[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82752-en":3,"doc-seo-82752-105":29,"detail-sidebar-cat-0-en-105":83},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82752,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","New bounds on randomized metric distortion of top-k voting","New bounds establish optimal worst-case and instance-specific metric distortion for randomized social choice mechanisms under first-choice voting. For voting where each voter reports only their top candidate, a proportional-to-the n1-th power of vote share rule achieves the optimal worst-case distortion of 3 − 2/n. The paper also derives a uniquely ν-optimal rule for instance distortion and extends results to top-k voting with closed-form worst-case distortion formulas for any k ≥ 2, improving known cyclic-profile lower bounds.","arXiv :2607 .03564v 1 [ cs .GT] 3 Jul 2026  \nNew bounds on randomized metric distortion of  \ntop-k voting  \nAlec Sun 1 and Daniel Zhu2  \n1 University of Chicago  \n2 The Harker School  \nAbstract. We prove new upper and lower bounds on metric distortion for randomized social choice mechanisms. Under first-choice voting where each voter reports only their most preferred candidate, we show that selecting a candidate with probability proportional to the n1-th power of their vote share achieves the optimal worst-case distortion of 3 − ~~2~~n .  \nThis is a simpler single-rule alternative to prior work. We also study instance-specific metric distortion of first-choice mechanisms in terms of the vote vector ν . We show that there is a uniquely optimal rule achieving distortion 1 + Pi~~ 2~~1~~i~~νi~~ ~~ . Finally, we extend our results to top-k voting where each voter reports their k nearest candidates. We derive a formula for the worst-case distortion for any k ≥ 2. For the cyclic profile family this improves the previously best known 3 − ⌊~~ 2n~~k⌋ lower bound.  \nKeywords: Voting rules · Metric distortion · Social choice  \n1 Introduction  \nSocial choice theory studies the aggregation of individual preferences into collective decisions. The distortion framework, introduced by [8], evaluates mechanisms by the worst-case ratio of their achieved social cost to the optimum, assuming agents have latent cardinal utilities, of which only ordinal rankings are revealed. This perspective, broadened by [3], treats voting rules as approximation algorithms for social welfare. Under the metric distortion model of [1], agents and candidates are embedded ina metric space, so each agent’s cost equals their distance to the elected candidate.  \nWhile mechanisms could in principle solicit full ordinal rankings, demanding a complete permutation of all n candidates imposes a large cognitive and communication burden. The first-choice voting setting—where each voter reports only their top choice—aligns with widely deployed plurality voting and dramatically reduces this overhead. [6] showed the worstcase distortion for randomized first-choice mechanisms is at least 3 − ~~2~~n , and [7] constructed a rule that achieves this bound.  \n2 Alec Sun and Daniel Zhu  \nOur contributions. Although [7] achieved optimal first-choice distortion 3 − ~~2~~n, they used a randomized combination of two rules; in this paper we provide a simpler rule achieving the same distortion. Moreover, classic distortion bounds evaluate the worst case uniformly without considering the specific vote vector ν, leaving open the instance-specific question posed by [2]: What is the best possible distortion for a given ν? We answer this question in our paper as well. Finally, we generalize our results to top-k voting where each voter reports an ordered list of their k nearest candidates. To summarize, our main contributions are:  \n– An exact instance-specific formula (Corollary 1): the exact dis  \ntortion of any first-choice mechanism, obtained by matching our new lower bound (Theorem 1) with a known upper bound.  \n– A simple proportional-to-powers mechanism (Theorem 2): the  \n– unique single qp rule achieving optimal metric distortion 3 − ~~2~~n . The uniquely ν-optimal rule (Theorem 3): proportional-to- ~~ ~~1~~i~~νi , which strictly dominates every other randomized first-choice mechanism for every ν .  \n– An exact top-k distortion formula (Theorem 5): for every k, the exact worst-case distortion of any mechanism, with a matching tight construction. For the cyclic profile family this gives a closed-form lower bound (Proposition 1) that improves the 3 − ⌊~~ 2n~~k⌋ bound of [6] .  \n1.1 Related work  \nUtilitarian social choice and distortion. [8] introduced distortion to quantify the information loss when mechanisms rely on ordinal preferences rather than cardinal utilities. [3] broadened this to analyze optimal social choice functions under worst-and average-case models, treating voting rules as ap","cbCairg58rpy7ICs","https://ap.wps.com/l/cbCairg58rpy7ICs","pdf",566655,1,18,"English","en",105,"# Abstract\n# Introduction\n## Social choice theory and distortion framework\n## Metric distortion model\n# Contributions\n## Main contributions overview\n# Related work\n## Utilitarian social choice and distortion\n## Metric distortion\n## Randomized mechanisms and first-choice voting\n# Preliminaries\n## Social cost and vote vectors\n## Randomized first-choice mechanisms\n## Metric consistency and distortion","[{\"question\":\"How is the top-k voting extension handled in the paper?\",\"answer\":\"For each k ≥ 2, the paper provides a formula for the worst-case distortion of any mechanism and shows that, for the cyclic profile family, the bound improves the previously best known lower bound.\"}]",1784182693,45,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":27},"new-bounds-on-randomized-metric-distortion-of-top-k-voting","",{"@graph":35,"@context":77},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/new-bounds-on-randomized-metric-distortion-of-top-k-voting/82752/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"How is the top-k voting extension handled in the paper?","Question",{"text":75,"@type":76},"For each k ≥ 2, the paper provides a formula for the worst-case distortion of any mechanism and shows that, for the cyclic profile family, the bound improves the previously best known lower bound.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]