Voting Methods (Stanford Encyclopedia of Philosophy)

2/6/2015 VotingMethods(StanfordEncyclopediaofPhilosophy)

http://plato.stanford.edu/entries/votingmethods/ 1/26

Stanford Encyclopedia of PhilosophyVotingMethodsFirstpublishedWedAug3,2011

Thinkbacktothelasttimeyouneededtomakeadecisionasamemberofagroup.Thismayhavebeenwhenyouvotedforyourfavoritepoliticalcandidateduringthelastelection.Onasmallerscale,itmayhavebeenwhenyoutookpartinacommitteethatneededtochoosethebestcandidateforajoborastudenttoreceiveaspecialaward.Whatmethod,orprocedure,didthegroupusetomakethefinaldecision?Manyinterestingissuesarisewhenwecarefullyexamineourgroupdecisionmakingprocesses.Considerasimpleexampleofagroupoffriendsdecidingwheretogofordinner.Ifeveryoneagreesonwhichrestaurantisbest,thenitisobviouswheretogo.Buthowshouldthefriendsdecidewheretogoiftheyhavedifferentopinionsaboutwhichrestaurantisbest?Istherealwaysachoicethatisfairtakingintoaccounteveryone'sopinions?Oraretheresituationsinwhichonepersonmustbechosentoactasadictatorbymakingaunilateraldecision?

Thisarticleintroducesandcriticallyexaminesanumberofdifferentvotingmethods.Thegoalisnottoprovideageneraloverviewofsocialchoicetheoryorevenacomprehensiveaccountofvotingtheory.Rather,myobjectiveistohighlightanddiscusskeyresultsandissuesthatunderliephenomenathatweobservewhendecisionmakerscometogethertomakeacollectivedecision.So,sometopicswillonlybrieflybementioned,whileotherswillnotbediscussedatall:Notableomissionsincludetheextensiveliteratureonthediscursivedilemma(seeList2006,andreferencestherein)andanoverviewoftheworkonvotingpowerindices(FelsenthalandMachover1998).Tolearnmoreaboutthesetopics,consultNurmi(1998)andSaari(2001)forgeneralintroductionstovotingtheoryandBramsandFishburn(2002)andSaari(1995)fortechnicalintroductionsandanalysisofthevastliterature.

1.TheProblem:WhoShouldbeElected?1.1Notation

2.ExamplesofVotingMethods3.VotingParadoxes

3.1Condorcet'sParadox3.1.1ElectingtheCondorcetWinner

3.2FailuresofMonotonicity3.3TheMultipleDistrictsParadox3.4TheMultipleElectionsParadox

4.TopicsinVotingTheory4.1Strategizing4.2CharacterizationResults4.3VotingtoTracktheTruth

5.ConcludingRemarks:fromTheorytoPracticeBibliographyAcademicToolsOtherInternetResourcesRelatedEntries



1.TheProblem:WhoShouldbeElected?

Thecentralquestionofthisarticleis:

Givenagroupofpeoplefacedwithsomedecision,howshouldacentralauthoritycombinetheindividualopinionssoastobestreflectthewillofthegroup?

Acompleteanalysisofthisquestionwouldincorporateanumberofdifferentissuesrangingfromcentraltopicsinpoliticalphilosophy(e.g.,howshouldwedefinethewillofthepeople?whatisademocracy?)tothepsychologyofdecisionmaking.Inthisarticle,Ifocusononeaspectofthisquestion:theformalanalysisofspecificvotingmethods(see,forexample,Riker1982Mackie2003,foramorecomprehensiveanalysisoftheabovequestion,incorporatingmanyoftheissuesraisedinthisarticle).

Istartwithaconcreteexampletoillustratethetypeofanalysissurveyedinthisarticle.Supposethatthereisagroupof21people,orvoters,whoneedtomakeadecisionaboutwhichoffourcandidates,oroptions,shouldbeelected,orchosen.LetA,B,CandDdenotethefourdifferentcandidates.Thefirststepistodecidehowtorepresentthevoters'opinionsaboutthesetofcandidates.Manydifferentapproacheshavebeenexploredinthevotingtheoryliterature.Oneapproachistoassumethateachvoterhasanordinalpreferenceorderingoverthesetofcandidates,describingtherelativerankingsofthecandidates.Asecondapproachassumesthatvotersassigntoeachcandidateacardinalvaluedescribinghowmuchthatvoterprefersorvaluesthecandidate.Finally,onecandescribeanunderlyingspaceofissues,howmucheachvotercaresabouteachissueandthedegreetowhicheachcandidatessupportsthedifferentissues.Unlessotherwisestated,Ifollowmuchofthevotingtheoryliteratureandassumethatthevoters'opinionsaredescribedbylinearrankingsofthesetofcandidates(describingthevoters'ordinalpreferenceorderings).

Forthisexample,assumethateachofthevotershasoneoffourpossiblerankingsofthecandidates.Theinformationabouttherankingsofeachvoterisgiveninthefollowingtable.

#Voters

3 5 7 6

A A B C

B C D B

C B C D

D D A A

Readthetableasfollows:Eachcolumnrepresentsarankinginwhichcandidatesinlowerrowsarerankedlower.Thenumbersatthetopofeachcolumnindicatethenumberofvoterswiththatparticularranking.Supposethatyouareanoutsideobserverwithoutanyinterestintheoutcomeofthiselection.Whichofthecandidatesbestrepresentsthewillofthisgroup?Iftherewereonlytwocandidatestochoosefrom,thereisaveryintuitiveanswer:Thewinnershouldbethecandidateoroptionthatissupportedbymorethan50percentofthevoters(cf.thediscussionbelowaboutMay'sTheoreminSection4.2).However,iftherearemorethantwocandidates,asintheaboveexample,thestatementthecandidatethatissupportedbymorethan50percentofthevoterscanbeinterpretedindifferentways,leadingtodifferentideasaboutwhoshouldwintheelection.

Onecandidatewho,atfirstsight,seemstobeagoodchoicetowintheelectioniscandidateA.Candidate



Aisrankedfirstinmoreofthevoters'rankingsthananyothercandidate.(Aisrankedfirstbyeightvoters,BisrankedfirstbysevenCisrankedfirstbysixandDisnotrankedfirstbyanyofthevoters.)Thatis,morepeoplethinkthatAisbetterthananyothercandidate.

Ofcourse,13peoplerankAlast,soamuchlargergroupofvoterswillbeunsatisfiedwiththeelectionofA.So,itseemsclearthatAshouldnotbeelected.NoneofthevotersrankDfirst,whichsuggeststhatDisalsonotagoodchoice.Thechoice,then,boilsdowntoBandC.Here,therearegoodargumentsforeachofBandCtobeelected.Thisechoesan18thcenturydebatebetweenthetwofoundingfathersofvotingtheory,JeanCharlesdeBorda(17331799)andM.J.A.N.deCaritat,MarquisdeCondorcet(17431794).Foraprecisehistoryofvotingtheoryasanacademicdiscipline,includingCondorcet'sandBorda'swritings,seeMcCleanandUrken(1995).IsketchtheintuitiveargumentsfortheelectionofBandCbelow.

CandidateCshouldwin.Initially,thismightseemlikeanoddchoicesinceCreceivedthefewestnumberoffirstplacerankings(6).However,Cisastrongchoicebecausehebeatseveryothercandidateinaoneononeelection.Toseethis,weneedtoexaminehowthepopulationwouldvoteinthevarioustwowayelections:

#Voters

3 5 7 6

A A B C

B C D B

C B C D

D D A A

#Voters

3 5 7 6

A A B C

B C D B

C B C D

D D A A

#Voters

3 5 7 6

A A B C

B C D B

C B C D

D D A A

13rankCaboveA8rankAaboveC

11rankCaboveB10rankBaboveC

15rankCaboveD7rankDaboveC

TheideaisthatCshouldbedeclaredthewinnersincehebeatseveryothercandidateinoneononeelections.AcandidatewiththispropertyiscalledaCondorcetwinner.(WecansimilarlydefineaCondorcetloser.Infact,intheaboveexample,candidateAistheCondorcetlosersinceshelosestoeveryothercandidateinheadtoheadelections.)

