{"id":"https://openalex.org/W7213527353","doi":"https://doi.org/10.48550/arxiv.2609.18710","title":"Ensuring proportionality: a logical model for compensatory seats added to multi-member constituencies","display_name":"Ensuring proportionality: a logical model for compensatory seats added to multi-member constituencies","publication_year":2026,"publication_date":"2026-09-16","ids":{"openalex":"https://openalex.org/W7213527353","doi":"https://doi.org/10.48550/arxiv.2609.18710"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.18710","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.18710","pdf_url":"https://arxiv.org/pdf/2609.18710","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.18710","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039959623","display_name":"Frederik Ravn Klausen","orcid":"https://orcid.org/0000-0001-7815-4315","observed_orcids":["https://orcid.org/0000-0001-7815-4315"]},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Klausen, Frederik Ravn","raw_affiliation_strings":[", University of Cambridge , DPMMS , Cambridge , United Kingdom"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":", University of Cambridge , DPMMS , Cambridge , United Kingdom","institution_ids":["https://openalex.org/I241749"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5039959623"],"corresponding_institution_ids":["https://openalex.org/I241749"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10108","display_name":"Electoral Systems and Political Participation","score":0.7817000150680542,"subfield":{"id":"https://openalex.org/subfields/3320","display_name":"Political Science and International Relations"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10108","display_name":"Electoral Systems and Political Participation","score":0.7817000150680542,"subfield":{"id":"https://openalex.org/subfields/3320","display_name":"Political Science and International Relations"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10991","display_name":"Game Theory and Voting Systems","score":0.2029000073671341,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11432","display_name":"Gender Politics and Representation","score":0.0015999999595806003,"subfield":{"id":"https://openalex.org/subfields/3318","display_name":"Gender Studies"},"field":{"id":"https://openalex.org/fields/33","display_name":"Social Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/proportionality","display_name":"proportionality","score":0.6230000257492065},{"id":"https://openalex.org/keywords/d-hondt-method","display_name":"D'Hondt method","score":0.9750000238418579},{"id":"https://openalex.org/keywords/seat-allocation","display_name":"seat allocation","score":0.5659999847412109},{"id":"https://openalex.org/keywords/closed-form-expressions","display_name":"closed-form expressions","score":0.6740000247955322},{"id":"https://openalex.org/keywords/portugal","display_name":"Portugal","score":0.41999998688697815},{"id":"https://openalex.org/keywords/poland","display_name":"Poland","score":0.9990000128746033},{"id":"https://openalex.org/keywords/spain","display_name":"Spain","score":0.9990000128746033},{"id":"https://openalex.org/keywords/turkey","display_name":"Turkey","score":0.9990000128746033}],"concepts":[{"id":"https://openalex.org/C2778337684","wikidata":"https://www.wikidata.org/wiki/Q3173700","display_name":"Apportionment","level":2,"score":0.8615999817848206},{"id":"https://openalex.org/C183763965","wikidata":"https://www.wikidata.org/wiki/Q603959","display_name":"Proportionality (law)","level":2,"score":0.6105999946594238},{"id":"https://openalex.org/C2781440851","wikidata":"https://www.wikidata.org/wiki/Q35749","display_name":"Parliament","level":3,"score":0.5393999814987183},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.5268999934196472},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.4108999967575073},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.38280001282691956},{"id":"https://openalex.org/C100001284","wikidata":"https://www.wikidata.org/wiki/Q2248246","display_name":"Public economics","level":1,"score":0.27489998936653137},{"id":"https://openalex.org/C144133560","wikidata":"https://www.wikidata.org/wiki/Q4830453","display_name":"Business","level":0,"score":0.2671999931335449},{"id":"https://openalex.org/C175444787","wikidata":"https://www.wikidata.org/wiki/Q39072","display_name":"Microeconomics","level":1,"score":0.265500009059906},{"id":"https://openalex.org/C48677424","wikidata":"https://www.wikidata.org/wiki/Q6888088","display_name":"Mode (computer interface)","level":2,"score":0.2624000012874603},{"id":"https://openalex.org/C114289077","wikidata":"https://www.wikidata.org/wiki/Q3284399","display_name":"Statistical model","level":2,"score":0.2624000012874603}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.18710","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.18710","pdf_url":"https://arxiv.org/pdf/2609.18710","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.18710","doi":"https://doi.org/10.48550/arxiv.2609.18710","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.18710","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.18710","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.18710","pdf_url":"https://arxiv.org/pdf/2609.18710","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","display_name":"Peace, justice, and strong institutions","score":0.6601999998092651}],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213527353.pdf","grobid_xml":"https://content.openalex.org/works/W7213527353.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W160958150","https://openalex.org/W3123178323","https://openalex.org/W3082726787","https://openalex.org/W3036361993","https://openalex.org/W2095111986","https://openalex.org/W2903625915","https://openalex.org/W2121571302","https://openalex.org/W4232877343","https://openalex.org/W4312204716","https://openalex.org/W4392546042"],"abstract_inverted_index":{"How":[0],"many":[1],"compensatory":[2,71,89,137,148,230],"seats":[3,26,72,90,138,231],"are":[4,91,115],"needed":[5],"to":[6,63,139,228,232],"ensure":[7,140],"(full)":[8],"proportionality":[9],"in":[10,66,131],"two-tier":[11,124],"electoral":[12],"systems":[13,251],"with":[14,157],"multi-member":[15,132],"constituencies?":[16],"This":[17,44],"question":[18],"is":[19,119,145,175],"answered":[20],"by":[21,203],"estimating":[22],"the":[23,38,67,85,99,106,109,158,163,166,172,179,191,198,206,221,233,250],"discrepancy":[24],"between":[25],"and":[27,31,57,95,105,162,212,220,242],"votes":[28],"rigorously,":[29],"probabilistically":[30],"empirically.":[32],"In":[33,50,142],"1919,":[34],"P\u00f3lya":[35],"showed":[36],"that":[37,122,144,155,197,224,245],"Jefferson/D'Hondt":[39],"method":[40],"favours":[41],"large":[42],"parties.":[43],"seat":[45,76,149],"surplus":[46],"accumulates":[47],"across":[48],"constituencies.":[49],"some":[51],"countries":[52],"(e.g.,":[53],"Portugal,":[54,209],"Poland,":[55,210],"Spain":[56,211],"Turkey),":[58],"this":[59,74],"accumulation":[60],"directly":[61],"leads":[62],"quantifiable":[64],"disproportionality":[65],"final":[68],"parliament.":[69],"Elsewhere":[70],"counteract":[73],"accumulated":[75],"surplus.":[77],"Using":[78],"a":[79,123,147,153,216],"logic-first":[80],"approach,":[81],"closed":[82],"formulas":[83,174],"for":[84],"required":[86],"number":[87,100,154,159],"of":[88,101,108,160,165,171,249],"derived":[92,120,173],"given":[93],"regional":[94],"national":[96],"apportionment":[97,130],"methods,":[98],"relevant":[102],"parties":[103,161],"$n$":[104],"size":[107,164],"largest":[110,167],"party":[111],"$p$.":[112],"If":[113],"there":[114],"$c$":[116],"constituencies,":[117],"it":[118],"logically":[121],"system":[125,219],"built":[126],"on":[127,190],"local":[128],"D'Hondt":[129,218],"constituencies":[133],"needs":[134],"around":[135],"$\\frac{c}{2}(n-\\frac{1}{p})$":[136],"proportionality.":[141],"words,":[143],"half":[146],"per":[150],"constituency":[151],"times":[152],"increases":[156],"party.":[168],"The":[169],"validity":[170],"analysed":[176],"empirically":[177],"using":[178],"extensive":[180],"recent":[181],"CLEA":[182],"dataset.":[183],"Many":[184],"quantities":[185],"do":[186],"not":[187],"depend":[188],"much":[189],"total":[192],"parliament":[193],"sizes,":[194],"which":[195],"means":[196],"variance":[199],"can":[200],"be":[201,238],"reduced":[202],"artificially":[204],"varying":[205],"house":[207],"sizes.":[208],"Turkey":[213],"all":[214],"employ":[215],"one-tier":[217],"analysis":[222],"shows":[223],"if":[225],"they":[226],"were":[227],"add":[229],"system,":[234],"reasonable":[235],"numbers":[236],"would":[237],"40,":[239],"75,":[240],"75":[241],"120,":[243],"assuming":[244],"no":[246],"other":[247],"properties":[248],"change.":[252]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-18T00:00:00"}
