[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85186-en":3,"doc-seo-85186-105":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},85186,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Best-of-Both-Worlds Fairness for Mixed Goods and Chores","Study of fair division of indivisible items with additive utilities under a mixed goods-and-chores setting, where items can give non-negative utility to some agents and negative utility to others. The work targets a “best-of-both-worlds” allocation: exactly fair ex ante and approximately fair ex post. It focuses on envy-freeness (EF) and EF1. Main result shows ex-ante EF and ex-post EF1 can be achieved simultaneously via probabilistic Hall-type matrix decomposition, solved using Sion’s minimax theorem and biased flow networks.","arXiv :2607 . 10232v 1 [ cs .GT] 11 Jul 2026  \nBest-of-Both-Worlds Fairness for Mixed Goods and Chores  \nHaris Aziz1, Xiaolin Bu2, Xinhang Lu3, Simon Mackenzie1, Mashbat Suzuki1,  \nBiaoshuai Tao2, and Toby Walsh1  \n1UNSW Sydney,  \n{haris . aziz,simon . w . mackenzie,[mashbat. suzuki](mashbat. suzuki) , [t. walsh}@unsw. edu. au](t. walsh}@unsw. edu. au)[ ](t. walsh}@unsw. edu. au)2 Shanghai Jiao Tong University, {lin_bu,[bstao}@sjtu. edu. cn](bstao}@sjtu. edu. cn)[ ](bstao}@sjtu. edu. cn)3 Kyushu University, [xinhang. lu@inf. kyushu-u. ac. jp](xinhang. lu@inf. kyushu-u. ac. jp)  \nAbstract  \nWe study the fundamental problem of fairly dividing indivisible items among agents with additive utilities. In our model, an item can be a good yielding non-negative utilities to some agents and simultaneously a chore yielding negative utilities to others. We take the best-ofboth-worlds perspective and our goal is to construct a randomized allocation that is exactly fair ex ante while also being supported on ex post approximately fair allocations. The fairness notions examined in this paper are envy-freeness (EF) and its well-known relaxation envyfreeness up to one item (EF1) . Our main result is that ex-ante EF and ex-post EF1 can be achieved simultaneously. To achieve this, we introduce a novel probabilistic Hall-type matrix decomposition that intricately correlates the fractional assignments of goods and chores. We resolve this decomposition problem by combining continuous minimax duality—via Sion’s minimax theorem—with carefully designed biased flow networks.  \n1 Introduction  \nThe fair division of valuable scarce resources or undesirable burden among agents is a common problem that arises frequently in society, and has attracted extensive and ongoing interest in the fields of mathematics, economics, operations research, and computer science [Brams and Taylor, 1996; Brandt et al., 2016; Moulin, 2003, 2019; Rothe, 2024] . The quintessential notion of fairness is arguably envy-freeness (EF), which stipulates that each agent values their own allocated bundle weakly more than any other bundle in the allocation [Foley, 1967] . In other words, each agent does not envy any other agent.  \nIn this paper, we are concerned with the fundamental problem of fairly dividing indivisible items. Due to its potential to capture a wide range of real-world applications, indivisible-items allocation has received significant attention in recent years, with focus mainly on allocating (purely) goods [Amanatidis et al., 2023] or (purely) chores [Guo et al., 2023] . Note that goods (resp., chores) are items yielding non-negative (resp., negative) utilities to the agents. In addition to theoretical research, the developed algorithms have already been deployed in real life and made accessible to the public through web applications; see, e.g., Course Match [Budish et al., 2017], Spliddit [Goldman and Procaccia, 2015] and Kajibuntan [Igarashi and Yokoyama, 2023] .  \nWe focus on a more general model that allows agents to report more expressive utilities over the set of indivisible items to be divided. More specifically, in our model of mixed goods and chores, an indivisible item can be a good to some agents and simultaneously a chore to other  \nagents. Use cases include household chore division between couples or tenants, the division of assets and liabilities between multiple parties, etc. In our first example, an agent may want to avoid cooking, while another agent may find cooking is desirable and relaxing. Similarly, in our second example, while some stakeholders seek risky assets as they have the potential to yield high returns, others may be risk-averse and would like to receive low-risk assets.  \nWhen items are indivisible, envy-freeness is too demanding to achieve, as envy-free allocations may not always exist. One common way to circumvent this issue is to weaken envy-freeness. For instance, envy-freeness up to one item (EF1) is a very well-known relaxation of ","cbCaig4mQ9wWSmMy","https://ap.wps.com/l/cbCaig4mQ9wWSmMy","pdf",390613,2,1,37,"English","en",105,"# Abstract\n# Introduction\n## Fair division and envy-freeness\n## Mixed goods and chores model\n## Relaxations and randomized fairness\n## Best-of-both-worlds approach","[{\"question\":\"What fairness concepts are analyzed for mixed goods and chores?\",\"answer\":\"The paper examines envy-freeness (EF) and its relaxation envy-freeness up to one item (EF1). EF1 allows envy to be removed by hypothetically removing one item from an agent’s bundle.\"},{\"question\":\"How can an item be both a good and a chore in this model?\",\"answer\":\"An indivisible item may yield non-negative utility (a good) to some agents while yielding negative utility (a chore) to other agents, enabling a unified mixed setting.\"},{\"question\":\"What is the main result about randomized allocations?\",\"answer\":\"Ex-ante EF and ex-post EF1 can be achieved simultaneously. The construction uses a probabilistic Hall-type matrix decomposition correlating assignments of goods and chores.\"}]",1784201610,93,{"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},"best-of-both-worlds-fairness-for-mixed-goods-and-chores","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/best-of-both-worlds-fairness-for-mixed-goods-and-chores/85186/",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","2026-07-24","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 fairness concepts are analyzed for mixed goods and chores?","Question",{"text":75,"@type":76},"The paper examines envy-freeness (EF) and its relaxation envy-freeness up to one item (EF1). EF1 allows envy to be removed by hypothetically removing one item from an agent’s bundle.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How can an item be both a good and a chore in this model?",{"text":80,"@type":76},"An indivisible item may yield non-negative utility (a good) to some agents while yielding negative utility (a chore) to other agents, enabling a unified mixed setting.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main result about randomized allocations?",{"text":84,"@type":76},"Ex-ante EF and ex-post EF1 can be achieved simultaneously. The construction uses a probabilistic Hall-type matrix decomposition correlating assignments of goods and chores.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]