CandidateBshouldwin.ConsiderB'sperformanceinheadtoheadelections.

#Voters

3 5 7 6

A A B C

B C D B

C B C D

#Voters

3 5 7 6

A A B C

B C D B

C B C D

#Voters

3 5 7 6

A A B C

B C D B

C B C D



D D A A D D A A D D A A

13rankBaboveA8rankAaboveB

10rankBaboveC11rankCaboveB

21rankBaboveD0rankDaboveB

CandidateBperformsthesameasCinaheadtoheadelectionwithA,losestoCbyonlyonevoteandbeatsDinalandslide(everyoneprefersBoverD).Arguably,weshouldtakeintoaccountallofthesefactswhendeterminingwhoshouldrepresentthewillofthepeople.Borda'sideaistoassigneachcandidateascorethatreflectsallofthisinformation.BothCondorcetandBordasuggestcomparingcandidatesinoneononeelectionsinordertodeterminethewinner.WhileCondorcettallieshowmanyoftheheadtoheadraceseachcandidatewins,Bordasuggeststhatoneshouldlookatthemarginofvictoryorloss.AccordingtoBorda,eachcandidateshouldbeassignedascorerepresentinghowmuchsupportheorshehasamongtheelectorate.Onewaytocalculatethescoreforeachcandidateisasfollows(Iwillgiveanalternativemethod,whichiseasiertouse,inthenextsection):

Areceives24points(8votesineachofthethreeheadtoheadraces)Breceives44points(13pointsinthecompetitionagainstA,plus10inthecompetitionagainstCplus21inthecompetitionagainstD)Creceives38points(13pointsinthecompetitionagainstA,plus11inthecompetitionagainstBplus14inthecompetitionagainstD)Dreceives20points(13pointsinthecompetitionagainstA,plus0inthecompetitionagainstBplus7inthecompetitionagainstC)

Thecandidatewiththehighestscore(inthiscase,B)istheonewhoshouldbeelected.

Theconclusionisthatinvotingsituationswithmorethantwocandidates,theremaynotalwaysbeoneobviouscandidatethatbestreflectsthewillofthepeople.Theremainderofthisentrywilldiscussdifferentmethods,orprocedures,thatcanbeusedtodeterminethewinnerofanelection.

1.1Notation

Inthisarticle,Iwillkeeptheformaldetailstoaminimumhowever,itisusefulatthispointtosettleonsometerminology.AssumethatthereisafinitesetofvotersVandafinitesetofcandidatesX.Iuselowercaselettersi,j,k,todenoteelementsofVanduppercaselettersA,B,C,todenoteelementsofX.Differentvotingmethodsrequiredifferenttypesofinformationfromthevotersasinput.Forexample,somemethodsaskvoterstoselectasinglecandidateorasetofcandidates,whileothermethodsaskvoterstolinearlyrankallofthecandidates.Theinputrequestedfromthevotersarecalledballots.Aprofileisasequenceofballots,onefromeachvoter.Thesecondcomponentofavotingprocedureisthemethodusedtocalculatethewinner,givenaprofileofballots.

Asnotedabove,oneunderlyingassumptionisthatthevoters'actualdesiresaboutwhoshouldwintheelectionarerepresentedaslinearpreferencerelationsoverthesetofcandidates.GivenasetofcandidatesX,letL(X)denotethesetoflinearorderingsonX(thatis,relationsonXthatareirreflexive,transitive,andcomplete).Theseorderingsareintendedtorepresentthevoters'ordinalpreferencesabouttherelativerankingsofeachofthecandidates(seetheentryonpreferences,HanssonandGrneYanoff2009,foranextendeddiscussionofthesepropertiesandotherissuessurroundingformalmodelingpreferences).WeusePitodenotevoteri'spreferenceorderingoverX.Itisimportanttonotethattheseorderingsdonot



reflectanycardinalinformation(forexample,theintensityofthepreferenceofonecandidateoveranother).Forinstance,supposethattherearethreecandidatesX={A,B,C}.Then,theassumptionisthatavoter'spreferencecanbeanyoneofthesixpossiblelinearorderingsoverX:

Preference P1 P2 P3 P4 P5 P6

A A B B C C

B C A C A B

C B C A B A

#Voters n1 n2 n3 n4 n5 n6

IcannowbemorepreciseaboutthedefinitionofaCondorcetwinner(loser).Thekeynotionhereisthemajorityrelation,whichistherankingofcandidatesintermsofhowtheyperforminoneononeelections.Formally,wewriteA>MB,providedthatmorevotersrankcandidateAabovecandidateBthantheotherwayaround(wewriteMifthereareties).So,ifthedistributionofpreferencesisgivenintheabovetable,wehave:

A>MBjustincasen1+n2+n5>n3+n4+n6(otherwiseBMA)A>MCjustincasen1+n2+n3>n4+n5+n6(otherwiseCMA)B>MCjustincasen1+n3+n4>n2+n5+n6(otherwiseBMC)

CandidateAiscalledtheCondorcetwinnerifAismaximalinthemajorityordering>M.TheCondorcetloseristhecandidatethatminimizesthisordering.

Iconcludethissectionwithafewcommentsontherelationshipbetweentheballotsandthevoters'opinionsaboutthecandidates.Twoissuesareimportanttokeepinmind.First,theballotsofaparticularvotingmethodareintendedtoreflectsomeaspectofthevoters'opinionsaboutthedesirabilityofthedifferentcandidates.Sometypesofballotsareintendedtorepresentallorpartofthevoter'spreferenceordering,whileothertypesrepresentinformationthatcannotbeinferreddirectlyfromthevoter'sordinalpreferenceordering(forexample,bydescribinghowmuchavoterlikesaparticularcandidate).Second,itisimportanttobepreciseaboutthetypeofconsiderationsvoterstakeintoaccountwhenselectingaballot.Oneapproachistoassumethatvoterschoosesincerelybyselectingtheballotthatbestreflectstheirviewaboutthedesirabilityofthedifferentcandidates.Asecondapproachassumesthatthatvoterschoosestrategically.Inthiscase,avoterselectsaballotthatsheexpectstoleadtohermostdesiredoutcomegiventheinformationshehasabouthowtheothermembersofthegroupwillvote.Strategicvotingisanimportanttopicinvotingtheoryandsocialchoicetheory(seeTaylor2005,foradiscussionandpointerstotheliterature),butinthisarticle,unlessotherwisestated,Iassumethatvoterschoosesincerely.

2.ExamplesofVotingMethods

Avotingprocedureisawayofaggregatingtheindividual'spreferencesinordertocometoacollectivedecision.Aquicksurveyofelectionsheldindifferentdemocraticsocietiesthroughouttheworldrevealsawidevarietyofmethods.Inthissection,Idiscusssomeofthekeyproceduresthathavebeenanalyzedinthevotingtheoryliterature.Theseproceduresmaybeofinterestbecausetheyarewidelyused(e.g.,pluralityruleorpluralityrulewithrunoff)orbecausetheyareoftheoreticalinterest(e.g.,Dodgson's



method).Idonotprovideacomprehensiveoverviewofthedifferentmethodsthathavebeendiscussedintheliterature(seeBramsandFishburn2002,forasystematicoverviewofdifferentvotingmethods).Rather,Ifocusonmethodsthateitherarefamiliarorhelpillustrateimportantideas.Istartwiththemostwidelyusedmethod:

PluralityRule:Eachvoterselectsonecandidate(ornoneifvoterscanabstain),andthecandidate(s)withthemostvoteswin.So,theballotsaresimplythesetofcandidatesXand,givenvoteri'struepreferenceorderingPi,theuniquesincereballotforvoteriistop(Pi)(themaximalelementintheorderingPi).

