{"title":"Impossibility theorem for two-tier electoral systems","authors":"Sebastian Tim Holdum, Frederik Ravn Klausen","doi":"10.1007/s10479-026-07344-1","DOIUrl":null,"url":null,"abstract":"<div><p>Two-tier electoral systems aim to balance regional representation and proportionality. Recently, the mathematical mechanisms of several Northern European two-tier systems have been challenged by increasing political party fragmentation, as evidenced by the expanding German parliament and the majority-deciding disproportionality of Denmark’s 2022 election. Regionality, proportionality, and two-tier systems are defined mathematically and used to identify the common cause behind these developments through the formulation of an impossibility theorem: no electoral system with fixed parliament size can guarantee both proportional and geographic representation. Finally, a new algorithm, <i>geographically ranked guaranteed proportionality</i>, is proposed to circumvent the impossibility theorem.</p></div>","PeriodicalId":8215,"journal":{"name":"Annals of Operations Research","volume":"364 3","pages":"2945 - 2970"},"PeriodicalIF":4.1000,"publicationDate":"2026-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10479-026-07344-1.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Operations Research","FirstCategoryId":"91","ListUrlMain":"https://link.springer.com/article/10.1007/s10479-026-07344-1","RegionNum":3,"RegionCategory":"管理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"OPERATIONS RESEARCH & MANAGEMENT SCIENCE","Score":null,"Total":0}
引用次数: 0
Abstract
Two-tier electoral systems aim to balance regional representation and proportionality. Recently, the mathematical mechanisms of several Northern European two-tier systems have been challenged by increasing political party fragmentation, as evidenced by the expanding German parliament and the majority-deciding disproportionality of Denmark’s 2022 election. Regionality, proportionality, and two-tier systems are defined mathematically and used to identify the common cause behind these developments through the formulation of an impossibility theorem: no electoral system with fixed parliament size can guarantee both proportional and geographic representation. Finally, a new algorithm, geographically ranked guaranteed proportionality, is proposed to circumvent the impossibility theorem.
期刊介绍：
The Annals of Operations Research publishes peer-reviewed original articles dealing with key aspects of operations research, including theory, practice, and computation. The journal publishes full-length research articles, short notes, expositions and surveys, reports on computational studies, and case studies that present new and innovative practical applications.
In addition to regular issues, the journal publishes periodic special volumes that focus on defined fields of operations research, ranging from the highly theoretical to the algorithmic and the applied. These volumes have one or more Guest Editors who are responsible for collecting the papers and overseeing the refereeing process.
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