[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83646-en":3,"doc-seo-83646-105":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},83646,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Facility Location Game with Envy Ratio","We study the one-facility location game on a real line using a new egalitarian objective, the envy ratio, defined as the maximum ratio between any two agents’ utilities. The goal is to design strategyproof and group strategyproof mechanisms that minimize this envy-ratio objective. Two interval-restriction settings are analyzed: both agents and facility constrained on a fixed interval, and agents unconstrained on the real line with the facility constrained relatively. Optimal solutions and best deterministic mechanisms are obtained, and bounds are derived for randomized mechanisms.","arXiv :2607 .02330v 1 [ cs .GT] 2 Jul 2026  \nFacility Location Game with Envy Ratio  \nYuan Dinga , Wenjing Liua,∗, Xin Chena , Qizhi Fanga , Qingqin Nongaa School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China  \nAbstract  \nWe study the one-facility location game on a real line with a new objective called envy ratio. The envy ratio, which is adopted from fair division and represents the egalitarianism, is defined as the maximum over the ratios between any two agents’ utilities. We are interested in strategyproof or group strategyproof mechanisms that can minimize the envy ratio objective.  \nWe consider the model in two settings that can capture natural scenarios: the facility location and all the agents’ locations are restricted on a fixed interval; every agent’s location can be any point on the real line but the facility location is restricted on a relative interval. In both settings, we obtain the optimal solution and the best deterministic strategyproof mechanism which is also group strategyproof. In the first setting, we provide a lower bound for randomized strategyproof mechanisms. In the second setting, we give a lower bound and two upper bounds for randomized strategyproof mechanisms. Keywords: approximate mechanism design, facility location, fairness, envy ratio, strategyproof  \n1. Introduction  \nApproximate mechanism design without money for facility location games has been extensively studied in the last ten years since the pathbreaking contribution of Procaccia & Tennenholtz (2009) . In the basic setting, the social planner (e.g., the local government) plans to design a mechanism which locates a facility (e.g., a library or a bus station) based on the reported locations  \n∗ Corresponding author.  \nEmail addresses: [dingyuan@ouc.edu.cn](dingyuan@ouc.edu.cn) (Yuan Ding), [liuwj@ouc.edu.cn](liuwj@ouc.edu.cn) (Wenjing  \nLiu), [cxin0307@163.com](cxin0307@163.com) (Xin Chen), [qfang@ouc.edu.cn](qfang@ouc.edu.cn) (Qizhi Fang),  \n[qqnong@ouc.edu.cn](qqnong@ouc.edu.cn) (Qingqin Nong)  \nfrom self-interested agents. Every agent has a connection cost which is usually her distance from the facility. On the one hand, the social planner wants to optimize a certain objective, such as minimizing the sum of individual costs (i.e., the social cost) . On the other hand, every agent who holds her location as private information, is self-interested and can misreport to decrease her own connection cost. Naturally, the social planner needs to design mechanisms which can optimize the objective while ensuring every agent’s truthful report (i.e., strategyproof or group strategyproof) .  \nBesides physical aspects, the facility location setting can be extended to many other applications. For example, every agent’s location can represent her qualitative political view and the facility location can be a political decision. In the foregoing scenarios, money or payment cannot be used as a medium of compensation due to ethical or legal considerations (Schummer & Vohra, 2007) . Moreover, considering that the optimal solution may be not strategyproof, the goal of the social planner becomes to design strategyproof mechanisms without money that have small approximation ratios. Our work belongs to the research agenda of approximate mechanism design without money.  \nMost of the existing work studies the objectives of minimizing the social cost (or equivalently, maximizing the social welfare) or minimizing the maximum cost (Procaccia & Tennenholtz, 2009; Lu et al., 2009, 2010; Chenget al. , 2013; Fotakis & Tzamos, 2014; Zhang & Li, 2014; Serafino & Ventre, 2014, 2015; Zou & Li, 2015; Yuan et al., 2016; Feigenbaum et al., 2017; Chen et al., 2018; Fong et al., 2018; Duan et al., 2019; Li et al., 2019; Mei et al., 2019) . From the perspective of economics, the social cost objective can represent the utilitarianism. By contrast, the maximum cost objective is often considered to be more fair and can represent the egalitarianism, which","cbCaimfMsxbfPBpY","https://ap.wps.com/l/cbCaimfMsxbfPBpY","pdf",398076,4,1,23,"English","en",105,"# Introduction\n## Our Results","[{\"question\":\"What is the envy ratio objective in the one-facility location game?\",\"answer\":\"The envy ratio measures egalitarianism by taking the maximum over ratios between any two agents’ utilities.\"},{\"question\":\"What does strategyproof (and group strategyproof) mean here?\",\"answer\":\"Mechanisms are designed so agents have no incentive to misreport their locations, and the property extends to groups of agents as well.\"},{\"question\":\"How do the paper’s two interval-restriction settings differ?\",\"answer\":\"In one setting, both all agents and the facility are restricted to a fixed interval; in the other, agents may be anywhere on the real line while only the facility is restricted by a relative interval.\"}]",1784189489,58,{"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},"facility-location-game-with-envy-ratio","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/facility-location-game-with-envy-ratio/83646/",{"url":52,"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","2026-07-25","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 envy ratio objective in the one-facility location game?","Question",{"text":75,"@type":76},"The envy ratio measures egalitarianism by taking the maximum over ratios between any two agents’ utilities.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does strategyproof (and group strategyproof) mean here?",{"text":80,"@type":76},"Mechanisms are designed so agents have no incentive to misreport their locations, and the property extends to groups of agents as well.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the paper’s two interval-restriction settings differ?",{"text":84,"@type":76},"In one setting, both all agents and the facility are restricted to a fixed interval; in the other, agents may be anywhere on the real line while only the facility is restricted by a relative interval.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"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":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"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"]