[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86459-en":3,"doc-seo-86459-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86459,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Fair Division with Binary Valuations: Characterizations","Fair allocation of indivisible goods under binary (0/1) valuations is characterized through axiomatic properties. For any number of agents, the maximum Nash welfare rule is shown to coincide with the leximin rule and with all additive welfarist rules using strictly concave functions. It is proven to be the unique rule satisfying envy-freeness up to one good, strategyproofness, neutrality, minimal completeness, and invariance under disapproving unassigned goods. For two agents, an alternative characterization replaces that condition with non-redundancy and resource-monotonicity, with necessity of every axiom.","arXiv :2607 . 10064v1 [ econ .TH] 11 Jul 2026  \nFair Division with Binary Valuations: Characterizations  \nFlorian Brandl University of Bonn  \nWarut Suksompong National University of Singapore  \nNicholas Teh  \nUniversity of Oxford  \nWe consider the fair allocation of indivisible goods with binary valuations. In this setting, the maximum Nash welfare rule, the leximin rule, and all additive welfarist rules with a strictly concave function coincide. We show that for any number of agents, this rule is the only rule that satisfies envyfreeness up to one good, strategyproofness, neutrality, minimal completeness, and invariance under disapproving unassigned goods (IDU) . Moreover, we present an alternative characterization for two agents, where we replace IDU with non-redundancy and resource-monotonicity. In both characterizations, all axioms are necessary.  \n1. Introduction  \nThe fair allocation of resources is a central problem at the intersection of economics and computer science, with applications ranging from divorce settlement to inheritance division to load balancing [Brams and Taylor, 1996, Robertson and Webb, 1998, Moulin, 2003] . A significant portion of recent work in the area investigates fairness considerations when allocating indivisible goods such as jewelry, artwork, electronics, toys, and furniture [Goldman and Procaccia, 2014, Amanatidis et al., 2023] .  \nA key fairness desideratum is envy-freeness, which states that each agent should value her own bundle of goods at least as much as every other agent’s bundle. However, the trivial instance of two agents and a single valuable good shows that envy-freeness cannot always be attained (if this good must be allocated), so a relaxation is necessary. One of the most intuitive relaxations is envy-freeness up to one good (EF1) . An allocation is said to be EF1 if any envy that an agent has towards another agent can be eliminated by removing a single good from the latter agent’s bundle. In addition to fairness, one may wish to achieve some form of efficiency. A commonly studied efficiency property is Pareto-optimality (PO), which stipulates that no other allocation makes at least one agent better off without making another agent worse off.  \nA popular method for simultaneously achieving these two properties is the maximum Nash welfare (MNW) rule, which selects an allocation maximizing the product of the agents’ values. In a far-reaching paper, Caragiannis et al. [2019] showed that under additive valuations, an MNW allocation always satisfies EF1 and PO. Subsequently, Yuen and Suksompong [2023] characterized MNW as the unique rule within the class of welfarist rules—rules that choose an allocation maximizing some increasing function of the agents’ values—that ensures EF1 under additive valuations. This extends an earlier result by Suksompong [2023], which provided this characterization only within the class of additive welfarist rules, where the function can be expressed as a sum of some function of each agent’s value (for MNW, this latter function is the logarithm function) .  \nDespite these advantages, MNW has its drawbacks too. For example, it fails to be strategyproof, which implies that an agent can sometimes benefit from misreporting her true preferences [Klaus and Miyagawa, 2002, Halpern et al., 2020] . It also violates resource-monotonicity, so an agent could become worse off when an extra good is added to the pool [Chakraborty et al., 2021] . Furthermore, finding or even approximating the maximum Nash welfare within a certain constant factor is a computationally hard problem [Lee, 2017] . Note that many rules other than MNW satisfy EF1 and PO. In particular, since EF1 and PO are properties of specific instances and do not connect different instances, we may consider an arbitrary instance and choose an EF1 and PO allocation different from the allocation chosen by MNW. It thus appears to be a challenging task to identify a set of axioms that MNW satisfies but no other r","cbCailfAuGyMVWfx","https://ap.wps.com/l/cbCailfAuGyMVWfx","pdf",396393,5,1,32,"English","en",105,"# Introduction\n## Fair allocation and fairness efficiency goals\n## Maximum Nash welfare and known properties\n## Binary valuations as an important special case\n## Main characterizations and uniqueness results","[{\"question\":\"What does envy-freeness up to one good (EF1) mean in this work?\",\"answer\":\"EF1 requires that any envy an agent has toward another can be removed by removing a single good from the other agent’s bundle.\"},{\"question\":\"Why is the study focused on binary valuations?\",\"answer\":\"With binary (0/1) valuations, the maximum Nash welfare rule becomes strategyproof, resource-monotone, and computable in polynomial time, addressing drawbacks present in more general settings.\"},{\"question\":\"What axioms uniquely identify the maximum Nash welfare rule for any number of agents?\",\"answer\":\"It is the only rule satisfying EF1 up to one good, strategyproofness, neutrality, minimal completeness, and invariance under disapproving unassigned goods.\"}]",1784211857,81,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"fair-division-with-binary-valuations-characterizations","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/fair-division-with-binary-valuations-characterizations/86459/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-28","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does envy-freeness up to one good (EF1) mean in this work?","Question",{"text":76,"@type":77},"EF1 requires that any envy an agent has toward another can be removed by removing a single good from the other agent’s bundle.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why is the study focused on binary valuations?",{"text":81,"@type":77},"With binary (0/1) valuations, the maximum Nash welfare rule becomes strategyproof, resource-monotone, and computable in polynomial time, addressing drawbacks present in more general settings.",{"name":83,"@type":74,"acceptedAnswer":84},"What axioms uniquely identify the maximum Nash welfare rule for any number of agents?",{"text":85,"@type":77},"It is the only rule satisfying EF1 up to one good, strategyproofness, neutrality, minimal completeness, and invariance under disapproving unassigned 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