Territorial Vote Inefficiency under District-Based D’Hondt Systems: A One-Parameter Comparative Analysis of Magnitude Rebalancing

Territorial Vote Inefficiency under District-Based D’Hondt Systems: A One-Parameter Comparative Analysis of Magnitude Rebalancing | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Territorial Vote Inefficiency under District-Based D’Hondt Systems: A One-Parameter Comparative Analysis of Magnitude Rebalancing Guillermo Cotrina Cuenca, Eva Del-Pozo García This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9277847/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 6 You are reading this latest preprint version Abstract This article proposes and cross-nationally evaluates a one-parameter territorial magnitude-rebalancing framework for district-based proportional representation systems using the D’Hondt rule. While D’Hondt is proportional conditional on fixed district magnitudes, heterogeneity in district size and the territorial distribution of party support generates systematic inequalities in vote—to—seat conversion, producing ineffective (“lost”) votes and dispersion in the effective cost of representation. The framework is applied to five countries sharing multi-member territorial constituencies with D’Hondt allocation—Spain, Portugal, Belgium, Finland, and Argentina—using the most recent lower-house elections with official district-level returns. For each country c , a scaling parameter α ∈ (0, 1) contracts baseline district magnitudes subject to a one-seat lower bound per district and fixed chamber size, reallocating residual seats through a deterministic largest-remainders rule based on ex ante territorial weights. Within-district seat assignment remains D’Hondt throughout. Outcomes are evaluated exhaustively over the grid A = {0.001, . . . , 0.999}, recomputing allocations and post-allocation diagnostics for each α. For each country and each disproportionality or seat-cost inequality metric, the analysis identifies the global minimizer(s) α ∗ Φ,c , thereby mapping the full response surface and exposing the trade-offs between proportionality and effective voting-power equality without collapsing them into a single composite objective. Physical sciences/Mathematics and computing Physical sciences/Physics Proportional Representation D’Hondt District Magnitude Apportionment Ineffective Votes Seat-Cost Inequality Territorial Bias Comparative Electoral Systems Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 27 Apr, 2026 Reviewers invited by journal 24 Apr, 2026 Editor assigned by journal 24 Apr, 2026 Editor invited by journal 17 Apr, 2026 Submission checks completed at journal 15 Apr, 2026 First submitted to journal 15 Apr, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9277847","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":630170103,"identity":"3a90e171-78fc-4c06-8de2-16ebf7e89430","order_by":0,"name":"Guillermo Cotrina Cuenca","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAr0lEQVRIiWNgGAWjYFACxgYQycNPopYEBh7JBtJsSmBgMDhArGL5iOTGDx9/3JMxvpH8dANDRR1hLYY3EpslZyQU85jdSDO7wXDmMBFaZiS2MfMkJAC15LDdYGwjwnlwLcYzQFr+EeEweQmoFgMJkJYGZsJaDHgeAv2SlsAjceaZ2Y2EY0T4Rb49/eGHDzYJ9vztyc9ufKghwmGo0ZFAWAPQlgZiVI2CUTAKRsHIBgAWUDebACRspAAAAABJRU5ErkJggg==","orcid":"","institution":"Complutense University of Madrid","correspondingAuthor":true,"prefix":"","firstName":"Guillermo","middleName":"Cotrina","lastName":"Cuenca","suffix":""},{"id":630170105,"identity":"72a78b45-5224-4d51-a9ee-3eab54c5fdb5","order_by":1,"name":"Eva Del-Pozo García","email":"","orcid":"","institution":"Complutense University of Madrid","correspondingAuthor":false,"prefix":"","firstName":"Eva","middleName":"Del-Pozo","lastName":"García","suffix":""}],"badges":[],"createdAt":"2026-03-31 09:40:01","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9277847/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9277847/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108181367,"identity":"e421b69a-ca7d-4b14-ba41-3fa825171c59","added_by":"auto","created_at":"2026-04-30 08:58:35","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1158301,"visible":true,"origin":"","legend":"","description":"","filename":"SPRINGER3.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9277847/v1_covered_1dc9127a-88f1-4f8a-80d1-83f08f62a40b.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Territorial Vote Inefficiency under District-Based D’Hondt Systems: A One-Parameter Comparative Analysis of Magnitude Rebalancing","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"humanities-and-social-sciences-communications","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"palcomms","sideBox":"Learn more about [Humanities \u0026 Social Sciences Communications](http://www.nature.com/palcomms/)","snPcode":"41599","submissionUrl":"https://submission.springernature.com/new-submission/41599/3","title":"Humanities and Social Sciences Communications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Nature AJ","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Proportional Representation, D’Hondt, District Magnitude, Apportionment, Ineffective Votes, Seat-Cost Inequality, Territorial Bias, Comparative Electoral Systems","lastPublishedDoi":"10.21203/rs.3.rs-9277847/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9277847/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis article proposes and cross-nationally evaluates a one-parameter territorial magnitude-rebalancing framework for district-based proportional representation systems using the D’Hondt rule. 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