[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86313-en":3,"doc-seo-86313-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86313,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Philosopher and Prophet Inequalities for Divisible Items","Online welfare maximization with divisible resources studies sequentially arriving agents who reveal multi-dimensional concave valuations over multiple divisible items drawn from known distributions. Each agent must be assigned an irrevocable fractional bundle under per-item unit-supply constraints. For monotone concave DR-submodular valuations, approximation algorithms achieve a 2/3 guarantee against the philosopher benchmark, using a capped online contention resolution via a low-dimensional concave relaxation. Hardness shows optimal online computation is #P-hard even with one divisible item. A tight prophet inequality follows from a fixed-price auction based on Aumann–Shapley supporting prices.","arXiv :2607 . 1 1742v 1 [ cs .DS] 13 Jul 2026  \nPhilosopher and Prophet Inequalities for Divisible Items  \nThiago Oliveira∗ Mohit Singh† Sahil Singla‡  \nJuly 14, 2026  \nAbstract  \nWe study online welfare maximization with divisible resources. A sequence of n players arrive one by one; upon arrival, each player draws a valuation function over m divisible items from a known distribution, reveals this valuation, and must be allocated an irrevocable fractional bundle subject to unit supply constraints. While online welfare maximization has been extensively studied for indivisible items and combinatorial valuations, much less is known when the resources are divisible and players have multi-dimensional concave valuations.  \nWe give approximation algorithms for monotone concave valuations satisfying diminishing returns. Our main result is a 2/3-approximation to the optimal online policy, also known as the philosopher benchmark. The algorithm is guided by a low-dimensional concave relaxation of the online benchmark and rounds it via a new single-item capped online contention resolution scheme. This Capped-OCRS problem allocates to each realized type no more than its prescribed fractional bundle while preserving a 2/3-fraction of that bundle in expectation. Its analysis uses a submartingale potential for the remaining supply. On the hardness side, we show that computing the optimal online policy is \\#P-hard even for a single divisible item.  \nWe also obtain a tight prophet inequality against the offline hindsight optimum. We show that a fixed-price auction with one linear per-unit price for each original divisible item achievesa 1/2-approximation to the offline/prophet benchmark. The prices are obtained by aggregating Aumann–Shapley supporting prices, a continuous analogue of supporting prices for submodular/XOS set functions, and yield simple item prices rather than copy-dependent prices arising from discretization. The factor 1/2 for the prophet benchmark is information-theoretically tight even for one item with linear valuations.  \n∗ ([toliveira9@gatech.edu](toliveira9@gatech.edu)) H. Milton Stewart School of Industrial and Systems Engineering, Georgia Tech. Supported in part by NSF awards CCF-2440113, 2504994, 2106444 and the 2026 Spring ARC-ACO Fellowship.  \n†([mohit.singh@isye.gatech.edu](mohit.singh@isye.gatech.edu)) H. Milton Stewart School of Industrial and Systems Engineering, Georgia Tech. Supported in part by NSF awards CCF-2504994, 2106444 .  \n‡([ssingla@gatech.edu](ssingla@gatech.edu)) School of Computer Science, Georgia Tech. Supported in part by NSF awards CCF-2327010 and CCF-2440113 .  \n1 Introduction  \nOnline welfare maximization asks how to allocate limited resources to agents arriving over time. In its Bayesian form, player i ∈ [n] draws a valuation Vi ∼ Di, reveals it upon arrival, and must be allocated an irrevocable bundle from the remaining resources. The goal is to maximize expected social welfare. This model is a common abstraction in online algorithms and also underlies static posted-price mechanisms in online market design [EIV23, Luc17 , FGL15 , DFKL20] .  \nMost of the literature on online welfare maximization focuses on indivisible items: each item can be allocated to at most one player, and players value subsets of items, e.g. , via additive, submodular, or XOS valuations. For example, fixed posted prices yield a 1/2-approximation to the offline (a.k.a. prophet) benchmark for submodular (or even XOS) valuations [FGL15], and recent work gives improved guarantees against the online (a.k.a. philosopher) benchmark for unit-demand and submodular valuations [PPSW24, BDP+25 , STW26] .  \nHowever, many modern resources are naturally divisible—e.g., CPU time, bandwidth, energy, or budget—and should be allocated fractionally. This motivates us to study the problem of maximizing Online Welfare with Divisible Items (OWDI): there are m divisible items, each with unit supply1 , and players arrive sequentially. When play","cbCair9jVTtbBk5C","https://ap.wps.com/l/cbCair9jVTtbBk5C","pdf",502230,3,1,30,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What is the online welfare maximization model for divisible items studied in the document?\",\"answer\":\"A sequence of n players arrives one by one, each drawing and revealing a valuation over m divisible items. The algorithm must allocate each player an irrevocable fractional bundle while respecting per-item unit supply constraints, aiming to maximize expected social welfare.\"},{\"question\":\"What approximation guarantee is provided against the philosopher benchmark?\",\"answer\":\"The main result gives a 2/3-approximation to the optimal online policy (the philosopher benchmark) for monotone concave valuations with diminishing returns, using a concave relaxation and a capped online contention resolution scheme.\"},{\"question\":\"How is the prophet inequality achieved and how tight is it?\",\"answer\":\"A fixed-price auction with one linear per-unit price per original item attains a 1/2-approximation to the offline hindsight (prophet) benchmark. The 1/2 factor is information-theoretically tight even for one item with linear valuations.\"}]",1784210415,76,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"philosopher-and-prophet-inequalities-for-divisible-items","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/philosopher-and-prophet-inequalities-for-divisible-items/86313/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the online welfare maximization model for divisible items studied in the document?","Question",{"text":75,"@type":76},"A sequence of n players arrives one by one, each drawing and revealing a valuation over m divisible items. The algorithm must allocate each player an irrevocable fractional bundle while respecting per-item unit supply constraints, aiming to maximize expected social welfare.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What approximation guarantee is provided against the philosopher benchmark?",{"text":80,"@type":76},"The main result gives a 2/3-approximation to the optimal online policy (the philosopher benchmark) for monotone concave valuations with diminishing returns, using a concave relaxation and a capped online contention resolution scheme.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the prophet inequality achieved and how tight is it?",{"text":84,"@type":76},"A fixed-price auction with one linear per-unit price per original item attains a 1/2-approximation to the offline hindsight (prophet) benchmark. 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