},"26"," develop DRO under Kullback–Leibler, ",[1897,2696],{"html":2697,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>χ\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\chi^2\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">χ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", and Hellinger ambiguity, with explicit worst-case expressions.",[1806,2700,2701,2704,2705,2708,2709,2711,2712,2715],{},[1824,2702,2703],{},"Wasserstein ambiguity."," Esfahani and Kuhn ",[1813,2706],{"n":2707},"27"," place a ",[1897,2710],{"html":2075,"display":-1},"-Wasserstein ball of radius ",[1897,2713],{"html":2714,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ε\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\varepsilon\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ε\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," around the empirical distribution and prove the equivalence",[1897,2717],{"html":2718,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>sup\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>Q\u003C\u002Fmi>\u003Cmo>:\u003C\u002Fmo>\u003Cmsub>\u003Cmi>W\u003C\u002Fmi>\u003Cmi>p\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>Q\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmover accent=\"true\">\u003Cmi>P\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>≤\u003C\u002Fmo>\u003Cmi>ε\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmsub>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmi>Q\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmrow>\u003Cmi>inf\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>λ\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi>λ\u003C\u002Fmi>\u003Cmsup>\u003Cmi>ε\u003C\u002Fmi>\u003Cmi>p\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo>+\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmfrac>\u003Cmunderover>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmunderover>\u003Cmunder>\u003Cmrow>\u003Cmi>sup\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">{\u003C\u002Fmo>\u003Cmi>L\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>λ\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∥\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmover accent=\"true\">\u003Cmi>x\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmsup>\u003Cmi mathvariant=\"normal\">∥\u003C\u002Fmi>\u003Cmi>p\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">}\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\sup_{Q : W_p(Q, \\hat P_n) \\le \\varepsilon} \\mathbb{E}_Q[L(a, X)] \\;=\\; \\inf_{\\lambda \\ge 0}\\,\\lambda \\varepsilon^p + \\frac{1}{n}\\sum_{i=1}^{n} \\sup_{x}\\big\\{L(a, x) - \\lambda \\|x - \\hat x_i\\|^p\\big\\},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:2.0705em;vertical-align:-1.3205em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4306em;\">\u003Cspan style=\"top:-1.9768em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">Q\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">:\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">W\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1645em;\">\u003Cspan style=\"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2819em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">Q\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord accent mtight\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan 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mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.143em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">≤\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">ε\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">sup\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3205em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">Q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.5617em;vertical-align:-0.8473em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-2.3479em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">λ\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">≥\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">in\u003Cspan style=\"margin-right:0.0778em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8473em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">ε\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7144em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:2.9291em;vertical-align:-1.2777em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3214em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.686em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.6514em;\">\u003Cspan style=\"top:-1.8723em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-4.3em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.2777em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4306em;\">\u003Cspan style=\"top:-2.2056em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">sup\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">{\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"mord\">∥\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2222em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">∥\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7144em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">}\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2720,2721,2722,2725,2726,2729,2730,2733],{},"which reduces Wasserstein DRO to a regularized empirical risk minimization. Blanchet, Murthy, and Si ",[1813,2723],{"n":2724},"28"," establish duality and rate results; Gao and Kleywegt ",[1813,2727],{"n":2728},"29"," give a comprehensive treatment. Wasserstein DRO has emerged as the unifying lens for both adversarial robustness in machine learning ",[1813,2731],{"n":2732},"30"," and operational hedging in supply chains.",[1806,2735,2736,2737,2740,2741,2744,2745,2747,2748,2751,2752,2755,2756,2759],{},"The Knightian distinction between ",[1809,2738,2739],{},"risk"," (probabilities known) and ",[1809,2742,2743],{},"uncertainty"," (probabilities themselves uncertain) ",[1813,2746],{"n":2161},", formalized by Ellsberg ",[1813,2749],{"n":2750},"31"," and Gilboa–Schmeidler ",[1813,2753],{"n":2754},"32",", is the philosophical foundation of this entire literature. Its decision-theoretic analogue — the ",[1809,2757,2758],{},"maxmin expected utility"," representation — is the link between behavioral economics and DRO.",[2231,2761,1502],{"id":1501},[1806,2763,2764,2765],{},"When decisions are sequential, the natural formalism is the Markov Decision Process. The Bellman equation ",[1813,2766],{"n":2153},[1897,2768],{"html":2769,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>V\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>a\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.8em\" maxsize=\"1.8em\">{\u003C\u002Fmo>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi>c\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>γ\u003C\u002Fmi>\u003Cmtext> \u003C\u002Fmtext>\u003Cmsub>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmrow>\u003Cmsup>\u003Cmi>s\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>∼\u003C\u002Fmo>\u003Cmi>P\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmtext> \u003C\u002Fmtext>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmsup>\u003Cmi>V\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsup>\u003Cmi>s\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmtext> \u003C\u002Fmtext>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.8em\" maxsize=\"1.8em\">}\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">V^{\\star}(s) \\;=\\; \\min_{a \\in \\mathcal{A}(s)} \\; \\Big\\{\\, c(s, a) + \\gamma\\, \\mathbb{E}_{s'\\sim P(\\cdot\\mid s,a)}\\,[V^{\\star}(s')]\\,\\Big\\},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7387em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:2.116em;vertical-align:-0.966em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.309em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">a\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∈\u003C\u002Fspan>\u003Cspan class=\"mord mathcal mtight\">A\u003C\u002Fspan>\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">s\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.966em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size2\">{\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.8em;vertical-align:-0.65em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3448em;\">\u003Cspan style=\"top:-2.5198em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6828em;\">\u003Cspan style=\"top:-2.786em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∼\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">P\u003C\u002Fspan>\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">⋅\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">s\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">a\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3552em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7387em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8019em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size2\">}\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2771,2772,2773,2776,2777,2780,2781,2784,2785,2788,2789,2792],{},"is the universal recursion. Bertsekas ",[1813,2774],{"n":2775},"33"," and Puterman ",[1813,2778],{"n":2779},"34"," are the standard references for finite-state MDPs. Powell ",[1813,2782],{"n":2783},"35"," develops ",[1809,2786,2787],{},"approximate dynamic programming"," with explicit treatment of the curse of dimensionality across resource allocation, energy, and freight applications. Sutton and Barto ",[1813,2790],{"n":2791},"36"," is the modern reinforcement-learning treatment.",[1806,2794,2795,2798,2799,2802,2803,2806,2807,2810],{},[1824,2796,2797],{},"Risk-sensitive DP."," Howard and Matheson ",[1813,2800],{"n":2801},"37"," introduced risk-sensitive Bellman recursions with exponential utility. Ruszczyński ",[1813,2804],{"n":2805},"38"," establishes the ",[1809,2808,2809],{},"time-consistency"," conditions under which a Markov risk-measure formulation yields a tractable DP. The risk-averse Bellman recursion is",[1897,2812],{"html":2813,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>V\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">{\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>γ\u003C\u002Fmi>\u003Cmtext> \u003C\u002Fmtext>\u003Cmsub>\u003Cmi>ρ\u003C\u002Fmi>\u003Cmrow>\u003Cmsup>\u003Cmi>s\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>∣\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmsup>\u003Cmi>V\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsup>\u003Cmi>s\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">}\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">V^{\\star}(s) \\;=\\; \\min_a \\big\\{ c(s,a) + \\gamma\\, \\rho_{s'\\mid s,a}[V^{\\star}(s')] \\big\\},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7387em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.55em;vertical-align:-0.7em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.4em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">{\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2052em;vertical-align:-0.3552em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">ρ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3448em;\">\u003Cspan style=\"top:-2.5198em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6828em;\">\u003Cspan style=\"top:-2.786em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">s\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3552em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7387em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8019em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">}\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2815,2816,2817,2819,2820,2822],{},"where ",[1897,2818],{"html":2517,"display":-1}," is a coherent (or convex) Markov risk measure. Time-consistency requires that ",[1897,2821],{"html":2517,"display":-1}," admits a translation-equivariant decomposition.",[1806,2824,2825,2828,2829,2832,2833,2836,2837,2840],{},[1824,2826,2827],{},"Constrained MDPs."," Altman ",[1813,2830],{"n":2831},"39"," develops the theory of constrained MDPs in which the agent maximizes a primary expected reward subject to expected-cost constraints. Lagrangian duality reduces the constrained MDP to an unconstrained one with cost ",[1897,2834],{"html":2835,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>c\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmsup>\u003Cmi>λ\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi>g\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>a\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">c(s,a) + \\lambda^\\top g(s,a)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0991em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", with ",[1897,2838],{"html":2839,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>λ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\lambda\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," the dual variable. This is the formal home of every \"budget-constrained\" decision system reviewed in Sections4, 8, 9, and 11.",[1806,2842,2843,2846,2847,2850,2851,2854,2855,2858,2859,2444],{},[1824,2844,2845],{},"Robust and DR MDPs."," Iyengar ",[1813,2848],{"n":2849},"40"," and Nilim and El Ghaoui ",[1813,2852],{"n":2853},"41"," develop robust MDPs with rectangular ambiguity in the transition kernel; Wiesemann, Kuhn, and Rustem ",[1813,2856],{"n":2857},"42"," extend to convex ambiguity. Distributionally robust dynamic programming has emerged as a unified language for safe RL ",[1813,2860],{"n":2861},"43",[2231,2863,1505],{"id":1504},[1806,2865,2866,2867,2275,2870,2873,2874,2877,2878,2881],{},"When the environment is observed sequentially and the agent must trade off exploration and exploitation, the relevant theory is online learning ",[1813,2868],{"n":2869},"44",[1813,2871],{"n":2872},"45"," and multi-armed bandits ",[1813,2875],{"n":2876},"46",". The canonical regret bound for the upper-confidence-bound (UCB) algorithm of Auer, Cesa-Bianchi, and Fischer ",[1813,2879],{"n":2880},"47"," is",[1897,2883],{"html":2884,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>R\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>O\u003C\u002Fmi>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmsqrt>\u003Cmrow>\u003Cmi>K\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsqrt>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">R_T \\;=\\; O\\!\\left( \\sqrt{KT \\log T} \\right),\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.8em;vertical-align:-0.65em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size2\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9839em;\">\u003Cspan class=\"svg-align\" style=\"top:-3.2em;\">\u003Cspan class=\"pstrut\" style=\"height:3.2em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:1em;\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">K\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.9439em;\">\u003Cspan class=\"pstrut\" style=\"height:3.2em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2561em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size2\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2886,2887,2888,2275,2891,2894,2895,2898,2899,2546,2902,2905,2906,2909,2910,2912,2913,2916],{},"matched up to constants by Thompson Sampling ",[1813,2889],{"n":2890},"48",[1813,2892],{"n":2893},"49",", which is empirically superior under model misspecification and delayed feedback ",[1813,2896],{"n":2897},"50",". Under budget constraints, ",[1809,2900,2901],{},"Bandits with Knapsacks",[1813,2903],{"n":2904},"51"," gives an ",[1897,2907],{"html":2908,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>O\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmrow>\u003Cmi mathvariant=\"normal\">O\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">P\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">T\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmsqrt>\u003Cmi>B\u003C\u002Fmi>\u003C\u002Fmsqrt>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">O(\\mathrm{OPT}\u002F\\sqrt{B})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1767em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">OPT\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9267em;\">\u003Cspan class=\"svg-align\" style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:0.833em;\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0502em;\">B\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.8867em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1133em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," regret bound directly relevant to advertising allocation ",[1813,2911],{"n":1815},". Under non-stationarity, dynamic-regret bounds of Besbes, Gur, and Zeevi ",[1813,2914],{"n":2915},"52"," yield",[1897,2918],{"html":2919,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>R\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>O\u003C\u002Fmi>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">(\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>K\u003C\u002Fmi>\u003Cmsub>\u003Cmi>V\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmrow>\u003Cmn>1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmsup>\u003Cmi>T\u003C\u002Fmi>\u003Cmrow>\u003Cmn>2\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo fence=\"false\" stretchy=\"true\" minsize=\"1.2em\" maxsize=\"1.2em\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">R_T \\;=\\; O\\big((K V_T)^{1\u002F3} T^{2\u002F3}\\big),\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.288em;vertical-align:-0.35em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">K\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.938em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u002F3\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.938em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u002F3\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"delimsizing size1\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2921,2816,2922,2924,2925,2928,2929,2932,2933,2935,2936,2939,2940,2943,2944,2275,2947,2950],{},[1897,2923],{"html":2037,"display":-1}," is the path variation; Chen, Lee, and Luo ",[1813,2926],{"n":2927},"53"," sharpen this to ",[1897,2930],{"html":2931,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>O\u003C\u002Fmi>\u003Cmo>~\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsqrt>\u003Cmrow>\u003Cmsubsup>\u003Cmi>V\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003Cmtext>res\u003C\u002Fmtext>\u003C\u002Fmsubsup>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsqrt>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tilde O(\\sqrt{V_T^{\\text{res}} T})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2803em;vertical-align:-0.3601em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9202em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.6023em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1667em;\">\u003Cspan class=\"mord\">~\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8799em;\">\u003Cspan class=\"svg-align\" style=\"top:-3.2em;\">\u003Cspan class=\"pstrut\" style=\"height:3.2em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:1em;\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6462em;\">\u003Cspan style=\"top:-2.4065em;margin-left:-0.2222em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.0448em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord text mtight\">\u003Cspan class=\"mord mtight\">res\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2935em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.8399em;\">\u003Cspan class=\"pstrut\" style=\"height:3.2em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3601em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," when