Yukio Koriyama Antonin Mace Jean Franc¸ois Laslier and Rafael Treibich
30 Pages
English
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Yukio Koriyama Antonin Mace Jean Franc¸ois Laslier and Rafael Treibich

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30 Pages
English

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Optimal Apportionment Yukio Koriyama?, Antonin Mace, Jean-Franc¸ois Laslier, and Rafael Treibich Ecole Polytechnique† November 6, 2011 Abstract This paper provides a theoretical foundation which supports the “degressive proportionality principle” in apportionment problems. The core of the argument is that each individual derives utility from the fact that the collective decision matches her own will with some frequency, with marginal utility being decreasing with respect to this frequency. Then classical utilitarianism at the social level recommends decision rules which exhibit degressive proportionality. The model is applied to the case of the 27 states of the European Union. 1 Introduction 1.1 Background Consider a situation in which repeated decisions have to be taken under the (possibly qualified) majority rule by representatives of groups (e.g. coun- tries) that differ in size. In that case, the principle of equal representation translates into a principle of proportional apportionment. In other words, if we require each representative to represent the same number of individuals, ? †Departement d'Economie, Palaiseau Cedex, 91128 France. For useful remarks, we thank Ani Guerdjikova, Annick Laruelle, Michel Le Breton, Eduardo Perez, Pierre Picard, Francisco Ruiz-Aliseda, Karine Van der Straeten, Jorgen Weibull, Stephane Zuber, and the participants of the 2011 workshop on Voting Power and Procedures at LSE, Advances in the Theory of Individual and Collective Decision-Making at Istanbul Bilgi University, SAET 2011, D-TEA Workshop 2011, seminar participants at Ecole Polytechnique, Paris School of Economics, University

  • big country

  • marginal social

  • utilitarian social

  • principle

  • utility ??

  • country

  • ??

