[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82578-en":3,"doc-seo-82578-105":29,"detail-sidebar-cat-0-en-105":95},{"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},82578,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Multiwinner Voting with Spatial Preferences under Incomplete Information","Multiwinner elections with many candidates cannot be fully assessed by each voter, even though standard proportional fairness axioms such as EJR+ assume fully specified approval ballots. The work studies whether strong proportional representation can be guaranteed when each voter reveals only limited information. It uses a spatial setting, the Axis-aligned Random Rectangle Voter (ARRV) model, with rectangular issue tolerances and planar comparison queries, yielding an EJR+ committee under broad rectangular-preference distributions.","arXiv :2607 .0 1036v 1 [ cs .GT] 1 Jul 2026  \nMultiwinner Voting with Spatial Preferences under Incomplete  \nInformation  \nDrew Springham 1 , Edith Elkind2 , Bart de Keijzer 1 , and Maria Polukarov 1  \n1 King’s College London, London, UK  \n2 Northwestern University, Evanston, IL, USA  \nAbstract. In multiwinner elections with many candidates, as in participatory budgeting or largescale recommendation, voters cannot plausibly evaluate every candidate, yet standard proportionalfairness guarantees such as EJR+ are stated for fully specified approval ballots. We ask whether strong proportional representation can still be guaranteed while eliciting only a little from each voter. We study this in a spatial model, the Axis-aligned Random Rectangle Voter (ARRV) model, in which candidates occupy a d-dimensional issue space and each voter approves an axis-aligned hyper-rectangle: a tolerance interval on every issue. Preferences are revealed only through Planar queries, each comparing a voter’s tolerance to a candidate on a single issue. We give an algorithm returning an EJR+ committee for any distribution over rectangular preferences, using only O (dlog dk) Planar queries per voter in expectation given a sufficiently large electorate, independent of the number of candidates m, where dis the number of issues and k the committee size. The algorithm rests on a dimension-agnostic verifyor-fallback framework whose query cost is governed by two properties supplied by interchangeable modules. We describe such modules, yielding end-to-end guarantees for known, unknown, and smooth distributions.  \nKeywords: Multiwinner voting · Proportional representation · Incomplete information · Spatial preferences  \n1 Introduction  \nMultiwinner elections arise in settings where the pool of candidates is so large that voters cannot reasonably assess every option, such as participatory budgeting, recommender systems, and large-scale committee selection. Real-world participatory-budgeting elections have featured more than 150 candidate projects [9], while [Pol.is](Pol.is) (an online platform where users vote on participant-submitted comments to surface representative viewpoints [25]) has hosted debates, such as a citizens’ assembly on climate legislation in Austria, generating more than 1,000 comments [27] . At this scale it is unrealistic to expect voters to evaluate every candidate. Yet the standard axiomatic guarantees in multiwinner voting, such as the proportional fairness notions PJR+ and EJR+ [3], are stated for fully specified approval ballots. This raises a question: can we guarantee strong proportional fairness while eliciting a small amount of information from each voter?  \nWe study this question in a spatial model. Candidates and voters live in an issue space [0 , 1] d: each of the d dimensions can be thought of as an issue, each candidate has a fixed position on every issue, and a voter approves a candidate exactly when it is acceptable to her on every issue. Concretely, each voter’s acceptable region is an axis-aligned hyper-rectangle: on each issue she has a lower and an upper tolerance, and she approves the candidates that fall within her range on every issue. As an illustration, in participatory budgeting each project sits in a space of policy issues (say, environmental–economic and centralised–community-led), and a resident supports the projects that are not too extreme for her on any issue, i.e. those lying within her acceptable band on every axis. We call the resulting random model the Axisaligned Random Rectangle Voter (ARRV) model. To elicit preferences we use Planar queries: a Planar query fixes one issue and asks whether the voter’s entire acceptable range lies to one side of a given threshold on that issue, e.g. “would you reject every project more economically focused than candidate c?”  \nOur results and techniques. Our main contribution is an affirmative answer to the question above. In the ARRV model, we guarantee an EJR+ committee fo","cbCaie2E8loexN31","https://ap.wps.com/l/cbCaie2E8loexN31","pdf",864277,1,26,"English","en",105,"# Introduction\n## Our results and techniques\n## Verify-or-fallback framework\n## Modules for known/unknown/smooth distributions\n# Spatial model and query elicitation","[{\"question\":\"What problem does the document address in multiwinner elections?\",\"answer\":\"It examines how to guarantee strong proportional representation, such as EJR+, when voters cannot evaluate all candidates and only reveal a small amount of information.\"},{\"question\":\"How are voter preferences modeled in the ARRV framework?\",\"answer\":\"Candidates and voters live in a d-dimensional issue space, and each voter approves candidates that fall within an axis-aligned hyper-rectangle of acceptable tolerances across all issues.\"},{\"question\":\"What information is elicited from voters, and how?\",\"answer\":\"Preferences are revealed through Planar queries that fix one issue and ask whether the voter’s acceptable range lies entirely on one side of a threshold for that issue.\"},{\"question\":\"What fairness guarantee and query complexity does the proposed approach achieve?\",\"answer\":\"The algorithm returns an EJR+ committee for any distribution over rectangular preferences while using an expected O(d log k) Planar queries per voter, independent of the number of candidates m under a sufficiently large electorate.\"}]",1784181617,66,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":27},"multiwinner-voting-with-spatial-preferences-under-incomplete-information","",{"@graph":35,"@context":89},[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/multiwinner-voting-with-spatial-preferences-under-incomplete-information/82578/",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",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the document address in multiwinner elections?","Question",{"text":75,"@type":76},"It examines how to guarantee strong proportional representation, such as EJR+, when voters cannot evaluate all candidates and only reveal a small amount of information.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are voter preferences modeled in the ARRV framework?",{"text":80,"@type":76},"Candidates and voters live in a d-dimensional issue space, and each voter approves candidates that fall within an axis-aligned hyper-rectangle of acceptable tolerances across all issues.",{"name":82,"@type":73,"acceptedAnswer":83},"What information is elicited from voters, and how?",{"text":84,"@type":76},"Preferences are revealed through Planar queries that fix one issue and ask whether the voter’s acceptable range lies entirely on one side of a threshold for that issue.",{"name":86,"@type":73,"acceptedAnswer":87},"What fairness guarantee and query complexity does the proposed approach achieve?",{"text":88,"@type":76},"The algorithm returns an EJR+ committee for any distribution over rectangular preferences while using an expected O(d log k) Planar queries per voter, independent of the number of candidates m under a sufficiently large electorate.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Story & 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