side information is available. Hazan ",[1813,2934],{"n":2872}," develops online convex optimization with ",[1897,2937],{"html":2938,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>O\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsqrt>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmsqrt>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">O(\\sqrt{T})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1767em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9267em;\">\u003Cspan class=\"svg-align\" style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:0.833em;\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.8867em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1133em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," regret via online gradient descent, ",[1897,2941],{"html":2942,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>O\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>T\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">O(\\log T)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," under strong convexity, and the mirror-descent generalization ",[1813,2945],{"n":2946},"54",[1813,2948],{"n":2949},"55"," that connects to natural gradients and Bregman divergences.",[2192,2952,2954,2955,2958,2959,2962,2963,2444],{"title":2953},"Theorem 3.3 — Lai–Robbins lower bound","For any consistent strategy on a parametric multi-armed bandit, the asymptotic regret satisfies ",[1897,2956],{"html":2957,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmrow>\u003Cmi mathvariant=\"normal\">lim inf\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>T\u003C\u002Fmi>\u003Cmo>→\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmsub>\u003Cmi>R\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>T\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmsub>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>:\u003C\u002Fmo>\u003Cmsub>\u003Cmi>μ\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo><\u003C\u002Fmo>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo>⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo>⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>μ\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmi>D\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>μ\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi mathvariant=\"normal\">∥\u003C\u002Fmi>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo>⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\liminf_{T\\to\\infty} R_T \u002F \\log T \\ge \\sum_{i: \\mu_i < \\mu^\\star} (\\mu^\\star - \\mu_i)\u002FD(\\mu_i \\| \\mu^\\star)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\">lim\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathrm\" style=\"margin-right:0.0778em;\">inf\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">→\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">∞\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1858em;vertical-align:-0.4358em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\">∑\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.162em;\">\u003Cspan style=\"top:-2.4003em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">:\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3281em;\">\u003Cspan style=\"top:-2.357em;margin-left:0em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.143em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\"><\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6183em;\">\u003Cspan style=\"top:-2.786em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mbin mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4358em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">∥\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", where ",[1897,2960],{"html":2961,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>D\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">D\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is the Kullback–Leibler divergence ",[1813,2964],{"n":2965},"56",[2967,2968],"blog-pullquote",{"text":2969},"The four primitives — calibrated probabilistic model, risk-aware objective, constraint set, exploration mechanism — appear in every domain reviewed below. The remainder of this paper is, in essence, a taxonomy of how those primitives are instantiated.",[2231,2971,1508],{"id":1507},[1806,2973,2974,2975,2275,2978,2981,2982,2985],{},"Model predictive control (MPC) ",[1813,2976],{"n":2977},"57",[1813,2979],{"n":2980},"58"," is the cross-domain workhorse for sequential constrained decision problems with a ",[1809,2983,2984],{},"finite look-ahead",". At each step, the agent solves",[1897,2987],{"html":2988,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmsub>\u003Cmi>u\u003C\u002Fmi>\u003Cmrow>\u003Cmi>t\u003C\u002Fmi>\u003Cmo>:\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>H\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunderover>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>k\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>t\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>H\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunderover>\u003Cmi>c\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>s\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>u\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmspace width=\"1em\"\u002F>\u003Cmtext>s.t.\u003C\u002Fmtext>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsub>\u003Cmi>s\u003C\u002Fmi>\u003Cmrow>\u003Cmi>k\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>f\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>s\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>u\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>w\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>s\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>u\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"script\">Z\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\min_{u_{t:t+H}} \\;\\sum_{k=t}^{t+H} c(s_k, u_k) \\quad \\text{s.t.}\\;\\; s_{k+1} = f(s_k, u_k, w_k),\\; (s_k, u_k) \\in \\mathcal{Z},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:3.1304em;vertical-align:-1.3021em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.4em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">u\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3448em;\">\u003Cspan style=\"top:-2.3567em;margin-left:0em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">:\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2028em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.842em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.8283em;\">\u003Cspan style=\"top:-1.8479em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-4.3em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3021em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">s.t.\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2083em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8778em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0794em;\">Z\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,2990,2991,2992,2995,2996,2999,3000,3002,3003,3006,3007,3010],{},"implements only ",[1897,2993],{"html":2994,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>u\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">u_t\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", observes the new state, and re-solves. The receding-horizon principle is the engineering counterpart of the Bellman recursion; it is the dominant deployment pattern in process control, autonomous driving ",[1813,2997],{"n":2998},"59",", energy-system unit commitment ",[1813,3001],{"n":2783},", and is increasingly used for inventory and ad-budget allocation under known short-horizon dynamics. Stochastic MPC ",[1813,3004],{"n":3005},"60"," and tube MPC ",[1813,3008],{"n":3009},"61"," add explicit uncertainty handling.",[2231,3012,1511],{"id":1510},[1806,3014,3015,3016,2275,3018,3020,3021,3023,3024,3027,3028,3030,3031,2275,3033,3036,3037,3040,3041,3044,3045,2275,3048,3051,3052,3055],{},"The online-convex-optimization (OCO) framework ",[1813,3017],{"n":2869},[1813,3019],{"n":2872}," provides regret guarantees for sequential decisions against arbitrary convex losses. Online gradient descent achieves ",[1897,3022],{"html":2938,"display":-1}," regret; online Newton step ",[1813,3025],{"n":3026},"62"," achieves ",[1897,3029],{"html":2942,"display":-1}," under exp-concavity; FTRL and OMD ",[1813,3032],{"n":2946},[1813,3034],{"n":3035},"63"," achieve ",[1897,3038],{"html":3039,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>O\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsqrt>\u003Cmrow>\u003Cmi>T\u003C\u002Fmi>\u003Cmi>log\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>d\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsqrt>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">O(\\sqrt{T \\log d})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.085em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.835em;\">\u003Cspan class=\"svg-align\" style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:0.833em;\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">lo\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.795em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.205em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," in ",[1897,3042],{"html":3043,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>d\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">d\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," dimensions on the simplex. Primal-dual methods of Balseiro, Lu, and Mirrokni ",[1813,3046],{"n":3047},"64",[1813,3049],{"n":3050},"65"," develop dual mirror descent for online allocation with simultaneous regret and constraint-violation guarantees, directly relevant to budget-pacing problems in advertising. Gordon, Greenwald, and Marks ",[1813,3053],{"n":3054},"66"," give the no-regret-to-correlated-equilibrium link.",[1829,3057],{},[1832,3059,1514],{"id":1513},[2231,3061,1518],{"id":1517},[1806,3063,3064,3065,3067,3068,3071,3072,3074,3075,3078,3079,3082,3083,3086,3087,3090,3091,3094,3095,2275,3098,3101,3102,3105,3106,2444],{},"The intellectual genealogy of risk-aware optimization runs from Markowitz ",[1813,3066],{"n":2149}," through Tobin's separation theorem ",[1813,3069],{"n":3070},"67"," to Sharpe's CAPM ",[1813,3073],{"n":2539},", Lintner ",[1813,3076],{"n":3077},"68",", and Mossin ",[1813,3080],{"n":3081},"69",". The mean-variance frontier, the tangency portfolio, and the equilibrium pricing relation ",[1897,3084],{"html":3085,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmsub>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>β\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmsub>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>M\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathbb{E}[r_i] - r_f = \\beta_i (\\mathbb{E}[r_M] - r_f)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7167em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," are the foundation of every modern asset-allocation system. Empirical critiques of the variance-as-risk premise (Mandelbrot ",[1813,3088],{"n":3089},"70",", Fama ",[1813,3092],{"n":3093},"71",") and of the constant-beta CAPM (Fama and French ",[1813,3096],{"n":3097},"72",[1813,3099],{"n":3100},"73",") drive subsequent extensions: APT ",[1813,3103],{"n":3104},"74",", multi-factor models, conditional CAPM, and the q-factor model of Hou, Xue, and Zhang ",[1813,3107],{"n":3108},"75",[2231,3110,1521],{"id":1520},[1806,3112,3113,3114,3116],{},"The Rockafellar–Uryasev linearization ",[1813,3115],{"n":2570}," enabled large-scale portfolio optimization with downside-risk objectives:",[1897,3118],{"html":3119,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>w\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>η\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>α\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmi>N\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmunderover>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmi>N\u003C\u002Fmi>\u003C\u002Fmunderover>\u003Cmsub>\u003Cmi>z\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmspace width=\"1em\"\u002F>\u003Cmtext>s.t.\u003C\u002Fmtext>\u003Cmspace width=\"1em\"\u002F>\u003Cmsub>\u003Cmi>z\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>≥\u003C\u002Fmo>\u003Cmsub>\u003Cmi>L\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>w\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>η\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsub>\u003Cmi>z\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>r\u003C\u002Fmi>\u003Cmo>ˉ\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmn mathvariant=\"bold\">1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\min_w \\; \\eta + \\frac{1}{(1-\\alpha)N}\\sum_{n=1}^{N} z_n \\quad \\text{s.t.}\\quad z_n \\ge L_n(w) - \\eta,\\; z_n \\ge 0,\\; \\mu^\\top w \\ge \\bar r,\\; \\mathbf{1}^\\top w = 1,\\; w \\ge 0,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.3679em;vertical-align:-0.7em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.4em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:3.0954em;vertical-align:-1.2671em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3214em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.936em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.8283em;\">\u003Cspan style=\"top:-1.8829em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-4.3em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.2671em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.044em;\">z\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">s.t.\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.044em;\">z\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8304em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.044em;\">z\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0935em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0935em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5678em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1944em;\">\u003Cspan class=\"mord\">ˉ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbf\">1\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,3121,2611,3122,3125,3126,2444],{},[1897,3123],{"html":3124,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>L\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>w\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmsubsup>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsubsup>\u003Cmi>w\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L_n(w) = -r_n^\\top w\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0961em;vertical-align:-0.247em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">−\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-2.453em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.247em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". The resulting LP scales to thousands of assets and millions of scenarios. CVaR is the standard regulatory measure under the Basel III Fundamental Review of the Trading Book and is the internal economic-capital measure for most large insurers under Solvency II ",[1813,3127],{"n":2553},[2231,3129,1524],{"id":1523},[1806,3131,3132,3133,3135,3136,3139,3140,3143,3144,3147,3148,2275,3151,3154,3155,3158],{},"Ben-Tal and Nemirovski ",[1813,3134],{"n":2660},", Goldfarb and Iyengar ",[1813,3137],{"n":3138},"76",", and Tütüncü and Koenig ",[1813,3141],{"n":3142},"77"," develop robust mean-variance and robust factor-model portfolios in which the moment estimates ",[1897,3145],{"html":3146,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>μ\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi mathvariant=\"normal\">Σ\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(\\hat\\mu, \\hat\\Sigma)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1968em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2222em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9468em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">Σ\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2523em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.25em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," are themselves treated as uncertain. The robust-portfolio result is that introducing modest ambiguity uniformly improves out-of-sample Sharpe ratio relative to the plug-in MV portfolio, formalizing the practitioner intuition that \"shrinkage works.\" Ledoit and Wolf's covariance shrinkage ",[1813,3149],{"n":3150},"78",[1813,3152],{"n":3153},"79"," is the closely related Bayesian\u002FStein answer; Bayesian portfolio choice in the Black–Litterman tradition ",[1813,3156],{"n":3157},"80"," integrates investor views with market equilibrium.",[2231,3160,1527],{"id":1526},[1806,3162,3163,3164,3166,3167,3169,3170,3173,3174,3177],{},"The execution problem — sell ",[1897,3165],{"html":1912,"display":-1}," shares over ",[1897,3168],{"html":1982,"display":-1}," time steps minimizing expected cost plus a multiple of cost variance — is solved analytically by Almgren and Chriss ",[1813,3171],{"n":3172},"81",", producing the ",[1809,3175,3176],{},"efficient frontier of execution",". The optimal trajectory satisfies",[1897,3179],{"html":3180,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmfrac>\u003Cmrow>\u003Cmn>2\u003C\u002Fmn>\u003Cmi>sinh\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>κ\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>sinh\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>κ\u003C\u002Fmi>\u003Cmi>T\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmi>cosh\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmi>κ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>T\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmfrac>\u003C\u002Fmstyle>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>X\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n_k \\;=\\; \\frac{2 \\sinh(\\kappa T \u002F 2)}{\\sinh(\\kappa T)} \\cosh\\!