Pluralityruleisaverysimplemethodthatiswidelyuseddespiteitsmanyproblems.ThemostpervasiveproblemisthefactthatpluralityrulecanelectaCondorcetloser.Borda(1784)observedthisphenomenoninthe18thcentury.

#Voters

1 7 7 6

A A B C

B C C B

C B A A

CandidateAistheCondorcetloser(bothBandCbeatcandidateA,138)however,Aisthepluralityrulewinner.Infact,thepluralityranking(Aisfirstwitheightvotes,BissecondwithsevenvotesandCisthirdwithsixvotes)reversesthemajorityorderingC>MB>MA.Butthereareother(morebasic)reasonstocriticizepluralityrule.Forinstance,theverysimplepluralityballotsseverelylimitwhatthevoterscanexpressabouttheiropinionsofthecandidates.Rankedvotingproceduresaskformuchmoreinformationfromthevoter:theballotsarelinearorderingsofthecandidates.ThemostwellknownexampleofsuchaprocedureisBordaCount:

BordaCount:Eachvoterprovidesalinearorderingofthecandidates.Eachcandidateisassignedascore(theBordascore)asfollows:Iftherearencandidates,given1pointstocandidatesrankedfirst,n2pointstocandidatesrankedsecond,,1pointtoacandidateranked2ndtolastand0pointstocandidatesrankedlast.So,theBordascoreofA,denotedBS(A),iscalculatedasfollows(where#UdenotesthenumberelementsinthesetU):

BS(A) = (n1)#{i|iranksAfirst}+(n2)#{i|iranksAsecond}++1#{i|iranksAsecondtolast}+0#{i|iranksAlast}

ThecandidatewiththehighestBordascorewins.

RecalltheexamplediscussedintheintroductiontoSection1.WecancalculatetheBordascoreforeachofthecandidatesasfollows:

BS(A)=38+20+10+013=24



BS(B)=37+29+15+00=44BS(C)=36+25+110+00=38BS(D)=30+27+16+08=20

BordaCountrequiresthevoterstocomeupwithalinearrankingofallthecandidates.Thiscanberatherdemandingwhentherearealargenumberofcandidates(asitcanbedifficultforvoterstomakedistinctionsbetweensomeofthemoreobscurecandidates).Asecondwaytomakeavotingmethodsensitivetomorethanthevoters'topchoiceistoholdmultistageelections.Thedifferentstagescancomeintheformofactualrunoffelectionsinwhichvotersareaskedtochoosefromareducedsetofcandidatesortheycanbebuiltintothewaythewinneriscalculatedbyaskingvoterstosubmitlinearorderingsoverthesetofallcandidates.Thefollowingarethemostwellknownexamplesofmultistagevotingmethods:

PluralitywithRunoff:Startwithapluralityvotetodeterminethetoptwocandidates(ormoreifthereareties).Then,thereisarunoffbetweenthesecandidates,andthecandidatewiththemostvoteswins.Sometimes,arunoffcanbeavoidedifthetopcandidategetsasufficientlylargepercentageofthevotes(forexample,ifshegetsanabsolutemajority:morethan50percentofthevotes).

Ratherthanfocusingonthetoptwocandidates,onecanalsoiterativelyremovethecandidate(s)withthefewestfirstplacevotes:

TheHareRule:Theballotsarelinearordersoverthesetofcandidates.Repeatedlydeletethecandidateorcandidatesthatreceivethefewestfirstplacevotes,withtheremainingcandidate(s)declaredthewinner(orwinnersinthecaseofties).

Ifthereareonlythreecandidates,thentheabovetwoproceduresarethesame(removingthecandidatewiththeleastnumberofvotesisthesameaskeepingthetoptwocandidates).Thefollowingexampleshowsthatthesetwoprocedurescanconflictwhentherearemorethanthreecandidates:

#Voters

7 5 4 3

A B D C

B C B D

C D C A

D A A B

CandidateAisthepluralitywithrunoffwinner:CandidatesAandBarethetoptwocandidates,receivingsevenandfivevotes,respectively,inthefirstround.Intherunoffelection,thegroupsvotingforcandidatesCandDgivetheirsupporttocandidateBandA,respectively,withAwinning109.

However,CandidateDwinswiththeHarerule:Inthefirstround,candidateCiseliminatedafterreceivingonlythreevotes.Butthenthisgroup'svotesaretransferredtoD,givinghersevenvotes.Thismeansthatinthesecondround,candidateBhasthefewestvotes(fivevotes)andsoiseliminated.AftertheeliminationofcandidateB,candidateDhasanabsolutemajoritywith12totalvotes(notethatinthisroundthegroupinthesecondcolumntransfersalltheirvotestoDsinceCwaseliminatedinanearlier



round).

OnefinalprocedureisCoombsrule,whichiterativelyremovesthecandidateswiththemostlastplacevotes.

CoombsRule:Eachvotersubmitsalinearorderingoverthesetofcandidates.Candidateswhoarerankedlastbythemostvotersareiterativelyremoved.Thelastcandidate(s)toberemovedarethewinner(s).

Intheaboveexample,candidateBwinstheelectionusingCoombsrule.Inthefirstround,A,withninelastplacevotes,iseliminated.ThenextcandidatetobeeliminatedisD,with12lastplacevotes.Finally,C,with16lastplacevotes,iseliminated.

Thenexttypeofproceduresaskvoterstosubmitballotsthatrepresentinformationthatcannotbeinferreddirectlyfromtheirordinalpreferenceorderings.Thefirstexamplegivesvoterstheoptiontoeitherselectacandidatethattheywanttovotefor(asinpluralityrule)ortoselectacandidatethattheywanttovoteagainst.

NegativeVoting:Eachvoterisallowedtochooseonecandidatetoeithervotefor(givingthecandidateonepoint)ortovoteagainst(givingthecandidate1points).Thewinner(s)is(are)thecandidate(s)withthehighestscore(s)(i.e.,themostpositivevotes).

Negativevotingistantamounttoallowingthevoterstosupporteitherasinglecandidateorallbutonecandidate(takingapointawayfromacandidateCisequivalenttogivingonepointtoallcandidatesexceptC).Thatis,thevotersareaskedtochooseasetofcandidatesthattheysupport,wherethechoiceisbetweensetsconsistingofsinglecandidatesorsetsconsistingofallexceptonecandidate.Thenextproceduregeneralizesthisideabyallowingvoterstochooseanysubsetofcandidates:

ApprovalVoting:Eachvoterselectsasubsetofthecandidates(wheretheemptysetmeansthevoterabstains)andthecandidate(s)withthemostvoteswins.

ApprovalvotinghasbeenextensivelydiscussedbyStevenBramsandPeterFishburn(BramsandFishburn2007Brams2008).See,also,therecentcollectionofarticlesdevotedtoapprovalvoting(LaslierandSanver2010).

Approvalvotingforcesvoterstothinkaboutthedecisionproblemdifferently:Theyareaskedtodeterminewhichcandidatestheyapproveofratherthandeterminingtherelativerankingofthecandidates.Thatis,thevoterisaskedwhichcandidatesareaboveacertainthresholdofacceptance.(SeeBramsandSanver2009,forexamplesofvotingproceduresthataskvoterstobothselectasetofofcandidatesthattheyapproveandto(linearly)rankthecandidates.)ThefinaltypeofproceduresIintroduceinthissectionallowvoterstoexpresstheirintensityofpreferenceamongthecandidates.