  • rather than

  • degressive proportionality

  • than when


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OptimalApportionmentYukioKoriyama,AntoninMac´e,Jean-Franc¸oisLaslier,andRafaelTreibichEcolePolytechniqueNovember6,2011AbstractThispaperprovidesatheoreticalfoundationwhichsupportsthe“degressiveproportionalityprinciple”inapportionmentproblems.Thecoreoftheargumentisthateachindividualderivesutilityfromthefactthatthecollectivedecisionmatchesherownwillwithsomefrequency,withmarginalutilitybeingdecreasingwithrespecttothisfrequency.Thenclassicalutilitarianismatthesociallevelrecommendsdecisionruleswhichexhibitdegressiveproportionality.Themodelisappliedtothecaseofthe27statesoftheEuropeanUnion.1Introduction1.1BackgroundConsiderasituationinwhichrepeateddecisionshavetobetakenunderthe(possiblyqualified)majorityrulebyrepresentativesofgroups(e.g.coun-tries)thatdifferinsize.Inthatcase,theprincipleofequalrepresentationtranslatesintoaprincipleofproportionalapportionment.Inotherwords,ifwerequireeachrepresentativetorepresentthesamenumberofindividuals,yukio.koriyama@polytechnique.eduDe´partementdE´conomie,PalaiseauCedex,91128France.Forusefulremarks,wethankAniGuerdjikova,AnnickLaruelle,MichelLeBreton,EduardoPerez,PierrePicard,FranciscoRuiz-Aliseda,KarineVanderStraeten,Jo¨rgenWeibull,Ste´phaneZuber,andtheparticipantsofthe2011workshoponVotingPowerandProceduresatLSE,AdvancesintheTheoryofIndividualandCollectiveDecision-MakingatIstanbulBilgiUniversity,SAET2011,D-TEAWorkshop2011,seminarparticipantsatEcolePolytechnique,ParisSchoolofEconomics,UniversityofEdinburghandBETA.1
thenumberofrepresentativesofagroupshouldbeproportionaltoitspop-ulation.Argumentshavebeenraisedagainstthisprincipleandinfavorofaprincipleofdegressiveproportionalityaccordingtowhichtheratioofthenumberofrepresentativestothepopulationsizeshoulddecreasewiththepopulationsizeratherthanbeconstant.ThedegressiveproportionalityprincipleisendorsedbymostpoliticiansandactuallyenforceduptosomequalificationsintheEuropeaninstitutions(Duff2010a,2010b,TEU2010).ItissometimestermedtheLamassoure-SeverinrequirementfollowingtheEuropeanParliamentResolutionon“Pro-posaltoamendtheTreatyprovisionsconcerningthecompositionoftheEu-ropeanParliament”whichwasadoptedonOctober11,2007afterthereportbyLamassoureandSeverin(2007).Onthatoccasion,itwasnotedthatthetreatiesandamendmentsoftheEuropeanUnionwerereferringtodegressiveproportionality“withoutdefiningthisterminanymorepreciseway.”ThentheOctober2007Resolutionstated:[TheEuropeanParliament]considersthattheprincipleofde-gressiveproportionalitymeansthattheratiobetweenthepopu-lationandthenumberofseatsofeachMemberStatemustvaryinrelationtotheirrespectivepopulationsinsuchawaythateachMemberfromamorepopulousMemberStaterepresentsmorecitizensthaneachMemberfromalesspopulousMemberStateandconversely,butalsothatnolesspopulousMemberStatehasmoreseatsthanamorepopulousMemberState.Itisknownthat,inthecaseofaParliament,inwhicheachmembermusthaveoneandonlyonevote,thedegressiveproportionalityrequirementisimpossibletosatisfyexactly,duetounavoidableroundingproblems(seeforinstanceCichockiandZ˙yczkowski,2010).Butifoneseekstorespecttheprinciple“uptoone”,or“beforerounding”,thenmanysolutionsbecomeavailable,amongwhichonehastochoose(Ramı´rez-Gonza´lez,PalomaresandMarquez2006;Martı´nez-ArozaandRamı´rez-Gonza´lez2008;Grimmetetal.2011).Suchisalsothecase(ratherobviously)ifoneallowsforfractionalweights.Thesameprinciplesformallyapplytothecasewhereacountryisrep-resentedbyanumberofrepresentatives,eachofwhomisgivenonevote,andtothecasewhereacountryisrepresentedbyasingledelegatewhomisgivenaweightinrelationwiththecountrysize.WeshallrefertothetwocasesasaParliamentandaCouncil.ThispaperappliesNormativeEconomicstoPolitics.Itsaimistojustifytheprincipleofdegressiveproportionalitybyanoptimalityargumentand2
therebytosuggestthecomputationofoptimalweightsinspecificinstances,optimalweightswhichwillbedegressivelyproportional.Hereisasketchoftheargument.1.2IllustrationoftheargumentTheargumentinfavorofdegressivelyproportionalapportionmentisbasedonthemaximizationofanexplicitutilitariansocialcriterion.Eachindivid-ualderivesutilityfromthefactthatthecollectivedecisionsoftenmatchherownwill.Thesocialobjectiveissimplythesumofsuchindividualutilities.Theargumentcanbeexplainedwithaverysimpleexample.Supposethereareonlytwocountries,ofsizen1andn2,withn1<n2.Then,themajorityrulegivesfullpowertothebigcountry.Whenthetwocountriesagreeonwhichdecisiontotake,theyarebothsatisfied,butwhentheydisagree,country1,thesmallone,isneversatisfied.Intuition,inthatcase,recommendsthatthepowertodecideshouldbeoccasionallygiventothesmallcountry.Tobemorespecific,supposethatbinarydecisionshavetobetakenaccordingtothesamedecisionrule.Amongthesedecisionsafractionαiscontroversialinthesensethatthetwocountriesdisagree.Supposealso,forthesimplicityoftheexample,thatthecitizenswithineachcountryalwaysagreeontheirbestchoice.Underthemajorityrule,acitizenofcountry2issatisfiedwithproba-bility1,andacitizenofcountry1issatisfiedwithprobability1α.Toevaluatethisruleatthecollectivelevel,onehastomakeanassumptionastohowthesocietyevaluatesthefactthewillofeachindividualisimplementedwithsomefrequency,sayp.Inthispaper,weshallmaketheassumptionthatthisevaluationisaconcavefunctionofp,sayψ(p),thesamefunctionforeverycitizen.Thismeansthattheindividualmaywellacceptthatinamoderateproportionofthecasesthecollectivedecisiondoesnotfollowherwill,butsheincursarelativelysignificantdisutilityifthatproportionbecomestoolarge.Forinstance,thedisutilityincuredbytheindividualandtakenintoaccountatthesocialevaluationlevelissmallerwhenherpde-creasesfrom1to.95thanwhenpdecreasesfrom.6to.55.Wefoundthishypothesispsychologicallysound,andwewilllaterexplainitsconnectiontotheliterature.Underthishypothesis,thesumofindividualutilitiesunderthemajorityruleis:n1ψ(1α)+n2ψ(1),becausethewillofthesmall(resp.big)country’scitizensisfulfilledwithprobability1α(resp.1).3