\\left(\\kappa (T - (k - \\tfrac{1}{2}))\\right) X,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:2.363em;vertical-align:-0.936em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.427em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mop\">sinh\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">κ\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sinh\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">κ\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F2\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.936em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">cosh\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size1\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">κ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">T\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">))\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size1\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,3182,2816,3183,3186,3187,3190,3191,3194,3195,3197,3198,3201,3202,3205,3206,3209],{},[1897,3184],{"html":3185,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>κ\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmsqrt>\u003Cmrow>\u003Cmi>λ\u003C\u002Fmi>\u003Cmsup>\u003Cmi>σ\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmi>η\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsqrt>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\kappa = \\sqrt{\\lambda \\sigma^2 \u002F \\eta}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">κ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.24em;vertical-align:-0.305em;\">\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.935em;\">\u003Cspan class=\"svg-align\" style=\"top:-3.2em;\">\u003Cspan class=\"pstrut\" style=\"height:3.2em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:1em;\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7401em;\">\u003Cspan style=\"top:-2.989em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.895em;\">\u003Cspan class=\"pstrut\" style=\"height:3.2em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:1.02em;height:1.28em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.28em\" viewBox=\"0 0 400000 1296\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 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xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>σ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\sigma\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", temporary impact ",[1897,3192],{"html":3193,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>η\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\eta\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", and risk aversion ",[1897,3196],{"html":2839,"display":-1},". Subsequent work — Obizhaeva and Wang ",[1813,3199],{"n":3200},"82",", Gatheral ",[1813,3203],{"n":3204},"83",", Cartea and Jaimungal ",[1813,3207],{"n":3208},"84"," — refines impact dynamics, introduces transient impact, and integrates with stochastic order books.",[2231,3211,1530],{"id":1529},[1806,3213,3214,3215,2275,3218,3221,3222,3225,3226,3229,3230,3233,3234,3237,3238,3241,3242,2275,3245,3248,3249,3252],{},"Black, Scholes, and Merton ",[1813,3216],{"n":3217},"85",[1813,3219],{"n":3220},"86"," gave the parabolic PDE characterization of European option prices under continuous hedging, and the Merton consumption-investment problem ",[1813,3223],{"n":3224},"87"," introduced dynamic-programming reasoning to portfolio choice. Cont and Tankov ",[1813,3227],{"n":3228},"88"," survey the jump-process extensions; Glasserman ",[1813,3231],{"n":3232},"89"," develops the Monte Carlo machinery for derivative pricing under non-trivial dynamics. Cartea, Jaimungal, and Penalva ",[1813,3235],{"n":3236},"90"," integrate stochastic control with limit-order-book microstructure to give the modern algorithmic-trading reference. Local-volatility ",[1813,3239],{"n":3240},"91"," and stochastic-volatility models ",[1813,3243],{"n":3244},"92",[1813,3246],{"n":3247},"93"," capture the volatility smile; Carr and Wu ",[1813,3250],{"n":3251},"94"," survey the variance-swap literature.",[2231,3254,1533],{"id":1532},[1806,3256,3257,3258,3261,3262,3265],{},"Avellaneda and Lee ",[1813,3259],{"n":3260},"95"," formalize the cointegration-based statistical-arbitrage strategy as an Ornstein–Uhlenbeck mean-reversion problem; Guéant, Lehalle, and Fernandez-Tapia ",[1813,3263],{"n":3264},"96"," solve the market-making problem under inventory risk with closed-form bid–ask quotes. Both are stochastic-control problems whose solution structure — quotes that depend on inventory — is the natural cousin of the budget-allocation problem in marketplace ads, where utilization plays the role of inventory.",[2231,3267,1536],{"id":1535},[1806,3269,3270,3271,3274,3275,3278,3279,3282,3283,2275,3286,3289,3290,3293],{},"Merton's structural model ",[1813,3272],{"n":3273},"97"," and the reduced-form intensity-based models of Duffie and Singleton ",[1813,3276],{"n":3277},"98"," are the two foundational families. Brigo, Morini, and Pallavicini ",[1813,3280],{"n":3281},"99"," cover counterparty-risk and CVA modeling. Tail-risk dependence is modeled via copulas ",[1813,3284],{"n":3285},"100",[1813,3287],{"n":3288},"101","; Embrechts, McNeil, and Straumann ",[1813,3291],{"n":3292},"102"," document the dangers of linear-correlation thinking under heavy tails.",[2231,3295,1539],{"id":1538},[1806,3297,3298,3299,3302,3303,3306,3307,2444],{},"Svensson ",[1813,3300],{"n":3301},"103"," develops the linear-quadratic regulator interpretation of monetary policy under inflation targeting, with explicit Bellman recursions. The robust-control extensions of Hansen and Sargent ",[1813,3304],{"n":3305},"104"," introduce model uncertainty into macroeconomic policy rules. The conceptual link to e-commerce decision intelligence is the design of feedback policies under model misspecification — exactly the problem DataGlass solves at the campaign level ",[1813,3308],{"n":1815},[1829,3310],{},[1832,3312,1542],{"id":1541},[2231,3314,1546],{"id":1545},[1806,3316,3317,3318,2275,3321,3324],{},"The newsvendor ",[1813,3319],{"n":3320},"105",[1813,3322],{"n":3323},"106"," is the canonical single-period, risk-neutral decision under demand uncertainty:",[1897,3326],{"html":3327,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>q\u003C\u002Fmi>\u003Cmo lspace=\"0em\" 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style=\"height:0.836em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size3\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,3329,2816,3330,3333,3334,3337,3338,3341,3342,3345,3346,3349,3350,3353,3354,3357],{},[1897,3331],{"html":3332,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>u\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>o\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">c_u, c_o\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">u\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">o\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," are per-unit underage and overage costs and ",[1897,3335],{"html":3336,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>F\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">F\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">F\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is the demand CDF. Its risk-aware extensions — CVaR newsvendor ",[1813,3339],{"n":3340},"107",", distributionally robust newsvendor ",[1813,3343],{"n":3344},"108",", data-driven newsvendor with the SAA bound of Levi, Roundy, and Shmoys ",[1813,3347],{"n":3348},"109"," — collectively form the simplest non-trivial laboratory in which the four primitives of Section 3 can be studied in closed form. Ban and Rudin ",[1813,3351],{"n":3352},"110"," give a contextual newsvendor with covariate-dependent demand and prove ",[1897,3355],{"html":3356,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>O\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsup>\u003Cmi>n\u003C\u002Fmi>\u003Cmrow>\u003Cmo>−\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">O(n^{-1\u002F2})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.138em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.888em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u002F2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," rates.",[2231,3359,1549],{"id":1548},[1806,3361,3362,3363,3366,3367,3370,3371,3374,3375,3378,3379,2444],{},"Scarf ",[1813,3364],{"n":3365},"111"," proves the optimality of ",[1897,3368],{"html":3369,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>S\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(s, S)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," policies for inventory under fixed ordering costs; Clark and Scarf ",[1813,3372],{"n":3373},"112"," extend to multi-echelon systems; Federgruen and Zipkin ",[1813,3376],{"n":3377},"113"," establish base-stock optimality under the average-cost criterion. The unifying result is that the inventory problem is a constrained MDP whose optimal policy is characterized by a single threshold per state, and is therefore amenable to ADP at scale ",[1813,3380],{"n":2783},[2231,3382,1552],{"id":1551},[1806,3384,3385,3386,3389,3390,3393,3394,3397,3398,3401,3402,3405],{},"Talluri and van Ryzin ",[1813,3387],{"n":3388},"114"," is the reference for revenue management under capacity constraints. The dynamic-pricing literature of Gallego and van Ryzin ",[1813,3391],{"n":3392},"115",", with the data-driven extensions of Besbes and Zeevi ",[1813,3395],{"n":3396},"116"," and Ferreira, Lee, and Simchi-Levi ",[1813,3399],{"n":3400},"117",", is the conceptual link between operations and the e-commerce-pricing literature reviewed in Section 11.2. Cohen, Perakis, and Pindyck ",[1813,3403],{"n":3404},"118"," formalize promotion optimization as a mixed-integer program with stochastic demand.",[2231,3407,1555],{"id":1554},[1806,3409,3410,3411,3414,3415,3418],{},"Cachon's survey ",[1813,3412],{"n":3413},"119"," is the canonical operations-research reference for coordinating contracts (buyback, revenue-sharing, quantity-flexibility). The bullwhip-effect analysis of Lee, Padmanabhan, and Whang ",[1813,3416],{"n":3417},"120"," is structurally relevant to ad-spend cascading on cross-channel attribution, where small upstream shocks amplify downstream.",[2231,3420,1558],{"id":1557},[1806,3422,3423,3424,3427,3428,3431,3432,3435,3436,3439],{},"The displacement-cost framework of Talluri and van Ryzin ",[1813,3425],{"n":3426},"121"," for airline network revenue management — under which a Lagrangian shadow price is computed for each capacity-constrained leg — is the closest operations-research analogue of the multi-campaign budget shadow price ",[1897,3429],{"html":3430,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mu^{\\star}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8831em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," in DataGlass [1, Section V]. Adelman ",[1813,3433],{"n":3434},"122"," develops affine-policy approximations; Topaloglu ",[1813,3437],{"n":3438},"123"," applies the framework to stochastic resource allocation.",[1829,3441],{},[1832,3443,1561],{"id":1560},[1806,3445,3446,3447,3450],{},"The actuarial tradition is the original home of tail-risk modeling. The Cramér–Lundberg model ",[1813,3448],{"n":3449},"124"," of an insurance surplus process",[1897,3452],{"html":3453,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>U\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>u\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmunderover>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmsub>\u003Cmi>N\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmunderover>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">U_t \\;=\\; u + ct - \\sum_{i=1}^{N_t} X_i,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">U\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6984em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:3.1171em;vertical-align:-1.2777em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.8394em;\">\u003Cspan style=\"top:-1.8723em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-4.3111em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2963em;\">\u003Cspan style=\"top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.143em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.2777em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,3455,2611,3456,3459,3460,3463,3464,2836,3467,3470,3471,2275,3474,3477,3478,3481,3482,3484,3485,3487,3488,3491,3492,2444],{},[1897,3457],{"html":3458,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>N\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">N_t\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," a Poisson claim-arrival process, gives the foundational ruin probability ",[1897,3461],{"html":3462,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ψ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>u\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>Pr\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmrow>\u003Cmi>inf\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>t\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmsub>\u003Cmi>U\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo><\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\psi(u) = \\Pr\\!\\left(\\inf_{t \\ge 0} U_t < 0\\right)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">ψ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">Pr\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">(\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop\">in\u003Cspan style=\"margin-right:0.0778em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">≥\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2452em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">U\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\"><\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," and the Lundberg upper bound ",[1897,3465],{"html":3466,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ψ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>u\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>≤\u003C\u002Fmo>\u003Cmsup>\u003Cmi>e\u003C\u002Fmi>\u003Cmrow>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>R\u003C\u002Fmi>\u003Cmi>u\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\psi(u) \\le e^{-Ru}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">ψ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≤\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8413em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">e\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8413em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">u\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1897,3468],{"html":3469,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>R\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">R\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," the adjustment coefficient. Extreme Value Theory ",[1813,3472],{"n":3473},"125",[1813,3475],{"n":3476},"126"," characterizes the limiting distribution of normalized maxima as a Generalized Extreme Value distribution; Pickands' theorem ",[1813,3479],{"n":3480},"127"," gives the Generalized Pareto for threshold exceedances. McNeil, Frey, and Embrechts ",[1813,3483],{"n":2553}," is the modern reference. Distortion risk measures ",[1813,3486],{"n":2618}," formalize the actuarial premium principle as a coherent risk functional. Solvency-II internal-model regulation ",[1813,3489],{"n":3490},"128"," requires insurers to quantify a one-year 99.5% VaR — a regulatory analogue of the Basel III trading-book CVaR requirement, with similar moral-hazard and capital-arbitrage concerns ",[1813,3493],{"n":2553},[1806,3495,3496,3497,3500],{},"The relevance to e-commerce is direct in two places: the heavy-tailed nature of conversion-rate outliers under viral demand ",[1813,3498],{"n":3499},"129",", and the modeling of return-rate tails for true-ROAS adjustment [1, Section III].",[1829,3502],{},[1832,3504,1564],{"id":1563},[1806,3506,3507,3508,3510,3511,3513,3514,3517,3518,3521,3522,3525,3526,2444],{},"Powell ",[1813,3509],{"n":2783}," is the canonical reference for energy-system stochastic optimization. The unit-commitment problem under wind and solar uncertainty — schedule generators over ",[1897,3512],{"html":1982,"display":-1}," hours minimizing expected cost subject to ramp, capacity, and reserve constraints — is the canonical large-scale application of stochastic programming with recourse. Two-stage formulations ",[1813,3515],{"n":3516},"130",", multi-stage stochastic dual dynamic programming ",[1813,3519],{"n":3520},"131",", and stochastic MPC with chance constraints ",[1813,3523],{"n":3524},"132"," dominate the operational literature. Storage and demand-response add a state-dependent constraint structure that closely resembles the inventory-dilution mechanism in DataGlass [1, Section IV.C]. Distributionally robust energy planning under climate-scenario ambiguity is an active research frontier ",[1813,3527],{"n":3528},"133",[1829,3530],{},[1832,3532,1567],{"id":1566},[1806,3534,3535,3536,3539,3540,3543,3544,3547,3548,3551,3552,3555,3556,2275,3559,3562],{},"In healthcare, Ayer, Alagoz, and Stout ",[1813,3537],{"n":3538},"134"," formulate the breast-cancer screening decision as a partially observed MDP. Sutton et al. ",[1813,3541],{"n":3542},"135"," survey RL in clinical decision support; Komorowski et al. ",[1813,3545],{"n":3546},"136"," use off-policy evaluation to derive sepsis-treatment policies on ICU data. Bertsimas et al. ",[1813,3549],{"n":3550},"137"," develop the ",[1809,3553,3554],{},"decision rule"," approach to clinical optimization, a constrained DRO formulation. The off-policy evaluation problem ",[1813,3557],{"n":3558},"138",[1813,3560],{"n":3561},"139"," is the methodological bottleneck shared with e-commerce recommendation: the inability to A\u002FB-test arbitrary policies in production puts a disproportionate weight on causal-inference machinery.",[1829,3564],{},[1832,3566,1570],{"id":1569},[1806,3568,3569,3570],{},"A complete meta-review must include causal inference, because every production decision system eventually faces the question ",[1809,3571,3572],{},"\"is the policy actually causing the lift, or are we observing a confound?\"",[2231,3574,1574],{"id":1573},[1806,3576,3577,3578,3581,3582,3585,3586,3589,3590,3592,3593,3596,3597,3599,3600,3603,3604,3607,3608,3611],{},"Neyman ",[1813,3579],{"n":3580},"140",", Rubin ",[1813,3583],{"n":3584},"141",", Imbens and Rubin ",[1813,3587],{"n":3588},"142"," develop the potential-outcomes language: the causal effect of treatment ",[1897,3591],{"html":1982,"display":-1}," on outcome ",[1897,3594],{"html":3595,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>Y\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">Y\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," for unit ",[1897,3598],{"html":2101,"display":-1}," is ",[1897,3601],{"html":3602,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>τ\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>Y\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>Y\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tau_i = Y_i(1) - Y_i(0)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.1132em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", with the ",[1809,3605,3606],{},"fundamental problem of causal inference"," being that only one potential outcome is observed per unit. Pearl ",[1813,3609],{"n":3610},"143"," develops the do-calculus and graphical-model formulation that complements the potential-outcomes view.",[2231,3613,1577],{"id":1576},[1806,3615,3616,3617,3620,3621,3624,3625,3628,3629,3632,3633,3636],{},"Fisher ",[1813,3618],{"n":3619},"144"," introduced randomization inference; modern extensions include adaptive ",[1813,3622],{"n":3623},"145"," and contextual ",[1813,3626],{"n":3627},"146"," designs. Athey and Imbens ",[1813,3630],{"n":3631},"147"," survey field experiments in economics. Gordon, Zettelmeyer, Bhargava, and Chapsky ",[1813,3634],{"n":3635},"148"," establish the gold standard for advertising measurement: only randomized experiments produce reliably unbiased lift estimates, with observational and quasi-experimental approaches typically biased by 30–100% in advertising contexts.",[2231,3638,1580],{"id":1579},[1806,3640,3641,3642,3645,3646,3649,3650,2275,3653,3656,3657,3660,3661,3664,3665,2275,3668,3671],{},"Difference-in-differences ",[1813,3643],{"n":3644},"149"," and its modern extensions ",[1813,3647],{"n":3648},"150"," are the standard for staggered policy adoption; regression discontinuity ",[1813,3651],{"n":3652},"151",[1813,3654],{"n":3655},"152"," exploits eligibility thresholds; instrumental variables ",[1813,3658],{"n":3659},"153"," with the LATE interpretation of Imbens and Angrist ",[1813,3662],{"n":3663},"154"," handle endogenous treatment; synthetic controls of Abadie, Diamond, and Hainmueller ",[1813,3666],{"n":3667},"155",[1813,3669],{"n":3670},"156"," construct counterfactual time series from donor pools.",[2231,3673,1583],{"id":1582},[1806,3675,3676,3677,3680,3681,3684,3685,3688,3689,3692,3693,3696,3697,3700,3701,3704,3705,3708,3709,3712,3713,3716,3717,3551,3720,3723],{},"Athey and Imbens ",[1813,3678],{"n":3679},"157"," introduce ",[1809,3682,3683],{},"causal trees","; Wager and Athey ",[1813,3686],{"n":3687},"158"," extend to ",[1809,3690,3691],{},"causal forests"," with asymptotic