CumulativeVoting:Eachvoterisaskedtodistributeafixednumberofpoints,sayten,amongthecandidatesinanywaytheyplease.Thecandidate(s)withthemostpointswinstheelection.

ThisgeneralideawastakenfurtherinarecentproposalforanewmethodofvotingbyMichelBalinksiandRidaLaraki(2007).Thegeneralideaoftheirnewmethod(majoritarianjudgment)isthatvotersassigngradestoeachcandidatefromacommonlyacceptedgradinglanguage.Oncethegradesare



assigned,eachcandidateisassignedhermediangrade.Thewinner(s)is(are)thecandidate(s)withthehighestmediangrade.Thedetailsofthisprocedurearebeyondthescopeofthisarticle,buttheycanbefoundalongwithaxiomaticcharacterizationsinBalinskiandLaraki(2010).

Thissectionintroducedanumberofdifferentproceduresthatcanbeusedtomakeagroupdecision.Onestrikingfactisthatmanyofthedifferentproceduresgiveconflictingresultsonthesameinput.Thisraisesanimportantquestion:Howshouldwecomparethedifferentprocedures?Canwearguethatsomeproceduresarebetterthanothers?Thereareanumberofdifferentcriteriathatcanbeusedtocompareandcontrastdifferentvotingmethods:

1. Pragmaticconcerns:Istheprocedureeasytouse?Isitlegaltouseaparticularvotingprocedureforanationalorlocalelection?Theimportanceofeaseofuseshouldnotbeunderestimated:Despiteitsmanyflaws,pluralityrule(arguablythesimplestvotingproceduretouseandunderstand)is,byfar,themostcommonlyusedmethod(cf.thediscussionbyLevinandNalebuff1995,p.19).

2. Behavioralconsiderations:Dothedifferentproceduresreallyleadtodifferentoutcomesinpractice?Aninterestingstrandofresearch,behavorialsocialchoice,incorporatesempiricaldataaboutactualelectionsintothegeneraltheoryofvoting(ThisisdiscussedbrieflyinSection5.SeeRegenwetteretal.2006,foranextensivediscussion).

3. Informationrequiredfromthevoters:Whattypeofinformationdotheballotsconvey?Whilerankedprocedures(e.g.,BordaCount)requirethevotertocompareallofthecandidates,itisoftenusefultoaskthevoterstoreportsomethingabouttheintensitiesoftheirpreferencesoverthecandidates.Ofcourse,thereisatradeoff:Limitingwhatvoterscanexpressabouttheiropinionsofthecandidatesoftenmakesaproceduremucheasiertouseandunderstand.

4. Axiomaticcharacterizationresultsandvotingparadoxes:Muchoftheworkinvotingtheoryhasfocusedoncomparingandcontrastingvotingproceduresintermsofabstractprinciplesthattheysatisfy.Thegoalistocharacterizethedifferentvotingproceduresintermsofnormativeprinciplesofgroupdecisionmaking.SeeSections3and5.2fordiscussions.

3.VotingParadoxes

Inthissection,Iintroduceanddiscussanumberofvotingparadoxesi.e.,anomaliesthathighlightproblemswithdifferentmethods.SeeSaari(1995,2001)andNurmi(1999)forpenetratinganalysesthatexplaintheunderlyingmathematicsbehindthedifferentvotingparadoxes.

3.1Condorcet'sParadox

Averycommonassumptionisthatarationalpreferenceorderingmustbetransitive(i.e.,ifAispreferredtoB,andBispreferredtoC,thenAmustbepreferredtoC.Seetheentryonpreferences(HanssonandGrneYanoff2009)foranextendeddiscussionoftherationalebehindthisassumption).Indeed,ifavoter'spreferenceorderingisnottransitive,allowingforcycles(A>B>C>A),thenthereisnocandidatethatthevotercanbesaidtoactuallysupport(foreachcandidate,thereisanothercandidatethatthevoterprefers).Suchvotershavecontradictoryopinionsaboutthecandidatesand,arguably,shouldbeignoredoreliminatedbyanyvotingsystem.Manyauthorsarguethatsuchvoterswithcyclicpreferenceorderingshaveinconsistentopinionsaboutthecandidatesandshouldbeignoredbyanyvotingprocedures(inparticular,Condorcetforcefullyarguedthispoint).AkeyobservationofCondorcet(whichhasbecomeknownasCondorcet'sParadox)isthatevenifeachvoter'spreferenceorderingistransitive,themajorityorderingmaynotbetransitive.

Condorcet'soriginalexamplewasmorecomplicated,butthefollowingsituationwiththreevotersand



threecandidatesillustratesthephenomenon:

#Voters

1 1 1

A C B

B A C

C B A

Notethatwehave:

CandidateAbeatscandidateB21inaoneononeelection.CandidateBbeatscandidateC21inaoneononeelection.CandidateCbeatscandidateA21inaoneononeelection.

Thus,wehaveamajoritycycleA>MB>MC>MA,andsothereisnoCondorcetwinner.Oneinterpretationisthat,althougheachoftheindividualvotershasarationalpreferenceordering,thegroup'spreferenceordering(definedasthemajorityordering)isnotrational.Thissimple,butfundamentalobservationhasbeenextensivelystudied(seeGehrlein2006,foranoverviewoftheliterature).

3.1.1ElectingtheCondorcetWinner

Condorcet'sParadoxshowsthattheremaynotalwaysbeaCondorcetwinnerinanelection.However,onenaturalrequirementforavotingruleisthatifthereisaCondorcetwinner,thenthatcandidateshouldbeelected.VotingproceduresthatsatisfythispropertyarecalledCondorcetconsistent.ManyoftheproceduresintroducedabovearenotCondorcetconsistent.IalreadypresentedanexampleshowingthatpluralityruleisnotCondorcetconsistent(infact,pluralityrulemayevenelecttheCondorcetloser).

TheexamplefromSection1showsthatBordaCountisnotCondorcetconsistent.Infact,thisisaninstanceofageneralphenomenonthatFishburn(1974)calledCondorcet'sotherparadox.Considerthefollowingvotingsituationwith81votersandthreecandidatesfromCondorcet(1785).

#Voters

30 1 29 10 10 1

A A B B C C

B C A C A B

C B C A B A

ThemajorityorderingisA>MB>MC,soAistheCondorcetwinner.UsingtheBordarule,wehave:

BS(A)=231+139+011=101BS(B)=239+131+011=109BS(C)=211+111+059=33

So,candidateBistheBordawinner.Condorcetpointedoutsomethingmore:Theonlywaytoelect



candidateAusinganyscoringmethodistoassignmorepointstocandidatesrankedsecondthantocandidatesrankedfirst.Ascoringmethod,whichgeneralizestheBordascore,isdefinedbyfirstfixinganondecreasingsequenceofrealnumberss0s1sn1withs0Score(B),wemusthave231+v39>239+v31,whichimpliesthatv>2.But,ofcourse,itiscounterintuitivetogivemorepointsforbeingrankedsecondthanforbeingrankedfirst.PeterFishburngeneralizedthisexampleasfollows:

Theorem(Fishburn1974).Forallm3,thereissomevotingsituationwithaCondorcetwinnersuchthateveryweightedscoringrulewillhaveatleastm2candidateswithagreaterscorethantheCondorcetwinner.

So,noscoringruleisCondorcetconsistent,butwhataboutothermethods?ThefollowingexamplefromStevenBrams(2008,Chapter3)showsthattherearesituationsinwhichnofixedvotingrulecanelectaCondorcetwinner.Afixedvotingrule(orkApprovalVoting)isamethodbywhichthevoterschooseapredeterminednumberofcandidates.Forexample,pluralityisavoteforonefixedrule.Considerthefollowingvotingsituationwithfivevotersandfourcandidates:

#Voters

2 2 1

A B C

D D A

B A B

C C D

CandidateAistheuniqueCondorcetwinner(themajorityorderingsisA>MB>MD>MC),butnofixedrulevotingprocedurewillguaranteethatAiselected.

