[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82744-en":3,"doc-seo-82744-105":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},82744,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Random Serial Dictatorship is √2-Envy-Free","House allocation problem with cardinal utilities is studied through random serial dictatorship (RSD), a central mechanism known for simplicity and classic exact property characterizations. A core unresolved issue is quantitative fairness: although RSD is often considered envy-free ex ante, it is not envy-free in expectation. The envy-ratio is introduced as a worst-case measure of expected-utility envy between agents. The main theorem gives an upper bound matching the prior factor-√2 lower bound, proving √2-envy-freeness. Extensions to randomized round-robin and iterated-RSD are analyzed across valuation classes with tight or nearly tight bounds.","arXiv :2607 .03463v 1 [ cs .GT] 3 Jul 2026  \nRandom Serial Dictatorship is √2-Envy-Free  \nFrank Connor 1 , Max Dupr´e la Tour 1 , Louis-Roy Langevin 1 , Vishnu V. Narayan2 ,  \nNdiam´e Ndiaye 1 , Neil Rahman 1 , and Adrian Vetta 1  \n1 McGill University  \n2 Indian Institute of Technology Bombay  \nJuly 3, 2026  \nAbstract  \nWe analyze the house allocation problem, in which a set of agents must be matched to a set of objects for which the agents have cardinal utilities. A central mechanism for this problem is random serial dictatorship (RSD), which has long served as a canonical subject of study due to its simplicity and the existence of exact characterizations by its properties. Despite this extensive understanding, a basic quantitative question about the fairness of this mechanism remains unresolved. Although RSD is often viewed as fair ex ante, surprisingly, it is not envyfree in expectation. We quantify its deviation from envy-freeness via the envy-ratio, which is defined as the maximum, over all instances and pairs of agents, of the ratio between an agent’s expected utility for another agent’s random object and for its own random object. Prior work shows a factor- √2 ≈ 1.414 lower bound on the envy-ratio of the RSD mechanism. Our headline result is an upper bound that matches this constant factor, proving that RSD is √2-envy-free in the house allocation problem.  \nWe further analyze the two natural extensions of RSD (the so-called randomized round-robin mechanism and the iterated-RSD mechanism) to settings with unequal numbers of agents and objects and with more general valuation classes. For additive valuations, we establish constantfactor bounds on the envy-ratio of both mechanisms, showing that this ratio increases to at least 1 .5 and at most 1 + (1/ √2) ≈ 1.707 for randomized round-robin, but remains exactly √2 for iterated-RSD. For submodular valuations, we prove constant-factor upper and lower bounds for both mechanisms, leaving only a small constant gap in both cases. For the more general classes of XOS and subadditive valuations, we present a tight analysis of the envy-ratio of both mechanisms, showing that this ratio is unbounded in the number of agents. These results provide the first tight or nearly tight quantitative guarantees on the extent to which random serial dictatorship and its natural generalizations approximate envy-freeness.  \n1 Introduction  \nHow should a collection of objects be fairly matched to a set of agents, who have varying preferences over the objects, so that each agent is assigned exactly one object? This matching problem, typically called the house allocation problem, arises in a multitude of situations such as the assignment of dorms to students, of hotel rooms to guests, and of parking spaces to employees.  \nWe are especially grateful to Simon Mauras, without whom this work would not have been possible. We thank Michal Feldman and Tomasz Ponitka for helpful discussions. This project was supported by NSERC Discovery Grant 2022-04191.  \n{frank.connor,louis-roy.langevin,vishnu.narayan,ndiame.ndiaye,[neil.rahman](neil.rahman}@mail.mcgill.ca)[}](neil.rahman}@mail.mcgill.ca)[@mail.mcgill.ca](neil.rahman}@mail.mcgill.ca) , [maxduprelatour@gmail.com](maxduprelatour@gmail.com) , [adrian.vetta@mcgill.ca](adrian.vetta@mcgill.ca).  \nPerhaps the simplest solution to the house allocation problem is the following: the agents line up in some order, and each agent in turn selects its most valuable object among the remaining objects until every agent is allocated an object. This straightforward mechanism, called a serial dictatorship, has been extensively studied over the past few decades. Despite its simplicity, this mechanism has several desirable properties, such as strategyproofness (no agent can benefit by misreporting its preferences), non-bossiness (no agent can change another agent’s object without changing its own object), and neutrality (when the objects are permuted, the assignment is permuted acc","cbCaimQYTjayZ726","https://ap.wps.com/l/cbCaimQYTjayZ726","pdf",532217,2,1,32,"English","en",105,"# Introduction\n## House allocation problem and fairness goals\n## Serial dictatorship and random serial dictatorship\n## Envy-freeness and quantitative deviation via the envy-ratio","[{\"question\":\"What is random serial dictatorship (RSD) in the house allocation setting?\",\"answer\":\"Agents are randomly ordered, and each agent sequentially selects its most valuable available object. This random priority extension of serial dictatorship defines RSD.\"},{\"question\":\"Why is RSD not envy-free in expectation?\",\"answer\":\"Even though RSD is often viewed as fair ex ante, the mechanism can produce envy when comparing agents’ expected utilities for each other’s assigned objects versus their own.\"},{\"question\":\"What does it mean that RSD is √2-envy-free?\",\"answer\":\"The envy-ratio, defined as the worst-case ratio of an agent’s expected utility for another agent’s random object to its expected utility for its own random object, is bounded above by √2. This matches an existing factor-√2 lower bound, proving √2-envy-freeness.\"}]",1784182630,81,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"random-serial-dictatorship-is-2-envy-free","",{"@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/random-serial-dictatorship-is-2-envy-free/82744/",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 random serial dictatorship (RSD) in the house allocation setting?","Question",{"text":75,"@type":76},"Agents are randomly ordered, and each agent sequentially selects its most valuable available object. This random priority extension of serial dictatorship defines RSD.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is RSD not envy-free in expectation?",{"text":80,"@type":76},"Even though RSD is often viewed as fair ex ante, the mechanism can produce envy when comparing agents’ expected utilities for each other’s assigned objects versus their own.",{"name":82,"@type":73,"acceptedAnswer":83},"What does it mean that RSD is √2-envy-free?",{"text":84,"@type":76},"The envy-ratio, defined as the worst-case ratio of an agent’s expected utility for another agent’s random object to its expected utility for its own random object, is bounded above by √2. This matches an existing factor-√2 lower bound, proving √2-envy-freeness.","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"]