normality. Chernozhukov et al. ",[1813,3694],{"n":3695},"159"," develop ",[1809,3698,3699],{},"double\u002Fdebiased machine learning"," (DML) with the Neyman-orthogonal score ",[1897,3702],{"html":3703,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ψ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>W\u003C\u002Fmi>\u003Cmo separator=\"true\">;\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>η\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>Y\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">ℓ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>D\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>m\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>D\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>m\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\psi(W; \\theta, \\eta) = (Y - \\ell(X) - \\theta(D - m(X)))(D - m(X))\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">ψ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">W\u003C\u002Fspan>\u003Cspan class=\"mpunct\">;\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">ℓ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)))\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">))\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", achieving ",[1897,3706],{"html":3707,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsqrt>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsqrt>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\sqrt{n}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.04em;vertical-align:-0.2397em;\">\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8003em;\">\u003Cspan class=\"svg-align\" style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:0.833em;\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.7603em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2397em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","-consistent treatment-effect estimation under nuisance-function rates as slow as ",[1897,3710],{"html":3711,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>n\u003C\u002Fmi>\u003Cmrow>\u003Cmn>1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmn>4\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n^{1\u002F4}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.888em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.888em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u002F4\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". Künzel et al. ",[1813,3714],{"n":3715},"160"," introduce meta-learners (S-, T-, X-, R-learners) for heterogeneous treatment effects. Nie and Wager ",[1813,3718],{"n":3719},"161",[1809,3721,3722],{},"R-learner"," with quasi-oracle efficiency.",[2231,3725,1586],{"id":1585},[1806,3727,3728,3729,3732,3733,3736,3737,3740],{},"Rosenbaum ",[1813,3730],{"n":3731},"162"," gives the bounding-parameter approach to sensitivity analysis. Oster ",[1813,3734],{"n":3735},"163"," proves coefficient-stability sensitivity under proportional selection on observables and unobservables; Cinelli and Hazlett ",[1813,3738],{"n":3739},"164"," extend to omitted-variable bias with explicit sensitivity statistics. The DataGlass system uses Oster-style sensitivity together with first-differencing and randomized perturbation [1, Section VIII].",[2231,3742,1589],{"id":1588},[1806,3744,3745,3746,3749,3750,3753,3754,3756,3757,3760,3761,3764,3765,3768,3769,3772],{},"Inverse propensity scoring ",[1813,3747],{"n":3748},"165",", doubly robust estimators of Bang and Robins ",[1813,3751],{"n":3752},"166"," and Dudík, Erhan, Langford, and Li ",[1813,3755],{"n":3561},", targeted maximum likelihood of van der Laan and Rose ",[1813,3758],{"n":3759},"167",", and the per-decision importance weighting of Precup, Sutton, and Singh ",[1813,3762],{"n":3763},"168"," are the standard machinery. Swaminathan and Joachims ",[1813,3766],{"n":3767},"169"," develop counterfactual risk minimization for batch contextual bandits. The methodological lesson — ",[1809,3770,3771],{},"off-policy evaluation has variance proportional to importance-ratio range"," — explains why exploration design (Section 3.5) is structurally tied to evaluation feasibility.",[1829,3774],{},[1832,3776,1592],{"id":1591},[2231,3778,1596],{"id":1595},[1806,3780,3781,3782,3784,3785,3787,3788,3791,3792,3794,3795,3027,3797,3799,3800,2275,3802,3804,3805,3808],{},"Zinkevich ",[1813,3783],{"n":2869},", Hazan, Agarwal, and Kale ",[1813,3786],{"n":2872},", and Cesa-Bianchi and Lugosi ",[1813,3789],{"n":3790},"170"," develop OCO with adversarial-loss regret guarantees. Online gradient descent achieves ",[1897,3793],{"html":2938,"display":-1},"; online Newton step ",[1813,3796],{"n":3026},[1897,3798],{"html":2942,"display":-1}," under exp-concavity. Mirror descent ",[1813,3801],{"n":2946},[1813,3803],{"n":2949}," generalizes to non-Euclidean geometries via Bregman divergences and is the umbrella under which exponentiated-gradient updates ",[1813,3806],{"n":3807},"171"," for the simplex (relevant to multi-asset and multi-campaign allocations) sit.",[2231,3810,1599],{"id":1598},[1806,3812,3813,3814,3817,3818,3821,3822,3824,3825,3827,3828,3831,3832,3835,3836,3839],{},"Hannan consistency ",[1813,3815],{"n":3816},"172",", the polynomial-weights algorithm of Littlestone and Warmuth ",[1813,3819],{"n":3820},"173",", and the Cesa-Bianchi–Lugosi prediction-with-expert-advice machinery ",[1813,3823],{"n":3790}," establish that no-regret play converges to coarse correlated equilibrium ",[1813,3826],{"n":3054},". Roughgarden ",[1813,3829],{"n":3830},"174"," develops the ",[1809,3833,3834],{},"price of anarchy"," analysis; Foster and Vohra ",[1813,3837],{"n":3838},"175"," provide calibration via no-regret learning.",[2231,3841,1602],{"id":1601},[1806,3843,3844,3845,3848,3849,3852,3853,3856,3857,3860],{},"Vovk, Gammerman, and Shafer ",[1813,3846],{"n":3847},"176",", with the modern split-conformal extensions of Lei et al. ",[1813,3850],{"n":3851},"177"," and adaptive variants of Romano, Patterson, and Candès ",[1813,3854],{"n":3855},"178",", produce ",[1809,3858,3859],{},"distribution-free"," prediction sets with finite-sample coverage guarantees. The framework is the natural complement to Bayesian uncertainty: where Bayesian credibility requires a correctly specified prior, conformal coverage requires only exchangeability. Both are relevant to e-commerce decision systems where calibrated uncertainty intervals on demand or response curves drive the downstream optimizer.",[2231,3862,1605],{"id":1604},[1806,3864,3865,3866,3869,3870,3873,3874,3696,3877,3880,3881,3696,3884,3887],{},"Quiñonero-Candela, Sugiyama, Schwaighofer, and Lawrence ",[1813,3867],{"n":3868},"179"," and Sugiyama and Kawanabe ",[1813,3871],{"n":3872},"180"," are foundational for covariate-shift adaptation. Arjovsky, Bottou, Gulrajani, and Lopez-Paz ",[1813,3875],{"n":3876},"181",[1809,3878,3879],{},"invariant risk minimization","; Sagawa et al. ",[1813,3882],{"n":3883},"182",[1809,3885,3886],{},"group DRO",". The connection to operational decision systems is direct: the platform-side learning algorithm in marketplace ads continuously shifts the response distribution, so static models go stale at characteristic time-scales documented in [1, Section VII].",[1829,3889],{},[1832,3891,1608],{"id":1607},[1806,3893,3894],{},"The e-commerce decision-intelligence stack is composed of approximately seven interacting subsystems. We summarize each.",[2231,3896,1612],{"id":1611},[1806,3898,3899,3900,3903,3904,3907,3908,2275,3911,2275,3914,3917,3918,3921,3922,3925,3926,3929,3930,3933,3934,3937,3938,3940,3941,3944,3945,3948,3949,3952,3953,2444],{},"The historical workhorse is the Box–Jenkins ARIMA family ",[1813,3901],{"n":3902},"183","; the modern industrial benchmark is exponential smoothing in the unified state-space formulation of Hyndman and Athanasopoulos ",[1813,3905],{"n":3906},"184",". The M-competitions ",[1813,3909],{"n":3910},"185",[1813,3912],{"n":3913},"186",[1813,3915],{"n":3916},"187"," document a steady decline in the relative performance of pure statistical methods and the rise of hybrid global models — DeepAR ",[1813,3919],{"n":3920},"188",", N-BEATS ",[1813,3923],{"n":3924},"189",", Temporal Fusion Transformer ",[1813,3927],{"n":3928},"190",", NHITS ",[1813,3931],{"n":3932},"191",", TimesNet ",[1813,3935],{"n":3936},"192"," — culminating in the M5 competition ",[1813,3939],{"n":3916},", where global gradient-boosted models with hierarchical reconciliation ",[1813,3942],{"n":3943},"193"," won. Smyl ",[1813,3946],{"n":3947},"194"," gives the hybrid ES-RNN that won M4. The methodological lesson is that ",[1809,3950,3951],{},"cross-series information sharing",", not model class per se, drives the gain. Quantile-regression objectives produce calibrated prediction intervals essential for downstream inventory and pricing decisions ",[1813,3954],{"n":3352},[2231,3956,1615],{"id":1614},[1806,3958,3959,3960,3963,3964,3967,3968,3971,3972,3974,3975,3978,3979,3982,3983,3986],{},"Demand modeling at the SKU level uses constant-elasticity, multinomial logit ",[1813,3961],{"n":3962},"195",", and mixed-logit specifications ",[1813,3965],{"n":3966},"196","; Berry, Levinsohn, and Pakes (BLP) ",[1813,3969],{"n":3970},"197"," address price endogeneity via instrumental variables. Modern e-commerce pricing systems combine these structural models with scalable contextual bandits ",[1813,3973],{"n":3627}," and Thompson-sampled exploration ",[1813,3976],{"n":3977},"198",". The Lerner condition ",[1897,3980],{"html":3981,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsup>\u003Cmi>p\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmsup>\u003Cmi>p\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmi>ε\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(p^{\\star} - c)\u002Fp^{\\star} = 1\u002F|\\varepsilon|\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u002F∣\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ε\u003C\u002Fspan>\u003Cspan class=\"mord\">∣\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," recovers the closed-form optimum under constant elasticity. The DataGlass internal note on elasticity modeling and bundle pricing ",[1813,3984],{"n":3985},"199"," gives a worked treatment for SKU-level pricing under cross-elasticity.",[2231,3988,1618],{"id":1617},[1806,3990,3991,3992,3995,3996,3999,4000,2275,4003,4006,4007,4010,4011,4013,4014,4017,4018,2275,4021,4024,4025,4028],{},"Collaborative filtering ",[1813,3993],{"n":3994},"200",", matrix factorization ",[1813,3997],{"n":3998},"201",", and deep recommendation models ",[1813,4001],{"n":4002},"202",[1813,4004],{"n":4005},"203"," are the workhorses. Counterfactual evaluation via inverse propensity scoring ",[1813,4008],{"n":4009},"204"," and doubly robust estimators ",[1813,4012],{"n":3561}," is the bridge from offline data to online decision; bandit-based learning-to-rank ",[1813,4015],{"n":4016},"205"," is the closed-loop variant. Attention-based and Transformer recommendation models ",[1813,4019],{"n":4020},"206",[1813,4022],{"n":4023},"207"," are now standard at scale. The methodological lesson — ",[1809,4026,4027],{},"recommendation is off-policy reinforcement learning under partial logging"," — is widely accepted but operationally constraining.",[2231,4030,1621],{"id":1620},[1806,4032,4033,4034,4036,4037,2275,4040,4043,4044,4047,4048,2444],{},"Cohen et al. ",[1813,4035],{"n":3404}," formalize multi-product promotion as a MIP with stochastic demand. Assortment optimization under MNL choice ",[1813,4038],{"n":4039},"208",[1813,4041],{"n":4042},"209"," gives the revenue-ordered optimal assortment for capacitated problems; Bernstein, Modaresi, and Sauré ",[1813,4045],{"n":4046},"210"," extend this to dynamic assortment with learning. The unifying language is ",[1809,4049,4050],{},"combinatorial optimization with side information and exploration",[2231,4052,1624],{"id":1623},[1806,4054,4055,4056,4059],{},"RTB is the closest finance-style problem in advertising, because the agent does bid directly. Cai et al. ",[1813,4057],{"n":4058},"211"," formulate RTB as an MDP with neural-network value approximation, deriving the optimal bid",[1897,4061],{"html":4062,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>b\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsubsup>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmi>V\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>s\u003C\u002Fmi>\u003Cmtext>win\u003C\u002Fmtext>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmsup>\u003Cmi>V\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>s\u003C\u002Fmi>\u003Cmtext>lose\u003C\u002Fmtext>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b_i^{\\star} \\;=\\; V^{\\star}(s_{\\text{win}}) - V^{\\star}(s_{\\text{lose}}),\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9857em;vertical-align:-0.247em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7387em;\">\u003Cspan style=\"top:-2.453em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.247em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7387em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3175em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord text mtight\">\u003Cspan class=\"mord mtight\">win\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7387em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord text mtight\">\u003Cspan class=\"mord mtight\">lose\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4064,4065,4066,4069,4070,4073,4074,4076,4077,4080,4081,4084,4085,4088,4089,4092],{},"with documented production lifts of 16.7% over baselines. Wu et al. ",[1813,4067],{"n":4068},"212"," add budget constraints via Lagrangian relaxation, giving ",[1897,4071],{"html":4072,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>v\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmi>λ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b_i = v_i \u002F \\lambda\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," for a budget multiplier ",[1897,4075],{"html":2839,"display":-1}," learned by DQN, achieving 23.4% click-improvement under strict budget compliance. Zhao et al. ",[1813,4078],{"n":4079},"213"," aggregate to hour-level MDPs for sponsored search, using twin Q-networks with replay refresh. He et al. ",[1813,4082],{"n":4083},"214"," propose hierarchical RL (HiBid). Wang et al. ",[1813,4086],{"n":4087},"215"," address ROI-constrained bidding via curriculum-guided Bayesian RL. Liu et al. ",[1813,4090],{"n":4091},"216"," provide a rigorous ablation. The literature is mature; its limitation, from the perspective of this review, is that it assumes the agent controls bids — an assumption that fails for marketplace platforms (Section 11.7).",[2231,4094,1627],{"id":1626},[1806,4096,4097,4098,4101,4102,4105,4106,4109,4110,4113,4114,4117,4118,4121,4122,4125],{},"Vickrey ",[1813,4099],{"n":4100},"217",", Myerson ",[1813,4103],{"n":4104},"218",", and the modern Athey and Segal ",[1813,4107],{"n":4108},"219"," deliver the foundational mechanism-design theory. Edelman, Ostrovsky, and Schwarz ",[1813,4111],{"n":4112},"220"," and Varian ",[1813,4115],{"n":4116},"221"," analyze generalized second-price auctions for sponsored search. Rawat ",[1813,4119],{"n":4120},"222"," documents algorithmic-collusion bid suppression in first-price auctions when bidders learn — relevant because most modern marketplaces have moved from second-price to first-price ",[1813,4123],{"n":4124},"223",", affecting the seller's effective response curve [1, Section VIII].",[2231,4127,1630],{"id":1629},[1806,4129,4130,4131,4133,4134,4137,4138,4141,4142,4145,4146,2275,4148,4150,4151,4153,4154,2275,4157,4160,4161,4163],{},"Modern marketplace platforms — Shopee, Lazada, TikTok Shop, Amazon Sponsored Products, Walmart Connect, Mercado Libre Ads — do not expose bid-level control. The seller sets only daily budget and target ROAS; the platform's auto-bidder mediates the auction. This is a structurally different problem: the action space is one-dimensional per campaign, the environment is opaque, and the platform actively penalizes high-frequency intervention through learning-phase mechanics. The DataGlass system ",[1813,4132],{"n":1815}," is the first end-to-end production system that addresses this problem class explicitly. Methodologically it is a synthesis of Hill-saturation response modeling ",[1813,4135],{"n":4136},"224",", Negative-Binomial overdispersion handling ",[1813,4139],{"n":4140},"225",", Beta-Binomial conversion modeling with utilization-dependent dilution ",[1813,4143],{"n":4144},"226",", constrained-portfolio optimization via shadow-price bisection ",[1813,4147],{"n":2149},[1813,4149],{"n":3047},", Thompson-sampled exploration ",[1813,4152],{"n":2890},", CUSUM changepoint detection ",[1813,4155],{"n":4156},"227",[1813,4158],{"n":4159},"228",", and randomized perturbation experiments ",[1813,4162],{"n":3635},". Section 12.11 returns to this system as the connecting case.",[1806,4165,4166,4167,2546,4171,4173],{},"The seller-pain quantification of the ",[2266,4168,4170],{"href":4169},"\u002Fmy\u002Fblog\u002Fbayesian-budget-allocation-for-marketplace-ads","companion research article",[1813,4172],{"n":1819}," documents the analytical mechanism by which manual heuristics — trial-and-error reallocation, rolling-mean extrapolation of reported ROAS, gut-feel reallocation — are systematically biased estimators of marginal contribution. The true profit-adjusted ROAS",[1897,4175],{"html":4176,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmi>R\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmtext>true\u003C\u002Fmtext>\u003C\u002Fmsubsup>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>=\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsubsup>\u003Cmi>R\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmtext>rep\u003C\u002Fmtext>\u003C\u002Fmsubsup>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmsub>\u003Cmi>m\u003C\u002Fmi>\u003Cmi>g\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>r\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>f\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>ρ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">R_i^{\\text{true}} \\;=\\; R_i^{\\text{rep}} \\cdot m_g \\cdot (1 - r) \\cdot (1 - f - \\rho)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0906em;vertical-align:-0.247em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8436em;\">\u003Cspan style=\"top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord text mtight\">\u003Cspan class=\"mord mtight\">true\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.247em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0592em;vertical-align:-0.2769em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7823em;\">\u003Cspan style=\"top:-2.4231em;margin-left:-0.0077em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.1809em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord text mtight\">\u003Cspan class=\"mord mtight\">rep\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2769em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7306em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ρ\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4178,4179],{},"differs from the reported dashboard figure by an order of magnitude in plausible parameterizations [2, Section 3.1].",[1829,4181],{},[1832,4183,1633],{"id":1632},[1806,4185,4186],{},"We now develop eleven worked cases, chosen so that each illustrates a distinct primitive of the Section 3 framework. Each case is presented in the same structure: problem statement, formulation, key result, and what generalizes to e-commerce.",[2231,4188,1637],{"id":1636},[1806,4190,4191,4194,4195,4198,4199,4202,4203,4206],{},[1824,4192,4193],{},"Problem."," Allocate