Usingvotefor1(pluralityrule),candidatesAandBaretiedforthewin.Usingvotefor2,candidateDiselected.Usingvotefor3,candidatesAandBaretiedforthewin.

Ofcourse,approvalvotingmayelectcandidateA(forexample,ifeveryoneapprovesofAandallcandidatestheyrankhigherthanA).Infact,Brams(2008,Chapter2)provesthatifthereisauniqueCondorcetwinner,thenthatcandidatemaybeelectedunderapprovalvoting(assumingthatallvotersvotesincerely:seeBrams2008,Chapter2,foradiscussion).Notethatapprovalvotingmayalsoelectothercandidates(perhapseventheCondorcetloser).

AnumberofvotingproceduresweredevisedspecificallytoguaranteethataCondorcetwinnerwillbe



elected,ifoneexists.IdiscussfourexamplestogiveaflavorofhowsuchCondorcetconsistentprocedureswork.(SeeBramsandFishburn2002,andTaylor2005formoreexamples.)

CondorcetRule:Eachvotersubmitsalinearorderingoverallthecandidates.IfthereisaCondorcetwinner,thenthatcandidatewinstheelection.Otherwise,allcandidatestieforthewin.

Copeland'sRule:Eachvotersubmitsalinearorderingoverallthecandidates.AwinlossrecordforcandidateBiscalculatedasfollows:

WL(B)=#{C|B>MC}#{C|C>MB}

TheCopelandwinneristhecandidatethatmaximizesWL.

ThenextmethodwasproposedbyCharlesDodgson(betterknownbythepseudonymLewisCarroll).Interestingly,thisisanexampleofaprocedureinwhichitiscomputationallydifficulttocomputethewinner(thatis,theproblemofcalculatingthewinnerisNPcomplete).SeeBartdholdietal.(1989)foradiscussion.

Dodgson'sMethod:Eachvotersubmitsalinearorderingoverallthecandidates.Foreachcandidate,determinethefewestnumberofpairwiseswapsneededtomakethatcandidatetheCondorcetwinner.Thecandidate(s)withthefewestswapsis(are)declaredthewinner(s).

Black'sProcedure:Eachvotersubmitsalinearorderingoverallthecandidates.IfthereisaCondorcetwinner,thenthatcandidateisthewinner.Otherwise,letthewinnersbetheBordaCountwinners.

Theseprocedures(andtheotherCondorcetconsistentprocedures)guaranteethataCondorcetwinner,ifoneexists,willbeelected.But,shouldaCondorcetwinnerbeelected?TherearestrongintuitionsthataCondorcetwinner(ifoneexists)isthecandidatethatbestreflectsthewillofthevotersandthatthereissomethingamisswithavotingprocedurethatdoesnotalwayselectsuchacandidate.However,thereareargumentsagainsttheseintuitions.ThemostpersuasiveargumentcomesfromtheworkofDonaldSaari(1995,2001).Considerthefollowingexampleof81voters(thisexamplewasoriginallydiscussedbyCondorcet).

#Voters

30 1 29 10 10 1

A A B B C C

B C A C A B

C B C A B A

ThisisanotherexamplethatshowsthatBorda'smethodneednotelecttheCondorcetwinner.Themajorityorderingis

A>MB>MC,



whiletherankinggivenbytheBordascoreis

B>BordaA>BordaC.

However,thereisanargumentthatcandidateBisthebestchoiceforthiselectorate.Saari'scentralobservationistonotethatthe81voterscanbedividedintothreegroups:

#Voters

10 10 10

A B C

B C A

C A B

#Voters

1 1 1

A C B

C B A

B A C

#Voters

20 28

A B

B A

C C

Group1 Group2 Group3

Groups1and2constitutemajoritycycleswiththevotersevenlydistributedamongthethreepossibleorderings.Thatis,thesegroupsformaperfectsymmetryamongthelinearorderings.So,withineachofthesegroups,thevoters'opinionscanceleachotherouttherefore,thedecisionshoulddependonlyonthevotersingroup3.Ingroup3,candidateBistheclearwinner.

3.2FailuresofMonotonicity

Avotingprocedureismonotonicprovidedthatmovingupintherankingsdoesnotadverselyaffectacandidate'schancestowinanelection.Thispropertycapturestheintuitionthatreceivingmoresupportfromthevotersisalwaysbetterforacandidate.Forexample,itiseasytoseethatpluralityruleismonotonic:Themorevotesacandidatereceives,thebetterchancethecandidatehastowin.Surprisingly,therearevotingmethodsthatdonotsatisfythisnaturalproperty.Themostwellknownexampleispluralitywithrunoff.Considerthetwotablesbelow.Notethattheonlydifferencebetweenthetwotablesisthepreferenceorderingsofthefourthgroupofvoters.ThisgroupoftwovotersranksBaboveAaboveCinthetableontheleftandswapsBandAinthetableontheright(so,AisnowtheirtoprankedcandidateBisrankedsecondandCisstillrankedthird).

#Voters

6 5 4 2

A C B B

B A C A

C B A C

#Voters

6 5 4 2

A C B A

B A C B

C B A C

CandidateAisthepluralitywithrunoffwinner CandidateCisthepluralitywithrunoffwinner

Intheelectionontheleft,candidateC,withfivevotes,iseliminatedinthefirstround.Then,C'svotesarealltransferredtocandidateA,givingheratotalof11towintheelection.However,intheelectionontheright,evenaftermovingupintherankingsofthefourthgroup(Aisnowrankedfirstbythisgroup),candidateAdoesnotwinthiselection.Infact,bytryingtogivemoresupporttothewinneroftheelectionontheleft,ratherthansolidifyingA'swin,thelastgroup'sleastpreferredcandidateendedup



winningtheelection!Intheelectionontheright,ratherthanCbeingeliminatedinthefirstround,itiscandidateB,withonlyfourvotes,whoiseliminated.OnceBiseliminated,candidateCbeatscandidateA(CreceivesninevoteswhileAreceiveseight).

Theaboveexampleissurprisingsinceitsuggeststhat,whenusingpluralitywithrunoff,itmaynotalwaysbebeneficialforacandidatetoreceiveextravotesinthefirstround.AsecondexampleofafailureofmontonicityisthenoshowparadoxofFishburnandBrams(1983),asthefollowingexampleillustrates.Supposethattherearethreecandidates,andthepopulationisdividedintothefollowinggroups:

#Voters

417 82 143 357 285 324

A A B B C C

B C A C A B

C B C A B A

Inthefirstround,candidateCwinstheelectionwith609votes(butthisisnotanabsolutemajority)candidateBreceives500votesandcandidateAreceives499votes.Thus,candidateAiseliminatedinthefirstround.Inthesecondround,417votesaretransferredtocandidateBand82votesaretransferredtocandidateC.Thus,candidateBwinstheelectionwith917votes(candidateCreceivesatotalof691votes).Now,supposethattherearetwovoterswiththerankingA>B>Cwhodidnottakepartintheaboveelection.ThesetwovotersrankAfirst,andso,theycertainlywouldpreferthattheirsupportforcandidateAbetakenintoaccount.But,considerwhathappenswhenthesetwovotersareaddedtothepopulation:

#Voters

419 82 143 357 285 324

A A B B C C

B C A C A B

C B C A B A

Inthiselection,candidateCstillwinsthefirstroundwith609votes,butcandidateBiseliminatedsinceAnowreceives501voteswhileBreceivesonly500votes.ButthismeansthatcandidateCwinstheelection(Creceives966votesandAreceives644votes).So,byshowinguptotheelection,thesetwoextravotersactuallycausedtheirleastpreferredcandidatetowin!