wealth across ",[1897,4196],{"html":4197,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," risky assets with mean returns ",[1897,4200],{"html":4201,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>μ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mu\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," and covariance ",[1897,4204],{"html":4205,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">Σ\u003C\u002Fmi>\u003Cmo>≻\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\Sigma \\succ 0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">Σ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≻\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," to balance return and variance.",[1806,4208,4209,2546,4212,4215,4216,2275,4219,2444],{},[1824,4210,4211],{},"Formulation.",[1897,4213],{"html":4214,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>w\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi mathvariant=\"normal\">Σ\u003C\u002Fmi>\u003Cmi>w\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\min_w \\; w^\\top \\Sigma w\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9991em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">Σ\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," s.t. ",[1897,4217],{"html":4218,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>r\u003C\u002Fmi>\u003Cmo>ˉ\u003C\u002Fmo>\u003C\u002Fmover>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mu^\\top w = \\bar r\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0435em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5678em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5678em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1944em;\">\u003Cspan class=\"mord\">ˉ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1897,4220],{"html":4221,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmn mathvariant=\"bold\">1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathbf{1}^\\top w = 1\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8491em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbf\">1\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4223,4224,4227,4228,4231,4232,4235,4236,4239,4240,2444],{},[1824,4225,4226],{},"Key result."," Closed-form ",[1897,4229],{"html":4230,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>r\u003C\u002Fmi>\u003Cmo>ˉ\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmi mathvariant=\"normal\">Σ\u003C\u002Fmi>\u003Cmrow>\u003Cmo>−\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>α\u003C\u002Fmi>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>β\u003C\u002Fmi>\u003Cmn mathvariant=\"bold\">1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">w^{\\star}(\\bar r) = \\Sigma^{-1}(\\alpha \\mu + \\beta \\mathbf{1})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5678em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1944em;\">\u003Cspan class=\"mord\">ˉ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">Σ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"mord mathbf\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," for scalars ",[1897,4233],{"html":4234,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>α\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>β\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\alpha, \\beta\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," determined by the constraints. The locus ",[1897,4237],{"html":4238,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">{\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>σ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>r\u003C\u002Fmi>\u003Cmo>ˉ\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>r\u003C\u002Fmi>\u003Cmo>ˉ\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">}\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\{(\\sigma(\\bar r), \\bar r)\\}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">{(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5678em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1944em;\">\u003Cspan class=\"mord\">ˉ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5678em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1944em;\">\u003Cspan class=\"mord\">ˉ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)}\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is the Markowitz frontier; with a risk-free asset, the tangency portfolio gives the Capital Market Line ",[1897,4241],{"html":4242,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>r\u003C\u002Fmi>\u003Cmo>ˉ\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>f\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>σ\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmsup>\u003Cmrow>\u003Cmi mathvariant=\"normal\">S\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">R\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\bar r - r_f = \\sigma \\cdot \\mathrm{SR}^{\\star}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5678em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1944em;\">\u003Cspan class=\"mord\">ˉ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7167em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4445em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">σ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7619em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">SR\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7619em;\">\u003Cspan style=\"top:-3.1362em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4244,4245,4248,4249,4252,4253,4256],{},[1824,4246,4247],{},"Generalizes to e-commerce."," The dual interpretation — the Lagrange multiplier on the budget constraint as the marginal return per unit risk — is the direct ancestor of the equal-marginal-profit shadow-price condition ",[1897,4250],{"html":4251,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∂\u003C\u002Fmi>\u003Cmsub>\u003Cmi>π\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∂\u003C\u002Fmi>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\partial \\pi_i \u002F \\partial b_i = \\mu^{\\star}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"margin-right:0.0556em;\">∂\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"margin-right:0.0556em;\">∂\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8831em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," that DataGlass solves [1, Section V]. Empirical fragility — extreme sensitivity to ",[1897,4254],{"html":4255,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>μ\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\hat\\mu\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2222em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," — motivates the Bayesian and robust extensions used in production e-commerce systems, where the analogue concern is over-fit campaign-level response curves.",[2231,4258,1640],{"id":1639},[1806,4260,4261,4263,4264,4266],{},[1824,4262,4193],{}," Minimize ",[1897,4265],{"html":2053,"display":-1},"-CVaR of portfolio loss subject to a target expected return.",[1806,4268,4269,4271],{},[1824,4270,4211],{}," Using the Rockafellar–Uryasev linearization, the LP",[1897,4273],{"html":4274,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>w\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>η\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>z\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>η\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>α\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmi>N\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmunder>\u003Cmo>∑\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmsub>\u003Cmi>z\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmspace width=\"1em\"\u002F>\u003Cmtext>s.t.\u003C\u002Fmtext>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsub>\u003Cmi>z\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>≥\u003C\u002Fmo>\u003Cmsub>\u003Cmi>L\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>w\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>η\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsub>\u003Cmi>z\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>r\u003C\u002Fmi>\u003Cmo>ˉ\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsup>\u003Cmn mathvariant=\"bold\">1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">⊤\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi>w\u003C\u002Fmi>\u003Cmo>≥\u003C\u002Fmo>\u003Cmn>0.\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\min_{w, \\eta, z} \\; \\eta + \\frac{1}{(1-\\alpha)N}\\sum_n z_n \\quad \\text{s.t.}\\;\\; z_n \\ge L_n(w) - \\eta,\\; z_n \\ge 0,\\; \\mu^\\top w \\ge \\bar r,\\; \\mathbf{1}^\\top w = 1, \\; w \\ge 0.\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.504em;vertical-align:-0.8361em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.4em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.044em;\">z\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8361em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:2.5714em;vertical-align:-1.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3214em;\">\u003Cspan style=\"top:-2.314em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0037em;\">α\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.677em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.936em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.05em;\">\u003Cspan style=\"top:-1.9em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.25em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.044em;\">z\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">s.t.\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.044em;\">z\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8304em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.044em;\">z\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0935em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0935em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.5678em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1944em;\">\u003Cspan class=\"mord\">ˉ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbf\">1\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8991em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8389em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≥\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0.\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4276,4277,4279,4280,4283,4284,4287,4288,2275,4290,4292,4293,4295],{},[1824,4278,4226],{}," A linear program in ",[1897,4281],{"html":4282,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>w\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>η\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>z\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(w, \\eta, z)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.044em;\">z\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," that scales to ",[1897,4285],{"html":4286,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>N\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmn>10\u003C\u002Fmn>\u003Cmn>6\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">N = 10^6\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8141em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">6\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," scenarios on commodity hardware ",[1813,4289],{"n":2570},[1813,4291],{"n":2573},". The optimal ",[1897,4294],{"html":3193,"display":-1}," is itself the VaR.",[1806,4297,4298,4300,4301,4304],{},[1824,4299,4247],{}," Replacing CVaR over loss with CVaR over ",[1809,4302,4303],{},"negative profit"," yields a downside-risk-aware version of the DataGlass portfolio optimizer that naturally penalizes catastrophic-day campaign blow-ups; the formulation is in the v2.0 roadmap [1, Section XI].",[2231,4306,1643],{"id":1642},[1806,4308,4309,4311,4312,3166,4314,4316],{},[1824,4310,4193],{}," Liquidate ",[1897,4313],{"html":1912,"display":-1},[1897,4315],{"html":1982,"display":-1}," time steps minimizing expected cost plus risk-aversion-weighted variance.",[1806,4318,4319,4321,4322,4325,4326,2275,4329,4332,4333,4336,4337,4340],{},[1824,4320,4211],{}," Let ",[1897,4323],{"html":4324,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x_k\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," be holdings at step ",[1897,4327],{"html":4328,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">k\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1897,4330],{"html":4331,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmrow>\u003Cmi>k\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n_k = x_{k-1} - x_k\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7917em;vertical-align:-0.2083em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">−\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2083em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," the trade, ",[1897,4334],{"html":4335,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>η\u003C\u002Fmi>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\eta n_k\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," the temporary impact, and ",[1897,4338],{"html":4339,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>γ\u003C\u002Fmi>\u003Cmsub>\u003Cmo>∑\u003C\u002Fmo>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\gamma \\sum_j n_j\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1858em;vertical-align:-0.4358em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\">∑\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.162em;\">\u003Cspan style=\"top:-2.4003em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0572em;\">j\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4358em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0572em;\">j\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," the permanent impact. The mean-variance objective is",[1897,4342],{"html":4343,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>C\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>λ\u003C\u002Fmi>\u003Cmtext> \u003C\u002Fmtext>\u003Cmrow>\u003Cmi mathvariant=\"normal\">V\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">a\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">r\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>C\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmspace width=\"1em\"\u002F>\u003Cmtext>s.t.\u003C\u002Fmtext>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmo>∑\u003C\u002Fmo>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmsub>\u003Cmi>n\u003C\u002Fmi>\u003Cmi>k\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">.\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\min_{(n_k)} \\; \\mathbb{E}[C(n)] + \\lambda \\,\\mathrm{Var}[C(n)] \\quad \\text{s.t.}\\;\\; \\sum_k n_k = X.\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.716em;vertical-align:-0.966em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.309em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3448em;\">\u003Cspan style=\"top:-2.3488em;margin-left:0em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1512em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.966em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">C\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:2.3521em;vertical-align:-1.3021em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">Var\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">C\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">s.t.\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.05em;\">\u003Cspan style=\"top:-1.8479em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3021em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0315em;\">k\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mord\">.\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4345,4346,4348,4349,2444],{},[1824,4347,4226],{}," Closed-form hyperbolic-cosine optimal trajectory; the efficient frontier is parameterized by ",[1897,4350],{"html":2839,"display":-1},[1806,4352,4353,4355],{},[1824,4354,4247],{}," Step-size limits and action-count caps in DataGlass [1, Section V] are the direct analogue: aggressive reallocation incurs platform-side learning-phase cost, slow reallocation incurs opportunity cost. The structural form of the trade-off is identical, and the solution structure — exponentially decaying departures from the unconstrained optimum — recurs.",[2231,4357,1646],{"id":1645},[1806,4359,4360,4362,4363,4366,4367,4370],{},[1824,4361,4193],{}," Decide a single-period order quantity ",[1897,4364],{"html":4365,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>q\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">q\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," given ",[1897,4368],{"html":4369,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>N\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">N\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," historical demand observations.",[1806,4372,4373,4375,4376,4379,4380,4383,4384,2444],{},[1824,4374,4211],{}," SAA: ",[1897,4377],{"html":4378,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>q\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>arg\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>q\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmi>N\u003C\u002Fmi>\u003C\u002Fmfrac>\u003Cmsub>\u003Cmo>∑\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmsub>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>u\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>D\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>q\u003C\u002Fmi>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>+\u003C\u002Fmo>\u003Cmsub>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>o\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>q\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>D\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\hat q = \\arg\\min_q \\frac{1}{N} \\sum_n [c_u (D_n - q)^+ + c_o (q - D_n)^+]\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1667em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">ar\u003Cspan style=\"margin-right:0.0139em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\">∑\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.0017em;\">\u003Cspan style=\"top:-2.4003em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2997em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">u\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0213em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">o\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0213em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". The solution is the empirical quantile ",[1897,4381],{"html":4382,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>q\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>D\u003C\u002Fmi>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmo stretchy=\"false\">⌈\u003C\u002Fmo>\u003Cmi>N\u003C\u002Fmi>\u003Cmi>τ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">⌉\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\hat q = D_{(\\lceil N \\tau \\rceil)}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">q\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1667em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0385em;vertical-align:-0.3552em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3448em;\">\u003Cspan style=\"top:-2.5198em;margin-left:-0.0278em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mopen mtight\">(⌈\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">⌉)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3552em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," for ",[1897,4385],{"html":4386,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>τ\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>u\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>u\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsub>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>o\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tau = c_u\u002F(c_u+c_o)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1132em;\">τ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">u\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">u\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">o\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4388,4389,4391,4392,4394,4395,4398,4399,4402,4403,4405,4406,4409,4410,4412,4413,4415],{},[1824,4390,4226],{}," Levi, Roundy, and Shmoys ",[1813,4393],{"n":3348}," prove a ",[1897,4396],{"html":4397,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>ϵ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">{1+\\epsilon}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ϵ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," approximation bound for ",[1897,4400],{"html":4401,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>N\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>O\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmsup>\u003Cmi>ϵ\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">N = O(1\u002F\\epsilon^2)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u002F\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">ϵ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". The DRO Wasserstein newsvendor of Esfahani–Kuhn ",[1813,4404],{"n":2707}," gives a regularized variant whose excess-risk bound is the classical ",[1897,4407],{"html":4408,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>O\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmsqrt>\u003Cmi>N\u003C\u002Fmi>\u003C\u002Fmsqrt>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">O(1\u002F\\sqrt{N})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1767em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u002F\u003C\u002Fspan>\u003Cspan class=\"mord sqrt\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9267em;\">\u003Cspan class=\"svg-align\" style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"padding-left:0.833em;\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.8867em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"hide-tail\" style=\"min-width:0.853em;height:1.08em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"400em\" height=\"1.08em\" viewBox=\"0 0 400000 1080\" preserveAspectRatio=\"xMinYMin slice\">\u003Cpath d=\"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1133em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," but with explicit dependence on the Wasserstein radius. Ban and Rudin ",[1813,4411],{"n":3352}," prove ",[1897,4414],{"html":3356,"display":-1}," rates for the contextual newsvendor with covariate-dependent demand.",[1806,4417,4418,4420],{},[1824,4419,4247],{}," The CVR-dilution mechanism [1, Section IV] is structurally a newsvendor problem at the campaign level, where utilization above threshold corresponds to over-ordering relative to the seller's effective service capacity.",[2231,4422,1649],{"id":1648},[1806,4424,4425,4427],{},[1824,4426,4193],{}," Compute the probability that an insurance surplus process becomes negative.",[1806,4429,4430,2546,4432,2275,4435,4438,4439,4441,4442,4444,4445,4448,4449,4451,4452,2444],{},[1824,4431,4211],{},[1897,4433],{"html":4434,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>U\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>u\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>S\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">U_t = u + ct - S_t\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">U\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6984em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1897,4436],{"html":4437,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>S\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsubsup>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmsub>\u003Cmi>N\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmsubsup>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">S_t = \\sum_{i=1}^{N_t} X_i\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2809em;vertical-align:-0.2997em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\">∑\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9812em;\">\u003Cspan style=\"top:-2.4003em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2029em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2963em;\">\u003Cspan style=\"top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.143em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2997em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," with ",[1897,4440],{"html":3458,"display":-1}," Poisson rate ",[1897,4443],{"html":2839,"display":-1}," and claim sizes ",[1897,4446],{"html":4447,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>X\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">X_i\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," iid with mean ",[1897,4450],{"html":4201,"display":-1}," and CDF ",[1897,4453],{"html":3336,"display":-1},[1806,4455,4456,4458,4459,4461,4462,4464,4465,4468,4469,2444],{},[1824,4457,4226],{}," Lundberg's inequality ",[1897,4460],{"html":3466,"display":-1},", where the adjustment coefficient ",[1897,4463],{"html":3469,"display":-1}," solves ",[1897,4466],{"html":4467,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>λ\u003C\u002Fmi>\u003Cmsubsup>\u003Cmo>∫\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmsubsup>\u003Cmsup>\u003Cmi>e\u003C\u002Fmi>\u003Cmrow>\u003Cmi>R\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>F\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmtext> \u003C\u002Fmtext>\u003Cmi mathvariant=\"normal\">d\u003C\u002Fmi>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\lambda \\int_0^\\infty e^{Rx}(1 - F(x))\\,\\mathrm{d}x = c\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.2151em;vertical-align:-0.3558em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop op-symbol small-op\" style=\"margin-right:0.1945em;position:relative;top:-0.0006em;\">∫\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8593em;\">\u003Cspan style=\"top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2579em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">∞\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3558em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">e\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8413em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0077em;\">R\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">x\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">F\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mclose\">))\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathrm\">d\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". For exponential claims, ",[1897,4470],{"html":4471,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ψ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>u\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmrow>\u003Cmi>λ\u003C\u002Fmi>\u003Cmi>μ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmfrac>\u003Cmi>exp\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmrow>\u003Cmo fence=\"true\">(\u003C\u002Fmo>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmi>μ\u003C\u002Fmi>\u003C\u002Fmfrac>\u003C\u002Fmstyle>\u003Cmo>−\u003C\u002Fmo>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmfrac>\u003Cmi>λ\u003C\u002Fmi>\u003Cmi>c\u003C\u002Fmi>\u003C\u002Fmfrac>\u003C\u002Fmstyle>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>u\u003C\u002Fmi>\u003Cmo fence=\"true\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\psi(u) = \\frac{\\lambda \\mu}{c} \\exp\\!\\left(-\\left(\\tfrac{1}{\\mu} - \\tfrac{\\lambda}{c}\\right) u\\right)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">ψ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.8em;vertical-align:-0.65em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9322em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.4461em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">λ\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">μ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">exp\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size2\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">\u003Cspan class=\"mopen delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size2\">(\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">μ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4811em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8801em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">λ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size2\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">u\u003C\u002Fspan>\u003Cspan class=\"mclose delimcenter\" style=\"top:0em;\">\u003Cspan class=\"delimsizing size2\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4473,4474,4476,4477,4479],{},[1824,4475,4247],{}," Conversion-rate \"viral demand\" tail risk and return-rate tails are heavy-tailed claim-like processes; the actuarial machinery for tail-quantile estimation ",[1813,4478],{"n":3473},"–:blog-cite transfers directly to the calibration of profit-adjusted ROAS confidence intervals when the seller portfolio includes rare but large-impact promotional events.",[2231,4481,1652],{"id":1651},[1806,4483,4484,4486,4487,4490,4491,4494],{},[1824,4485,4193],{}," Choose one of ",[1897,4488],{"html":4489,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>K\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">K\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">K\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," articles to display, given user context ",[1897,4492],{"html":4493,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmsup>\u003Cmi mathvariant=\"double-struck\">R\u003C\u002Fmi>\u003Cmi>d\u003C\u002Fmi>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x \\in \\mathbb{R}^d\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8491em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">d\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", to maximize click-through.",[1806,4496,4497,4499,4500,4503,4504,2444],{},[1824,4498,4211],{}," Linear payoff model ",[1897,4501],{"html":4502,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmsub>\u003Cmi>r\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>∣\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo 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style=\"top:-2.357em;margin-left:0em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.143em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2501em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". 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class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". The constrained MDP has expected-cost constraint ",[1897,4551],{"html":4552,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmsub>\u003Cmo>∑\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmsub>\u003Cmrow>\u003Cmi mathvariant=\"normal\">c\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">o\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">s\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">t\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmo>≤\u003C\u002Fmo>\u003Cmi>B\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\mathbb{E}[\\sum_t \\mathrm{cost}_t] \\le B\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0497em;vertical-align:-0.2997em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\">∑\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1308em;\">\u003Cspan style=\"top:-2.4003em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2997em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathrm\">cost\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≤\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0502em;\">B\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". Lagrangian relaxation gives ",[1897,4555],{"html":4556,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>b\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>v\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>v\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmsup>\u003Cmi>λ\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">b^{\\star}(v) = v \u002F \\lambda^{\\star}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1897,4558],{"html":4559,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>λ\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\lambda^{\\star}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," is the budget shadow price ",[1813,4562],{"n":4068},[1806,4564,4565,4567,4568,4570,4571,4573,4574,4576],{},[1824,4566,4226],{}," Cai et al. ",[1813,4569],{"n":4058}," report 16.7% lift; Wu et al. ",[1813,4572],{"n":4068}," report 23.4% lift with strict budget compliance; Liu et al. ",[1813,4575],{"n":4091}," ablate that budget\u002Ftime-ratio features dominate.",[1806,4578,4579,4581,4582,4584,4585,4587],{},[1824,4580,4247],{}," The dual variable ",[1897,4583],{"html":4559,"display":-1}," in RTB and the shadow price ",[1897,4586],{"html":3430,"display":-1}," in DataGlass are the same object — the marginal value of an additional dollar of advertising — at different temporal granularities. RTB controls bids per impression; DataGlass controls budgets per day; the underlying primal-dual structure is invariant.",[2231,4589,1658],{"id":1657},[1806,4591,4592,4594],{},[1824,4593,4193],{}," Forecast hierarchical Walmart unit sales at SKU-store-day granularity for 28 days.",[1806,4596,4597,4599],{},[1824,4598,4211],{}," A panel forecasting problem with hierarchical reconciliation across SKU\u002Fcategory\u002Fdepartment\u002Fstore\u002Fstate aggregation levels.",[1806,4601,4602,4604,4605,4607],{},[1824,4603,4226],{}," Winning models were global gradient-boosted regressors (LightGBM) with hand-crafted lag and rolling features, tweaked by hierarchical reconciliation methods ",[1813,4606],{"n":3943},". Pure deep models (DeepAR, NBEATS) under-performed at SKU level despite winning M4. Quantile-regression objectives produce the calibrated prediction intervals essential for downstream inventory decisions.",[1806,4609,4610,4612,4613,4615,4616,4619],{},[1824,4611,4247],{}," The DataGlass response-curve estimation problem is ",[1809,4614,2183],{}," a standard forecasting problem (it is a budget→outcome mapping, not time-series extrapolation), but the M5 lesson — ",[1809,4617,4618],{},"calibrated uncertainty intervals matter more than headline point accuracy for downstream decisions"," — translates directly. DataGlass uses Negative-Binomial likelihoods explicitly to produce the calibrated posterior the optimizer requires [1, Section IV].",[2231,4621,1661],{"id":1660},[1806,4623,4624,4626,4627,4630,4631,2444],{},[1824,4625,4193],{}," Estimate the average treatment effect ",[1897,4628],{"html":4629,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi>Y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>Y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\theta = \\mathbb{E}[Y(1) - Y(0)]\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," from observational data with high-dimensional confounders ",[1897,4632],{"html":1912,"display":-1},[1806,4634,4635,4637,4638,2275,4641,4644,4645,2444],{},[1824,4636,4211],{}," Partially linear model ",[1897,4639],{"html":4640,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>Y\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>D\u003C\u002Fmi>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>g\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>ε\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">Y = D\\theta + g(X) + \\varepsilon\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ε\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1897,4642],{"html":4643,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>D\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>m\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>v\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">D = m(X) + v\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",". The Neyman-orthogonal score is ",[1897,4646],{"html":4647,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ψ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>W\u003C\u002Fmi>\u003Cmo separator=\"true\">;\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>η\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>Y\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>g\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>D\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>m\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>D\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>m\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\psi(W; \\theta, \\eta) = (Y - g(X) - \\theta(D - m(X)))(D - m(X))\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">ψ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">W\u003C\u002Fspan>\u003Cspan class=\"mpunct\">;\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">η\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">)))\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">D\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mclose\">))\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4649,4650,4652,4653,4655,4656,3599,4659,4661,4662,4665,4666,4669],{},[1824,4651,4226],{}," Chernozhukov et al. ",[1813,4654],{"n":3695}," prove that with cross-fitting, ",[1897,4657],{"html":4658,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>θ\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\hat\\theta\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9579em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9579em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2634em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1667em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1897,4660],{"html":3707,"display":-1},"-consistent and asymptotically normal as long as the nuisance functions ",[1897,4663],{"html":4664,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>g\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>m\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">g, m\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," are estimated at rate ",[1897,4667],{"html":4668,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>o\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsup>\u003Cmi>n\u003C\u002Fmi>\u003Cmrow>\u003Cmo>−\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmn>4\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">o(n^{-1\u002F4})\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.138em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">o\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.888em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u002F4\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," — a rate achieved by most modern machine learning methods.",[1806,4671,4672,4674],{},[1824,4673,4247],{}," Causal estimation of the budget-response relationship in the presence of high-dimensional confounders (calendar, product, market, competitor) is a direct application. The DataGlass system uses first-differencing plus randomized perturbation as the primary identification strategy [1, Section VIII], but DML provides the standard observational fallback when randomization is infeasible.",[2231,4676,1664],{"id":1663},[1806,4678,4679,4681],{},[1824,4680,4193],{}," Train a classifier robust to small perturbations of the input distribution.",[1806,4683,4684,4686],{},[1824,4685,4211],{}," Wasserstein DRO",[1897,4688],{"html":4689,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>min\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmi>θ\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmrow>\u003Cmi>sup\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>Q\u003C\u002Fmi>\u003Cmo>:\u003C\u002Fmo>\u003Cmsub>\u003Cmi>W\u003C\u002Fmi>\u003Cmi>p\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>Q\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmover accent=\"true\">\u003Cmi>P\u003C\u002Fmi>\u003Cmo>^\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>≤\u003C\u002Fmo>\u003Cmi>ε\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsub>\u003Cmi mathvariant=\"double-struck\">E\u003C\u002Fmi>\u003Cmi>Q\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">[\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">ℓ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>θ\u003C\u002Fmi>\u003Cmo separator=\"true\">;\u003C\u002Fmo>\u003Cmi>X\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>Y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">]\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\min_\\theta \\; \\sup_{Q : W_p(Q, \\hat P_n) \\le \\varepsilon} \\;\\mathbb{E}_Q[\\ell(\\theta; X, Y)],\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:2.0705em;vertical-align:-1.3205em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-2.3479em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">min\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7521em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4306em;\">\u003Cspan style=\"top:-1.9768em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">Q\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">:\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">W\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1645em;\">\u003Cspan style=\"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\">p\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2819em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen mtight\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">Q\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord accent mtight\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9468em;\">\u003Cspan style=\"top:-2.7em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.1389em;\">P\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-2.9523em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1667em;\">\u003Cspan class=\"mord mtight\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1645em;\">\u003Cspan style=\"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.143em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose mtight\">)\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">≤\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">ε\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">sup\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.3205em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">E\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3283em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">Q\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord\">ℓ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">θ\u003C\u002Fspan>\u003Cspan class=\"mpunct\">;\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0785em;\">X\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">Y\u003C\u002Fspan>\u003Cspan class=\"mclose\">)]\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4691,4692,4693,2444],{},"with the Esfahani–Kuhn dual representation as a regularized empirical-risk minimization ",[1813,4694],{"n":2707},[1806,4696,4697,4699,4700,4703,4704,4707,4708,2546,4710,4712,4713,4716],{},[1824,4698,4226],{}," For ",[1897,4701],{"html":4702,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p = \\infty\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∞\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," Wasserstein and Lipschitz loss, the worst-case is exactly equivalent to adversarial training with ",[1897,4705],{"html":4706,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi mathvariant=\"normal\">ℓ\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\ell_\\infty\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">ℓ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">∞\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," perturbations of magnitude ",[1897,4709],{"html":2714,"display":-1},[1813,4711],{"n":2732},". For ",[1897,4714],{"html":4715,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>p\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">p = 2\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">p\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",", the dual is a Tikhonov-style regularization with explicit penalty.",[1806,4718,4719,4721,4722,4724],{},[1824,4720,4247],{}," Marketplace platforms continuously shift the response distribution as competitor behavior, platform algorithms, and seasonal demand co-evolve. A Wasserstein-DRO formulation of the DataGlass response model, with ",[1897,4723],{"html":2714,"display":-1}," tuned to historical drift, is among the open problems identified in Section 15.",[2231,4726,1667],{"id":1666},[1806,4728,4729,4731],{},[1824,4730,4193],{}," Allocate daily budgets across a seller's portfolio of campaigns on a platform-controlled marketplace, maximizing expected profit subject to operational constraints.",[1806,4733,4734,4736,4737,4739,4740,4742],{},[1824,4735,4211],{}," As in ",[1813,4738],{"n":1815}," and the ",[2266,4741,4170],{"href":4169}," [2, Section 4]:",[1897,4744],{"html":4745,"display":2430},"\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi>max\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmsub>\u003Cmi mathvariant=\"bold\">b\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmunder>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunderover>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmi>N\u003C\u002Fmi>\u003C\u002Fmunderover>\u003Cmsub>\u003Cmi>π\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmspace width=\"1em\"\u002F>\u003Cmtext>s.t.\u003C\u002Fmtext>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmo>∑\u003C\u002Fmo>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmunder>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>≤\u003C\u002Fmo>\u003Cmsub>\u003Cmi>B\u003C\u002Fmi>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>∈\u003C\u002Fmo>\u003Cmsub>\u003Cmi mathvariant=\"script\">F\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\max_{\\mathbf b_t} \\;\\sum_{i=1}^{N} \\pi_i(b_{i,t}) \\quad\\text{s.t.}\\;\\;\\sum_i b_{i,t} \\le B_t,\\;\\; b_{i,t} \\in \\mathcal{F}_{i,t},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:3.106em;vertical-align:-1.2777em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4306em;\">\u003Cspan style=\"top:-2.3479em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathbf mtight\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2963em;\">\u003Cspan style=\"top:-2.357em;margin-left:0em;margin-right:0.0714em;\">\u003Cspan class=\"pstrut\" style=\"height:2.5em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size3 size1 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.143em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">max\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8522em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.8283em;\">\u003Cspan style=\"top:-1.8723em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-4.3em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.109em;\">N\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.2777em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:1em;\">\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\">s.t.\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.05em;\">\u003Cspan style=\"top:-1.8723em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.2777em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≤\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9805em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0502em;\">B\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.9694em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathcal\" style=\"margin-right:0.0993em;\">F\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mpunct mtight\">,\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1806,4747,2611,4748,4751,4752,4755],{},[1897,4749],{"html":4750,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>π\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\pi_i\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," derived from a Negative-Binomial click model with conditional Hill saturation, a Beta-Binomial conversion model with utilization-dependent dilution, and a true-profit adjustment ",[1897,4753],{"html":4754,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>m\u003C\u002Fmi>\u003Cmi>g\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>r\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>f\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmi>ρ\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">m_g(1-r)(1-f-\\rho)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ρ\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," that corrects the gap between reported and contribution-margin-adjusted ROAS [2, Section 3.1].",[1806,4757,4758,4760,4761,4764,4765,4767],{},[1824,4759,4226],{}," Optimal solution is characterized by the equal-marginal-profit condition ",[1897,4762],{"html":4763,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∂\u003C\u002Fmi>\u003Cmsub>\u003Cmi>π\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi mathvariant=\"normal\">\u002F\u003C\u002Fmi>\u003Cmi mathvariant=\"normal\">∂\u003C\u002Fmi>\u003Cmsubsup>\u003Cmi>b\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsubsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmi>μ\u003C\u002Fmi>\u003Cmo lspace=\"0em\" rspace=\"0em\">⋆\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\partial \\pi_i \u002F \\partial b_i^{\\star} = \\mu^{\\star}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0087em;vertical-align:-0.2587em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"margin-right:0.0556em;\">∂\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord\" style=\"margin-right:0.0556em;\">∂\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-2.4413em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2587em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8831em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">μ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," on the active set; solved by bisection on ",[1897,4766],{"html":4201,"display":-1},". Empirical lifts of 21.3% offline and 21.6% online with reallocation frequency reduced by 43.8% [1, Section X].",[1806,4769,4770,4773,4774,4777],{},[1824,4771,4772],{},"Synthesis."," This case is the connecting tissue of the meta-review. It inherits Markowitz's portfolio framing (Section 12.1), Almgren–Chriss's cost-of-thrashing (Section 12.3), the newsvendor's capacity-constrained service-level structure (Section 12.4), Cramér–Lundberg's tail-risk machinery (Section 12.5), contextual bandits' calibrated exploration (Section 12.6), RTB's Lagrangian shadow-price interpretation (Section 12.7), the M5 competition's emphasis on calibrated uncertainty (Section 12.8), and double machine learning's causal-identification machinery (Section 12.9). None of these primitives is novel to e-commerce. The novelty is in the ",[1809,4775,4776],{},"composition"," — and in the explicit treatment of the platform as an opaque, constraint-imposing intermediary rather than a transparent auction.",[2967,4779],{"text":4780},"None of these primitives is novel to e-commerce. The novelty is in the composition — and in the explicit treatment of the platform as an opaque, constraint-imposing intermediary rather than a transparent auction.",[1829,4782],{},[1832,4784,1670],{"id":1669},[2231,4786,1674],{"id":1673},[1806,4788,4789],{},"Across the eleven cases and the broader literature surveyed in Sections4–11, every mature decision system instantiates the same four primitives. Table 1 summarizes the instantiation across selected domains.",[1806,4791,4792],{},[1809,4793,4794],{},"Table 1 — Cross-domain instantiation of the four primitives.",[1873,4796,4797,4816],{},[1876,4798,4799],{},[1879,4800,4801,4804,4807,4810,4813],{},[1882,4802,4803],{},"Domain",[1882,4805,4806],{},"Probabilistic model",[1882,4808,4809],{},"Risk objective",[1882,4811,4812],{},"Constraint set",[1882,4814,4815],{},"Exploration",[1889,4817,4818,4835,4851,4867,4884,4900,4917,4937,4954,4973,4990],{},[1879,4819,4820,4823,4826,4829,4832],{},[1894,4821,4822],{},"Mean-variance portfolio",[1894,4824,4825],{},"Multivariate normal returns",[1894,4827,4828],{},"Variance",[1894,4830,4831],{},"Budget, no-short",[1894,4833,4834],{},"Static (none)",[1879,4836,4837,4840,4843,4846,4848],{},[1894,4838,4839],{},"CVaR portfolio",[1894,4841,4842],{},"Empirical scenarios",[1894,4844,4845],{},"CVaR",[1894,4847,4831],{},[1894,4849,4850],{},"Static",[1879,4852,4853,4856,4859,4862,4865],{},[1894,4854,4855],{},"Almgren–Chriss execution",[1894,4857,4858],{},"Brownian + impact",[1894,4860,4861],{},"Mean-variance of cost",[1894,4863,4864],{},"Liquidation horizon",[1894,4866,4850],{},[1879,4868,4869,4872,4875,4878,4881],{},[1894,4870,4871],{},"Newsvendor",[1894,4873,4874],{},"Demand CDF",[1894,4876,4877],{},"Expected cost (or CVaR)",[1894,4879,4880],{},"Capacity",[1894,4882,4883],{},"Empirical \u002F DRO",[1879,4885,4886,4889,4892,4895,4898],{},[1894,4887,4888],{},"Cramér–Lundberg",[1894,4890,4891],{},"Compound Poisson",[1894,4893,4894],{},"Ruin probability",[1894,4896,4897],{},"Capital floor",[1894,4899,4850],{},[1879,4901,4902,4905,4908,4911,4914],{},[1894,4903,4904],{},"LinUCB news rec.",[1894,4906,4907],{},"Linear payoff",[1894,4909,4910],{},"Expected reward",[1894,4912,4913],{},"Slot capacity",[1894,4915,4916],{},"UCB",[1879,4918,4919,4922,4925,4928,4931],{},[1894,4920,4921],{},"RTB MDP",[1894,4923,4924],{},"Q-network value",[1894,4926,4927],{},"Expected click value",[1894,4929,4930],{},"Daily budget",[1894,4932,4933,4936],{},[1897,4934],{"html":4935,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>ϵ\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\epsilon\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">ϵ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","-greedy \u002F TS",[1879,4938,4939,4942,4945,4948,4951],{},[1894,4940,4941],{},"M5 forecasting",[1894,4943,4944],{},"GBM ensemble",[1894,4946,4947],{},"Pinball loss",[1894,4949,4950],{},"None operational",[1894,4952,4953],{},"Hyperparameter",[1879,4955,4956,4959,4962,4967,4970],{},[1894,4957,4958],{},"DML causal",[1894,4960,4961],{},"Cross-fit nuisances",[1894,4963,4964,4966],{},[1897,4965],{"html":3707,"display":-1}," ATE bias",[1894,4968,4969],{},"None",[1894,4971,4972],{},"None (observational)",[1879,4974,4975,4978,4981,4984,4987],{},[1894,4976,4977],{},"Wasserstein DRO",[1894,4979,4980],{},"Empirical + ball",[1894,4982,4983],{},"Worst-case loss",[1894,4985,4986],{},"Hypothesis class",[1894,4988,4989],{},"Implicit (adversary)",[1879,4991,4992,4997,5002,5007,5012],{},[1894,4993,4994],{},[1824,4995,4996],{},"DataGlass",[1894,4998,4999],{},[1824,5000,5001],{},"NB2 × Beta-Binomial",[1894,5003,5004],{},[1824,5005,5006],{},"Expected profit + posterior",[1894,5008,5009],{},[1824,5010,5011],{},"Six op. constraints",[1894,5013,5014],{},[1824,5015,5016],{},"TS + perturbation",[2231,5018,1677],{"id":1676},[1806,5020,5021],{},"Table 2 collates headline empirical lifts for the production deployments included in the corpus, where reported.",[1806,5023,5024],{},[1809,5025,5026],{},"Table 2 — Production deployment lifts (where reported).",[1873,5028,5029,5044],{},[1876,5030,5031],{},[1879,5032,5033,5036,5038,5041],{},[1882,5034,5035],{},"System",[1882,5037,4803],{},[1882,5039,5040],{},"Headline lift",[1882,5042,5043],{},"Source",[1889,5045,5046,5061,5075,5090,5104,5120,5135,5150,5171],{},[1879,5047,5048,5051,5054,5057],{},[1894,5049,5050],{},"Cai et al. RTB",[1894,5052,5053],{},"Display advertising",[1894,5055,5056],{},"+16.7% clicks",[1894,5058,5059],{},[1813,5060],{"n":4058},[1879,5062,5063,5066,5068,5071],{},[1894,5064,5065],{},"Wu et al. CMDP-RTB",[1894,5067,5053],{},[1894,5069,5070],{},"+23.4% clicks",[1894,5072,5073],{},[1813,5074],{"n":4068},[1879,5076,5077,5080,5083,5086],{},[1894,5078,5079],{},"Zhao et al. RTB",[1894,5081,5082],{},"Sponsored search",[1894,5084,5085],{},"−15.4% CPC",[1894,5087,5088],{},[1813,5089],{"n":4079},[1879,5091,5092,5095,5097,5100],{},[1894,5093,5094],{},"Wang et al. ROI-RTB",[1894,5096,5053],{},[1894,5098,5099],{},"+14.0% ROI",[1894,5101,5102],{},[1813,5103],{"n":4087},[1879,5105,5106,5109,5112,5115],{},[1894,5107,5108],{},"Jauvion et al. SSP",[1894,5110,5111],{},"Header bidding",[1894,5113,5114],{},"Reported significant",[1894,5116,5117],{},[1813,5118],{"n":5119},"230",[1879,5121,5122,5125,5128,5131],{},[1894,5123,5124],{},"Ferreira–Lee–Simchi-Levi",[1894,5126,5127],{},"Online retail pricing",[1894,5129,5130],{},"+9.7% revenue",[1894,5132,5133],{},[1813,5134],{"n":3400},[1879,5136,5137,5140,5143,5146],{},[1894,5138,5139],{},"Li et al. LinUCB",[1894,5141,5142],{},"News recommendation",[1894,5144,5145],{},"+12.5% CTR",[1894,5147,5148],{},[1813,5149],{"n":3627},[1879,5151,5152,5157,5162,5167],{},[1894,5153,5154],{},[1824,5155,5156],{},"DataGlass (offline)",[1894,5158,5159],{},[1824,5160,5161],{},"Marketplace ads",[1894,5163,5164],{},[1824,5165,5166],{},"+21.3% profit",[1894,5168,5169],{},[1813,5170],{"n":1815},[1879,5172,5173,5178,5182,5187],{},[1894,5174,5175],{},[1824,5176,5177],{},"DataGlass (online A\u002FB)",[1894,5179,5180],{},[1824,5181,5161],{},[1894,5183,5184],{},[1824,5185,5186],{},"+21.6% profit",[1894,5188,5189],{},[1813,5190],{"n":1815},[2231,5192,1680],{"id":1679},[1806,5194,5195],{},"Table 3 evaluates the strength of cross-domain transfer between finance\u002Foperations primitives and e-commerce decision intelligence.",[1806,5197,5198],{},[1809,5199,5200],{},"Table 3 — Transfer strength of selected primitives into e-commerce decision systems.",[1873,5202,5203,5219],{},[1876,5204,5205],{},[1879,5206,5207,5210,5213,5216],{},[1882,5208,5209],{},"Primitive",[1882,5211,5212],{},"Source domain",[1882,5214,5215],{},"Transfer to e-commerce",[1882,5217,5218],{},"Strength",[1889,5220,5221,5235,5249,5262,5274,5286,5301,5314,5326,5339,5350,5362,5374,5387,5401],{},[1879,5222,5223,5226,5229,5232],{},[1894,5224,5225],{},"Equal-marginal-return optimization",[1894,5227,5228],{},"Finance (MV)",[1894,5230,5231],{},"Budget allocation across campaigns",[1894,5233,5234],{},"Strong",[1879,5236,5237,5240,5243,5246],{},[1894,5238,5239],{},"Mean-variance objective",[1894,5241,5242],{},"Finance",[1894,5244,5245],{},"Risk-aware budget allocation",[1894,5247,5248],{},"Partial",[1879,5250,5251,5254,5256,5259],{},[1894,5252,5253],{},"CVaR linearization",[1894,5255,5242],{},[1894,5257,5258],{},"Downside-risk ad allocation",[1894,5260,5261],{},"Strong (untapped)",[1879,5263,5264,5267,5269,5272],{},[1894,5265,5266],{},"Almgren–Chriss cost-of-thrashing",[1894,5268,5242],{},[1894,5270,5271],{},"Action-count caps",[1894,5273,5234],{},[1879,5275,5276,5279,5281,5284],{},[1894,5277,5278],{},"Newsvendor capacity logic",[1894,5280,356],{},[1894,5282,5283],{},"Inventory-dilution mechanism",[1894,5285,5234],{},[1879,5287,5288,5294,5296,5299],{},[1894,5289,5290,5293],{},[1897,5291],{"html":5292,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>s\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>S\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(s,S)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," threshold policies",[1894,5295,356],{},[1894,5297,5298],{},"Bid-budget revision triggers",[1894,5300,5248],{},[1879,5302,5303,5306,5309,5312],{},[1894,5304,5305],{},"Cramér–Lundberg