3.3MultipleDistrictsParadox

Supposethatapopulationisdividedintodistricts.Ifacandidatewinseachofthedistricts,onewouldexpectthatcandidatetowintheelectionovertheentirepopulationofvoters.Thisiscertainlytrueforpluralityvote:Ifacandidateisrankedfirstbyamajorityofthevotersinineachofthedistricts,thenthatcandidatewillalsoberankedfirstbyamajorityofvotersovertheentirepopulation.Interestingly,though,thisisnottrueforpluralityrulewithrunoff,asthefollowingexamplefromFishburnandBrams(1983)shows.



District1

#Voters

160 0 143 0 0 285

A A B B C C

B C A C A B

C B C A B A

District2

#Voters

257 82 0 357 285 39

A A B B C C

B C A C A B

C B C A B A

CandidateAwinsbothdistricts:

District1:Thereareatotalof588votersinthisdistrict.CandidateBreceivesthefewestfirstplacevotes,andsoiseliminatedinthefirstround.Inthesecondround,candidateAisnowthepluralitywinnerwith303totalvotes.

District2:Thereareatotalof1020votersinthisdistrict.CandidateCreceivesthefewestfirstplacevotes(324),andsoiseliminatedinthefirstround.Inthesecondround,285votesaretransferredtocandidateAand39aretransferredtocandidateC.Inthesecondround,CandidateAisthepluralitywinnerwith644votes.

However,notethatifyoucombinethetwodistricts,thenCandidateBisthewinner(thecombineddistrictsgiveustheexamplediscussedaboveinSection3.2).

ThisparadoxisanexampleofamoregeneralphenomenonknownasSimpson'sParadox(MalinasandBigelow2009).SeeSaari(2001,Section4.2)foradiscussionofSimpson'sParadoxinthecontextofvotingtheory.

3.4TheMultipleElectionsParadox

Thisparadox,firstintroducedbyBrams,KilgourandZwicker(1998),hasasomewhatdifferentstructurefromtheparadoxesdiscussedabove.Votersaretakingpartinareferendum,wheretheyareaskedtheiropiniondirectlyaboutvariouspropositions.So,votersmustselecteitheryes(Y)orno(N)foreachproposition.Supposethatthereare13voterswhocastthefollowingvotesforthreepropositions(sovoterscancastoneofeightpossiblevotes):

Propositions YYY YYN YNY YNN NYY NYN NNY NNN

#Votes 1 1 1 3 1 3 3 0

Whenthevotesaretalliedforeachpropositionseparately,theoutcomeisNforeachproposition(Nwins76forallthreepropositions).Puttingthisinformationtogether,thismeansthatNNNistheoutcomeofthiselection.However,thereisnosupportforthisoutcomeinthispopulationofvoters.



AsimilarissueisraisedbyAnscombe'sparadox(Anscombe1976),inwhich:

Itispossibleforamajorityofvoterstobeonthelosingsideofamajorityofissues.

Thisphenomenonisillustratedbythefollowingexamplewithfivevotersvotingonthreedifferentissues(thevoterseithervoteryesornoonthedifferentissues).

Issue1 Issue2 Issue3

Voter1 yes yes no

Voter2 no no no

Voter3 no yes yes

Voter4 yes no yes

Voter5 yes no yes

Majority yes no yes

However,amajorityofthevoters(voters1,2and3)donotsupportthemajorityoutcomeonamajorityoftheissues(notethatvoter1doesnotsupportthemajorityoutcomeonissues2and3voter2doesnotsupportthemajorityoutcomeonissues1and3andvoter3doesnotsupportthemajorityoutcomeonissues1and2)!

Theissueismoreinterestingwhenthevotersdonotvotedirectlyontheissues,butoncandidatesthattakepositionsonthedifferentissues.SupposetherearetwocandidatesAandBwhotakethefollowingpositionsonthethreeissues:

Issue1 Issue2 Issue3

CandidateA yes no yes

CandidateB no yes no

CandidateAtakesthemajorityposition,agreeingwithamajorityofthevotersoneachissue,andcandidateBtakestheopposite,minorityposition.Underthenaturalassumptionthatvoterswillvoteforthecandidatewhoagreeswiththeirpositiononamajorityoftheissues,candidateBwillwintheelection(eachofthevoters1,2and3agreewithBontwoofthethreeissues,soBwinstheelection32)!ThisversionoftheparadoxisknownasOstrogorski'sParadox(Ostrogorski1902).(SeeKelly1989RaeandDaudt1976Wagner1983,1984andSaari2001,Section4.6foranalysesofthisparadoxandPigozzi2005,forrelationshipstojudgmentaggregationliterature.)

4.TopicsinVotingTheory

4.1Strategizing

Inthediscussionabove,Ihaveassumedthatvotersselectballotssincerely.Thatis,thevotersaresimplytryingtocommunicatetheiropinionsaboutthecandidatesundertheconstraintsofthechosenvotingmethod.However,inmanycontexts,voterswouldratherchoosestrategically.OneneedonlylooktorecentU.S.electionstoseeconcreteexamplesofstrategicvoting.Themostoftencitedexampleisthe



2000U.S.election:ManyvoterswhorankedthirdpartycandidateRalphNaderfirstvotedfortheirsecondchoice(typicallyAlGore).Adetailedoverviewoftheliteratureonstrategicvotingisbeyondthescopeofthisarticle(seeTaylor2005foradiscussionandpointertotherelevantliteraturealsoseePoundstone2008foranentertainingandinformativediscussionoftheoccurrenceofthisphenomnoninmanyactualelections).Iwillexplainthemainissues,focusingonspecificvotingrules.

Ingeneral,therearetwogeneraltypesofmanipulationthatcanbestudiedinthecontextofvoting.Thefirstismanipulationbyachairmanoroutsidepartythathastheauthoritytosettheagendaorselectthevotingmethodthatwillbeused.So,theoutcomeofanelectionisnotmanipulatedfromwithinbyunhappyvoters,but,rather,itiscontrolledbyanoutsideauthorityfigure.Toillustratethistypeofcontrol,considerapopulationwiththreevoterswhosepreferencesoverfourcandidatesaregiveninthetablebelow:

#Voters

1 1 1

B A C

D B A

C D B

A C D

NotethateveryonepreferscandidateBovercandidateD.Nonetheless,achairmancanasktherightquestionssothatcandidateDendsupbeingelected.Thechairmanproceedsasfollows:First,askthevotersiftheyprefercandidateAorcandidateB.SincethevoterspreferAtoBbyamarginoftwotoone,thechairmandeclaresthatcandidateBisnolongerintherunning.ThechairmanthenasksvoterstochoosebetweencandidateAandcandidateC.CandidateCwinsthiselection21,socandidateAisremoved.Finally,inthelastroundthechairmanasksvoterstochoosebetweencandidateCandcandidateD.CandidateDwinsthiselection21andisdeclaredthewinner.

Asecondtypeofmanipulationfocusesonhowthevotersthemselvescanmanipulatetheoutcomeofanelectionbymisrepresentingtheirpreferences.Considerthefollowingtwosevenvoter,threecandidateelectionscenarios:

#Voters

3 3 1

A B C

B A A

C C B

#Voters

3 3 1

A B C

B C A

C A B

ElectionScenario1 ElectionScenario2

Theonlydifferencebetweenthetwoscenariosisthatthemiddlegroupofvotersswappedtheirorderingoftheirbottomrankedcandidates(AandC).Inthefirstelectionscenario,candidateAistheBordacountwinner.However,inthesecondelectionscenario,candidateBistheBordacountwinner.So,ifweassumethatscenario1representsthetruepreferencesoftheelectorate,itisintheinterestofthemiddlegrouptomisrepresenttheirpreferenceandrankCsecond,followedbyA,sincetheoutcomewillresultin



theirmostpreferredcandidate(B)beingelected.ThisisaninstanceofageneralresultknownastheGibbardSatterthwaiteTheorem(Gibbard1973Satterthwaite1975):Undernaturalassumptions,thereisnovotingmethodthatguaranteesthatvoterswillchoosetheirballotssincerely(foraprecisestatementofthistheoremandanextensiveanalysis,seeTaylor2005).

Thereisagrowingliteraturethatcharacterizesvotingmethodsintermsofhowcomputationallycomplextheyaretomanipulate.Adiscussionofthisliteratureisbeyondthescopeofthisarticlehowever,IreferthereadertoBartholdietal.(1989)Conitzeretal.(2007)FaliszewskiandProcaccia(2010)andFaliszewskietal.(2010)foranintroductionandpointerstotherelevantliterature.

4.2CharacterizationResults

Muchoftheliteratureonvotingtheory(and,moregenerally,socialchoicetheory)isfocusedonsocalledaxiomaticcharacterizationresults.Themaingoalhereistocharacterizedifferentvotingmethodsintermsofabstractnormativeprinciplesofcollectivedecisionmaking.So,theaxiomsdiscussedinthisliteratureareintendedtodescribepropertiesthatagroupdecisionmethodshouldsatisfy.Itisworthpointingoutthatthisisdifferentfromthewayamathematicianorlogicianusesthewordaxiom:Tomathematiciansorlogicians,axiomsarebasicprinciplesthatamathematicaltheoryorlogicalsystemdosatisfy.Thatis,axiomsarebeingusedinadescriptivesense.(SeeEndriss2011,foraninterestingdiscussionofcharacterizationresultsfromalogician'spointofview.)

Iwillnotattempttoprovideageneraloverviewofaxiomaticcharacterizationsinsocialchoicetheoryhere(seeGaertner2006,foranintroductiontothisvastliterature).Rather,Iinformallydiscussafewkeyaxiomsandresultsandhowtheyrelatetothevotingmethodsandparadoxesdiscussedabove.Istartwiththreecoreproperties.

Anonymity:Thenamesofthevotersdonotmatter:Iftwovoterschangevotes,thentheoutcomeoftheelectionisunaffected.Neutrality:Thenamesofthecandidates,oroptions,donotmatter:Iftwocandidatesareexchangedineveryranking,thentheoutcomeoftheelectionchangesaccordingly.UniversalDomain:Thevotersarefreetohaveanyopinionaboutthecandidates.Inotherwords,nopreferenceorderingoverthecandidatescanbeignoredbyavotingmethod.Formally,thismeansthatvotingmethodsmustbetotalfunctionsonthespaceofallprofiles(recallthataprofileisasequenceofballots,onefromeachvoter.Here,Iamassuming,asistypicalforthisliterature,thattheballotsarethelinearorderingsoverthesetofcandidates).

Thesepropertiesensurethattheoutcomeofanelectiondependsonlyonthevoters'opinions,withallthevotersbeingtreatedequally.Otherpropertiesareintendedtoruleoutsomeoftheparadoxesandanomaliesdiscussedabove.Insection4.1,thereisanexampleofasituationinwhichacandidateiselected,eventhoughallthevoterspreferadifferentcandidate.Thenextprinciplerulesoutsuchsituations:

Unanimity:IfcandidateAispreferredtocandidateBbyallvoters,thencandidateBshouldnotwintheelection.

Section3.2discussedexamplesinwhichcandidatesenduplosinganelectionasaresultofmoresupportfromsomeofthevoters.Intuitively,avotingprocedureismonotonicifmovingupintherankings(allelsebeingequal)shouldnotcauseacandidatetolosetheelection.Therearemanywaystomakethisprecise.Thefollowingstrongversion(calledPositiveResponsivenessintheliterature)isusedtocharacterizemajorityrulewhenthereareonlytwocandidates:



PositiveResponsiveness:IfcandidateAistiedforthewinandmovesupintherankings,thencandidateAistheuniquewinner.

Icannowstateourfirstcharacterizationresult.Notethatinalloftheexamplesabove,itiscrucialthattherearethreeormorecandidates(forexample,Condorcet'sparadoxdependscriticallyontherebeginthreeormorecandidates).Infact,whenthereareonlytwocandidates,oroptions,thenmajorityrule(choosetheoptionwiththemostvotes)canbesingledoutasbest:

Theorem(May1952).Asocialdecisionmethodforchoosingbetweentwocandidatessatisfiesneutrality,anonymityandpositiveresponsivenessifandonlyifthemethodismajorityrule.

SeeMay(1952)foraprecisestatementofthistheoremandAsanandSanver(2002),Maskin(1995),andWoeginger(2003)forgeneralizationsandalternativecharacterizationsofmajorityrule.Withmorethantwocandidates,themostimportantresultisKenArrow'scelebratedimpossibilitytheorem(1963).Arrowshowedthatthereisnosocialwelfarefunction(asocialchoicefunctionmapsthevoters'linearpreferenceorderingstoasinglesocialpreferenceordering)satisfyinguniversaldomain,unanimity,nondictatorship(thesocialorderingisdefinedtobetheorderingofasingleindividual)andthefollowingkeyproperty:

IndependenceofIrrelevantAlternatives:Thesocialranking(higher,lower,orindifferent)oftwocandidatesAandBdependsonlyontherelativerankingsofAandBforeachindividual.

Thismeansthatifthevoters'rankingsoftwocandidatesAandBarethesameintwodifferentelectionscenarios,thenthesocialrankingsofAandBmustbethesame.Thisisaverystrongpropertythathasbeenextensivelycriticized(seeGaertner2006,forpointerstotherelevantliterature).ItisbeyondthescopeofthisarticletogointodetailabouttheproofandtheramificationsofArrow'stheorem,butInotethatmanyofthevotingmethodswehavediscusseddonotsatisfytheaboveproperty.AstrikingexampleofavotingmethodthatdoesnotsatisfyindependenceofirrelevantalternativesisBordacount.Considerthefollowingtwoelectionscenarios:

#Voters

3 2 2

A B C

B C A

C A B

#Voters

3 2 2

A B C

B C X

C X A

X A B

ElectionScenario1 ElectionScenario2

NoticethattherelativerankingsofcandidatesA,BandCarethesameinbothelectionscenarios.Inthesecondscenario,anew(undesirable)candidateisadded(i.e.,anirrelevantalternative).TherankingofthecandidatesaccordingtotheirBordascoreinscenario1putsAfirstwitheightpoints,BsecondwithsevenpointsandClastwithsixpoints.WithcandidateXintheelection(scenario2),thisrankingisreversed:CandidateCisfirstwith13voterscandidateBissecondwith12pointscandidateAisthirdwith11pointsandcandidateXislastwithsixpoints.So,eventhoughtherelativerankingsofcandidatesA,BandCdonotdifferinthetwoscenarios,thepresenceofcandidateXreversestheBordarankings.



Finally,Idiscusscharacterizationsofallscoringrules(anymethodthatcalculatesascorebasedonweightsgiventodifferentcandidatesaccordingtowheretheyfallintherankingseeSection3.1.1foradefinition)andApprovalvoting.Onedefiningpropertyofthesemethodsisthattheydonotsufferfromthemultipledistrictsparadox.

Reinforcement:SupposethatN1andN2aredisjointsetsofvotersfacingthesamesetofcandidates.Further,supposethatW1isthesetofwinnersforthepopulationN1,andW2isthesetofwinnersforthepopulationN2.Ifthereisatleastonecandidatethatwinsbothelections,thenthewinner(s)fortheentirepopulation(includingvotersfrombothN1andN2)isW1W2.