ruin",[1894,5307,5308],{},"Insurance",[1894,5310,5311],{},"Tail-risk calibration",[1894,5313,5248],{},[1879,5315,5316,5319,5321,5324],{},[1894,5317,5318],{},"EVT tail estimation",[1894,5320,5308],{},[1894,5322,5323],{},"Conversion outlier modeling",[1894,5325,5248],{},[1879,5327,5328,5331,5334,5337],{},[1894,5329,5330],{},"LinUCB \u002F Thompson Sampling",[1894,5332,5333],{},"Online learning",[1894,5335,5336],{},"Campaign exploration",[1894,5338,5234],{},[1879,5340,5341,5343,5345,5348],{},[1894,5342,2901],{},[1894,5344,5333],{},[1894,5346,5347],{},"Constrained exploration",[1894,5349,5234],{},[1879,5351,5352,5355,5357,5360],{},[1894,5353,5354],{},"DML",[1894,5356,1788],{},[1894,5358,5359],{},"Observational lift estimation",[1894,5361,5234],{},[1879,5363,5364,5367,5369,5372],{},[1894,5365,5366],{},"Synthetic controls",[1894,5368,1788],{},[1894,5370,5371],{},"Holdout-region lift",[1894,5373,5248],{},[1879,5375,5376,5378,5381,5384],{},[1894,5377,4977],{},[1894,5379,5380],{},"Stochastic optim.",[1894,5382,5383],{},"Drift-robust response curves",[1894,5385,5386],{},"Open",[1879,5388,5389,5392,5395,5398],{},[1894,5390,5391],{},"MPC",[1894,5393,5394],{},"Control",[1894,5396,5397],{},"Intra-day budget pacing",[1894,5399,5400],{},"Partial (open)",[1879,5402,5403,5406,5408,5411],{},[1894,5404,5405],{},"Conformal prediction",[1894,5407,5333],{},[1894,5409,5410],{},"Calibrated CIs on recommendations",[1894,5412,5400],{},[2231,5414,1683],{"id":1682},[1806,5416,5417,5418,5421],{},"Across the 213 primary papers, the median rubric score (formal rigor, empirical validation, reproducibility, deployment) is ",[1897,5419],{"html":5420,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(2, 1, 1, 0)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">2\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>"," on the 0–2 scale. Deployment evidence is the rubric dimension with the lowest mean score, consistent with the well-known publication-bias toward methodologically novel but production-untested results.",[1829,5423],{},[1832,5425,1686],{"id":1685},[2231,5427,1690],{"id":1689},[1806,5429,5430],{},"Cross-domain heterogeneity is structural. Finance papers favor closed-form, asymptotic, and worst-case results; operations papers favor approximation algorithms with provable bounds; machine-learning papers favor empirical benchmarks; e-commerce papers favor production-deployment lifts. The four-primitive taxonomy is the unifying frame; the calibration choices (which risk measure, which ambiguity set, which exploration schedule) are where heterogeneity concentrates.",[2231,5432,1693],{"id":1692},[1806,5434,5435],{},"Three risk-of-bias considerations apply across domains.",[1806,5437,5438,5441,5442,5444],{},[1824,5439,5440],{},"Selection bias toward positive results."," Production-deployment papers self-select for lifts large enough to publish. The reported 18–24% lifts in advertising RL ",[1813,5443],{"n":4058},"–:blog-cite should be read with this caveat.",[1806,5446,5447,5450],{},[1824,5448,5449],{},"Outcome-definition bias."," Finance papers report Sharpe ratios; e-commerce papers report CTR or ROAS; both are gameable proxies for the underlying contribution. We recommend that future cross-domain reviews explicitly require contribution-margin-adjusted reporting [2, Section 3.1].",[1806,5452,5453,5456],{},[1824,5454,5455],{},"Comparator bias."," \"Static baseline\" in advertising RL is rarely defined precisely; the magnitude of reported lifts depends on the strength of the baseline.",[2231,5458,1696],{"id":1695},[1806,5460,5461,5462,5464],{},"We did not run a formal funnel-plot analysis because effect sizes are not commensurable across domains. We note three indicators of publication bias: (i) the under-representation of negative-result papers in advertising RL; (ii) the gap between the strong methodological literature on robust optimization and the relatively sparse production-deployment literature in the same area; (iii) the absence of replication studies of headline RTB results ",[1813,5463],{"n":4058},"–:blog-cite. These observations suggest reported lifts are likely upper-bounded estimates.",[1829,5466],{},[1832,5468,1699],{"id":1698},[1806,5470,5471],{},"We identify nine open problems at the methodological and operational frontier, expanding on the five raised in the prior version of this review.",[2231,5473,1703],{"id":1702},[1806,5475,5476,5477,5479],{},"Ruszczyński's ",[1813,5478],{"n":2805}," formalism is mature for finance and operations but not deployed at scale in e-commerce. A practical CVaR-Bellman recursion for budget allocation, with explicit treatment of attribution-window non-Markovianity, is open.",[2231,5481,1706],{"id":1705},[1806,5483,5484,5485,5487],{},"Esfahani–Kuhn DRO ",[1813,5486],{"n":2707}," is a natural fit for marketplace ad budget allocation; whether the DRO regularization radius can be tuned from data without sacrificing bandit-style regret is open.",[2231,5489,1709],{"id":1708},[1806,5491,5492],{},"Marketplace sellers control two levers; current systems, including DataGlass v1.0, optimize budget conditional on a fixed target ROAS [1, Section XI]. Joint optimization raises identifiability questions (the platform-side response model must include the seller-side target as a covariate) and is on the v2.0 roadmap.",[2231,5494,1712],{"id":1711},[1806,5496,5497,5498,5500,5501,2444],{},"Standard randomized experimentation ",[1813,5499],{"n":3635}," generates clean variation, but the platform's attribution window introduces a convolutional smoothing kernel that complicates inference. Deconvolution-style estimators may transfer from neuroimaging ",[1813,5502],{"n":5503},"231",[2231,5505,1715],{"id":1714},[1806,5507,5508],{},"Sellers operating across Shopee, Lazada, TikTok Shop, Amazon, and Walmart have structurally similar but parameter-different response curves. Hierarchical Bayesian or domain-adaptation approaches that share information across marketplaces are an open frontier.",[2231,5510,1718],{"id":1717},[1806,5512,5513,5514,5516],{},"Distribution-free prediction sets ",[1813,5515],{"n":3847},"–:blog-cite are an attractive complement to Bayesian posteriors but have not been integrated into production budget-optimization systems with theoretical guarantees on the resulting decisions.",[2231,5518,1721],{"id":1720},[1806,5520,5521,5522,2275,5525,5528,5529,5531],{},"Fairness-aware optimization ",[1813,5523],{"n":5524},"232",[1813,5526],{"n":5527},"233"," is mature for classification but underdeveloped for sequential decision systems. The seller-platform power asymmetry ",[1813,5530],{"n":1819}," is a natural site for fairness-constrained allocation.",[2231,5533,1724],{"id":1723},[1806,5535,5536,5537,2275,5539,5541],{},"As marketplaces shift from second-price to first-price ",[1813,5538],{"n":4120},[1813,5540],{"n":4124},", the seller's optimal budget response becomes more dependent on competitor behavior. Game-theoretic counter-strategies under partial information are open.",[2231,5543,1727],{"id":1726},[1806,5545,5546,5547,2275,5550,5553],{},"Listing copy, image quality, and review content are unstructured signals that affect the budget→outcome curve but are not usually included in response-curve estimation. Foundation-model embeddings ",[1813,5548],{"n":5549},"234",[1813,5551],{"n":5552},"235"," are a candidate input; the calibration problem is open.",[1829,5555],{},[1832,5557,1730],{"id":1729},[1806,5559,5560,5561,5563,5564,5566],{},"We acknowledge five limitations. First, the corpus is heavily weighted toward English-language publications; the substantial Chinese-language literature on marketplace advertising (especially Taobao and JD) is under-represented, despite citations to ",[1813,5562],{"n":4144}," and others. Second, the search period ends in April 2026, so the most recent 2026 conference proceedings (KDD'26, NeurIPS'26) are not included. Third, the review is structured-narrative rather than meta-analytic, so we cannot pool effect sizes; cross-domain effect-size comparison is a known limitation. Fourth, the included e-commerce production literature is biased toward systems with publishable academic affiliations; pure-industry systems without academic write-ups are systematically under-represented. Fifth, the authors are affiliated with one of the production systems reviewed (DataGlass ",[1813,5565],{"n":1815},"), introducing a potential conflict of interest documented in Section 18.",[1829,5568],{},[1832,5570,1733],{"id":1732},[1806,5572,5573],{},"The primary inputs to this review are public-domain academic publications, listed exhaustively in the References. The internal DataGlass reports cited are cross-referenced to the corresponding public-domain sources where they exist. The search strings of Appendix B reproduce the candidate-record search; the screening protocol of Section 2.4 is documented in writing and available on request. The structured-extraction database used to generate Tables 1–3 is available on request, subject to redaction of any individually-identifying seller or campaign data.",[1829,5575],{},[1832,5577,1736],{"id":1735},[1806,5579,5580,5581,5583],{},"DataGlass Labs Research is the institutional author of this meta-review. The DataGlass production system ",[1813,5582],{"n":1815}," is one of the systems reviewed; this introduces a potential conflict of interest. We have attempted to mitigate this in two ways: (i) by citing the relevant academic literature comprehensively rather than selectively; and (ii) by explicitly identifying open problems and limitations of the DataGlass system in Section 15 and Section 16. This is a working paper, not a peer-reviewed publication; the synthesis it presents is internal research output and the reader should treat the DataGlass-specific empirical figures it cites accordingly.",[1806,5585,5586,5587,5589],{},"The review was funded internally by DataGlass Labs Research; no external funding was received. No participating sellers were identifiable in the data underlying Section 11.7 or ",[1813,5588],{"n":1815},"; all empirical data has been aggregated and anonymized prior to analysis.",[1829,5591],{},[1832,5593,1739],{"id":1738},[1806,5595,5596,5597,5599],{},"This meta-review has argued that the literature on prediction and risk optimization under uncertainty is more unified than its domain-fragmented appearance suggests. The same four primitives — calibrated probabilistic models, coherent risk-aware objectives, explicit operational constraint sets, and principled exploration mechanisms — underlie every mature decision system in finance, operations, insurance, energy, healthcare, causal inference, and e-commerce. Eleven detailed case studies have shown how distinct-looking problems instantiate the same underlying framework. The connecting case — DataGlass marketplace ad budget allocation ",[1813,5598],{"n":1815}," — inherits its formal structure from Markowitz, Almgren–Chriss, Rockafellar–Uryasev, the newsvendor, Cramér–Lundberg, contextual bandits, the M5 forecasting consensus, and double machine learning. The novelty is composition, calibration, and the explicit treatment of the platform-imposed constraint structure — not the underlying primitives.",[1806,5601,5602,5603,5606],{},"For researchers, the implication is that cross-domain reading is ",[1809,5604,5605],{},"under-weighted"," in current practice and that the next wave of e-commerce decision-intelligence systems will be built by engineers fluent in the finance and operations literatures of the prior generation. For practitioners, the implication is that the choice of risk measure, the calibration of ambiguity, and the design of exploration are not implementation details: they are the system. We hope this review can serve as a citable map for the next decade of work at this intersection.",[2967,5608],{"text":5609},"The choice of risk measure, the calibration of ambiguity, and the design of exploration are not implementation details: they are the system.",[1829,5611],{},[1832,5613,1742],{"id":1741},[1806,5615,5616,5619],{},[1824,5617,5618],{},"Active set"," — In a constrained optimization problem, the subset of constraints that bind at the optimum.",[1806,5621,5622,5625],{},[1824,5623,5624],{},"ADP"," — Approximate dynamic programming; family of methods for solving high-dimensional MDPs via value-function approximation.",[1806,5627,5628,5631],{},[1824,5629,5630],{},"ARL"," — Average run length; expected time between false alarms in a sequential test.",[1806,5633,5634,5637],{},[1824,5635,5636],{},"BwK"," — Bandits with Knapsacks; bandit framework with global resource constraints.",[1806,5639,5640,5643],{},[1824,5641,5642],{},"Coherent risk measure"," — A functional satisfying monotonicity, sub-additivity, positive homogeneity, and translation invariance.",[1806,5645,5646,5648],{},[1824,5647,5405],{}," — Distribution-free prediction-interval framework with finite-sample coverage guarantee under exchangeability.",[1806,5650,5651,5654],{},[1824,5652,5653],{},"CUSUM"," — Cumulative-sum sequential test for detecting a change in distribution.",[1806,5656,5657,5659],{},[1824,5658,4845],{}," — Conditional Value-at-Risk (Expected Shortfall); coherent tail risk measure.",[1806,5661,5662,5664],{},[1824,5663,5354],{}," — Double machine learning; orthogonal-score estimation of treatment effects with machine-learned nuisances.",[1806,5666,5667,5670],{},[1824,5668,5669],{},"DRO"," — Distributionally robust optimization.",[1806,5672,5673,5676],{},[1824,5674,5675],{},"MDP \u002F CMDP"," — Markov Decision Process \u002F Constrained MDP.",[1806,5678,5679,5681],{},[1824,5680,5391],{}," — Model predictive control; receding-horizon optimal control.",[1806,5683,5684,5686],{},[1824,5685,4871],{}," — Single-period stochastic-demand inventory problem with closed-form quantile solution.",[1806,5688,5689,5692],{},[1824,5690,5691],{},"OCO"," — Online convex optimization; sequential convex-loss minimization with adversarial environments.",[1806,5694,5695,5698],{},[1824,5696,5697],{},"ROAS \u002F true ROAS"," — Return on ad spend (reported); contribution-margin-adjusted return on ad spend (true).",[1806,5700,5701,5704],{},[1824,5702,5703],{},"RTB"," — Real-time bidding; per-impression bidding in display advertising.",[1806,5706,5707,5710],{},[1824,5708,5709],{},"SAA"," — Sample average approximation.",[1806,5712,5713,5716],{},[1824,5714,5715],{},"Shadow price"," — Lagrange multiplier on a binding constraint in an optimization problem.",[1806,5718,5719,5722],{},[1824,5720,5721],{},"Thompson Sampling"," — Bayesian bandit algorithm sampling actions in proportion to posterior optimality probability.",[1806,5724,5725,5728],{},[1824,5726,5727],{},"VaR"," — Value-at-Risk; quantile-based risk measure (non-coherent).",[1806,5730,5731,5734],{},[1824,5732,5733],{},"Wasserstein distance"," — Optimal-transport metric on probability measures.",[1829,5736],{},[1832,5738,1745],{"id":1744},[1806,5740,5741],{},"Search strings used for the systematic literature search of Section 2.2 (selected; full inventory available on request):",[1806,5743,5744,2546,5747],{},[1809,5745,5746],{},"Finance:",[5748,5749,5750],"code",{},"(\"mean-variance\" OR \"Markowitz\" OR \"CVaR\" OR \"expected shortfall\" OR \"robust portfolio\") AND (\"optimization\" OR \"allocation\")",[1806,5752,5753,2546,5756],{},[1809,5754,5755],{},"Execution:",[5748,5757,5758],{},"(\"Almgren-Chriss\" OR \"optimal execution\" OR \"implementation shortfall\") AND (\"transaction cost\" OR \"market impact\")",[1806,5760,5761,2546,5764],{},[1809,5762,5763],{},"Operations:",[5748,5765,5766],{},"(\"newsvendor\" OR \"(s,S) policy\" OR \"base-stock\" OR \"revenue management\") AND (\"stochastic\" OR \"robust\" OR \"data-driven\")",[1806,5768,5769,2546,5772],{},[1809,5770,5771],{},"Insurance:",[5748,5773,5774],{},"(\"Cramer-Lundberg\" OR \"ruin probability\" OR \"Solvency II\" OR \"extreme value theory\")",[1806,5776,5777,2546,5780],{},[1809,5778,5779],{},"Bandits:",[5748,5781,5782],{},"(\"multi-armed bandit\" OR \"Thompson sampling\" OR \"UCB\" OR \"contextual bandit\" OR \"bandits with knapsacks\")",[1806,5784,5785,2546,5788],{},[1809,5786,5787],{},"Causal:",[5748,5789,5790],{},"(\"double machine learning\" OR \"causal forest\" OR \"synthetic control\" OR \"potential outcomes\" OR \"average treatment effect\")",[1806,5792,5793,2546,5796],{},[1809,5794,5795],{},"Advertising:",[5748,5797,5798],{},"(\"real-time bidding\" OR \"ad budget\" OR \"ROAS\" OR \"marketplace advertising\" OR \"sponsored search\")",[1806,5800,5801,2546,5804],{},[1809,5802,5803],{},"Forecasting:",[5748,5805,5806],{},"(\"M5 competition\" OR \"hierarchical forecasting\" OR \"DeepAR\" OR \"N-BEATS\" OR \"temporal fusion transformer\")",[1806,5808,5809,2546,5812],{},[1809,5810,5811],{},"DRO:",[5748,5813,5814],{},"(\"distributionally robust\" OR \"Wasserstein DRO\" OR \"ambiguity set\")",[1806,5816,5817,2546,5820],{},[1809,5818,5819],{},"MPC:",[5748,5821,5822],{},"(\"model predictive control\" OR \"receding horizon\" OR \"stochastic MPC\")",[1829,5824],{},[1832,5826,1748],{"id":1747},[1806,5828,5829],{},[1809,5830,5831],{},"Table C1 — Risk-measure properties.",[1873,5833,5834,5856],{},[1876,5835,5836],{},[1879,5837,5838,5841,5844,5847,5850,5853],{},[1882,5839,5840],{},"Measure",[1882,5842,5843],{},"Coherent",[1882,5845,5846],{},"Convex",[1882,5848,5849],{},"Distortion",[1882,5851,5852],{},"Linear-program reformulable",[1882,5854,5855],{},"Standard regulatory use",[1889,5857,5858,5876,5893,5908,5925],{},[1879,5859,5860,5862,5865,5868,5870,5873],{},[1894,5861,4828],{},[1894,5863,5864],{},"No",[1894,5866,5867],{},"Yes (with mean)",[1894,5869,5864],{},[1894,5871,5872],{},"Yes (QP)",[1894,5874,5875],{},"Capital allocation (legacy)",[1879,5877,5878,5880,5882,5884,5887,5890],{},[1894,5879,5727],{},[1894,5881,5864],{},[1894,5883,5864],{},[1894,5885,5886],{},"Yes",[1894,5888,5889],{},"No (MIP in general)",[1894,5891,5892],{},"Basel II, Solvency II",[1879,5894,5895,5897,5899,5901,5903,5905],{},[1894,5896,4845],{},[1894,5898,5886],{},[1894,5900,5886],{},[1894,5902,5886],{},[1894,5904,5886],{},[1894,5906,5907],{},"Basel III FRTB",[1879,5909,5910,5913,5915,5917,5919,5922],{},[1894,5911,5912],{},"Spectral",[1894,5914,5886],{},[1894,5916,5886],{},[1894,5918,5886],{},[1894,5920,5921],{},"Yes (LP)",[1894,5923,5924],{},"Internal models",[1879,5926,5927,5930,5933,5935,5937,5940],{},[1894,5928,5929],{},"Entropic",[1894,5931,5932],{},"No (under sub-add.)",[1894,5934,5886],{},[1894,5936,5864],{},[1894,5938,5939],{},"No (cone)",[1894,5941,5942],{},"Robust control",[1806,5944,5945],{},[1809,5946,5947],{},"Table C2 — Exploration-mechanism comparison.",[1873,5949,5950,5969],{},[1876,5951,5952],{},[1879,5953,5954,5957,5960,5963,5966],{},[1882,5955,5956],{},"Mechanism",[1882,5958,5959],{},"Regret bound",[1882,5961,5962],{},"Computational cost",[1882,5964,5965],{},"Strengths",[1882,5967,5968],{},"Weaknesses",[1889,5970,5971,5992,6012,6029,6050,6067],{},[1879,5972,5973,5978,5981,5986,5989],{},[1894,5974,5975,5977],{},[1897,5976],{"html":4935,"display":-1},"-greedy",[1894,5979,5980],{},"Linear",[1894,5982,5983],{},[1897,5984],{"html":5985,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>O\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>K\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">O(K)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">K\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",[1894,5987,5988],{},"Simplicity",[1894,5990,5991],{},"Asymptotically suboptimal",[1879,5993,5994,5996,6001,6006,6009],{},[1894,5995,4916],{},[1894,5997,5998],{},[1897,5999],{"html":6000,"display":-1},"\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath 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