Thereinforcementpropertyexplicitlyrulesoutmultipledistrictsparadoxes(so,candidatesthatwinallsubelectionsareguaranteedtowinthefullelection).Inordertocharacterizeallscoringrules,oneadditionaltechnicalpropertyisneeded:

Continuity:SupposethatagroupofvotersN1electsacandidateAandadisjointgroupofvotersN2electsadifferentcandidateB.ThentheremustbesomenumbermsuchthatthepopulationconsistingofthesubgroupN2togetherwithmcopiesofN1willelectA.

Wethenhave:

Theorem(Young1975).Asocialdecisionmethodsatisfiesanonymity,neutrality,reinforcementandcontinuityifandonlyifthemethodisascoringrule.

ThisresultwasgeneralizedbyMyerson(1995)bydroppingtherequirementthatvotershavelinearpreferences.AdditionalaxiomshavebeensuggestedthatsingleoutBordacountamongallscoringmethods(Young1974NitzanandRubinstein1981).Infact,SaarihasarguedthatanyfaultorparadoxadmittedbyBorda'smethodalsomustbeadmittedbyallotherpositionalvotingmethods(Saari1989,pg.454).Forexample,itisoftenremarkedthatBordacount(andallscoringrules)canbeeasilymanipulatedbythevoters.Saari(1995,Section5.3.1)showsthatamongallscoresrulesBordacountistheleastsusceptibletomanipulation(inthesensethatithasthefewestprofileswhereasmallpercentageofvoterscanmanipulatetheoutcome).

IconcludethisbriefdiscussionofcharacterizationresultswithFishburn'scharacterizationofapprovalvoting(seeXu2010,foranoverviewofthedifferentcharacterizationsofapprovalvoting).

Theorem(Fishburn1978).Asocialdecisionmethodisapprovalvotingifandonlyifthemethodsatisfiesanonymity,neutrality,reinforcementandthefollowingtechnicalproperty:

Ifthereareexactlytwovoterswhoapproveofdisjointsetsofcandidates,thenthemethodsselectsaswinnersallthecandidateschosenbythetwovoters(i.e.,theunionoftheballotschosenbythevoters).

4.3VotingtoTracktheTruth

Thevotingmethodsdiscussedabovehavebeenjudgedonproceduralgrounds.ThisproceduralistapproachtocollectivedecisionmakingisdefinedbyColemanandFerejohn(1986,p.7)asonethatidentifiesasetofidealswithwhichanycollectivedecisionmakingprocedureoughttocomply.[A]processofcollectivedecisionmakingwouldbemoreorlessjustifiabledependingontheextenttowhich



itsatisfiesthem.Theauthorsaddthatadistinguishingfeatureofproceduralismisthatwhatjustifiesa[collective]decisionmakingprocedureisstrictlyanecessarypropertyoftheprocedureoneentailedbythedefinitionoftheprocedurealone.Indeed,thecharacterizationtheoremsdiscussedintheprevioussectioncanbeviewedasanimplementationofthisidea(cf.Riker1982).Thegeneralviewistoanalyzevotingmethodsintermsoffairnesscriteriathatensurethatagivenmethodissensitivetoallofthevoters'opinionsintherightway.

However,onemaynotbeinterestedonlyinwhetheracollectivedecisionwasarrivedatintherightway,butinwhetherornotthecollectivedecisioniscorrect.ThisepistemicapproachtovotingisnicelyexplainedbyJoshuaCohen(1986):

Anepistemicinterpretationofvotinghasthreemainelements:(1)anindependentstandardofcorrectdecisionsthatis,anaccountofjusticeorofthecommongoodthatisindependentofcurrentconsensusandtheoutcomeofvotes(2)acognitiveaccountofvotingthatis,theviewthatvotingexpressesbeliefsaboutwhatthecorrectpoliciesareaccordingtotheindependentstandard,notpersonalpreferencesforpoliciesand(3)anaccountofdecisionmakingasaprocessoftheadjustmentofbeliefs,adjustmentsthatareundertakeninpartinlightoftheevidenceaboutthecorrectanswerthatisprovidedbythebeliefsofothers.(p.34)

Underthisinterpretationofvoting,agivenmethodisjudgedonhowwellittracksthetruthofsomeobjectivefact(thetruthofwhichisindependentofthemethodbeingused).Acomprehensivecomparisonofthesetwoapproachestovotingtouchesonanumberofissuessurroundingthejustificationofdemocracy(cf.Christiano2008)however,Iwillnotfocusonthesebroaderissueshere.Instead,Ibrieflydiscussananalysisofmajorityrulethattakesthisepistemicapproach.

ThemostwellknownanalysiscomesfromthewritingsofCondorcet(1785).Thefollowingtheorem,whichisattributedtoCondorcetandwasfirstprovedformallybyLaplace,showsthatifthereareonlytwooptions,thenmajorityruleis,infact,thebestprocedurefromanepistemicpointofview.Thisisinterestingbecauseitalsoshowsthataproceduralistanalysisandanepistemicanalysisbothsingleoutmajorityrulesasthebestvotingmethodwhenthereareonlytwocandidates.

Assumethattherearenvotersthathavetodecidebetweentwoalternatives.Exactlyoneofthesealternativesis(objectively)correctorbetter.Thetypicalexamplehereisajurydecidingwhetherornotadefendantisguilty.ThetwoassumptionsoftheCondorcetjurytheoremare:

Independence:Thevoters'opinionsareprobabilisticallyindependent(so,theprobabilitythattwoormorevotersarecorrectistheproductoftheprobabilitythateachindividualvoteriscorrect).VoterCompetence:Theprobabilitythatavotermakestherightdecisionisgreaterthan1/2,andthisprobabilityisthesameforallvoters.

SeeDietrich(2008)foracriticaldiscussionofthesetwoassumptions.Theclassictheoremis:

CondorcetJuryTheorem.SupposethatIndependenceandVoterCompetencearebothsatisfied.Then,asthegroupsizeincreases,theprobabilitythatthemajoritychoosesthecorrectoptionincreasesandconvergestocertainty.

SeeNitzan(2010)foramodernexpositionofthistheorem.Forageneralizationofthistheorembeyondtwocandidates,seeYoung(1995)andListandGoodin(2001).ConitzerandSandholm(2005)taketheseideasfurtherbyclassifyingdifferentvotingmethodsaccordingtowhetherornotthemethodscanbe



viewedasamaximumlikelihoodestimator(foranoisemodel).

5.ConcludingRemarks:fromTheorytoPractice

Aswithanymathematicalanalysisofsocialphenomena,questionsaboundaboutthereallifeimplicationsofthetheoreticalanalysisofthevotingmethodsgivenabove.Themaindifficultyiswhetherthevotingparadoxesaresimplyfeaturesoftheformalframeworkusedtorepresentanelectionscenarioorformalizationsofreallifephenomena.Thisraisesanumberofsubtleissuesaboutthescopeofmathematicalmodelinginthesocialsciences,manyofwhichfalloutsidethescopeofthisarticle.Iconcludewithabriefdiscussionoftwoquestionsthatshedsomelightonhowoneshouldinterprettheaboveanalysis.

HowlikelyisaCondorcetParadoxoranyoftheothervotingparadoxes?Therearetwowaystoapproachthisquestion.Thefirstistocalculatetheprobabilitythatamajoritycyclewilloccurinanelectionscenario.Thereisasizableliteraturedevotedtoanalyticallyderivingtheprobabilityofamajoritycycleoccurringinelectionscenariosofvaryingsizes(seeGehrlein2006,andRegenwetteretal.2006,foroverviewsofthisliterature).Thecalculationsdependonassumptionsaboutthedistributionofpreferenceorderingsamongthevoters.Onedistributionthatistypicallyusedisthesocalledimpartialculture,whereeachpreferenceorderingispossibleandoccurswithequalprobability.Forexample,iftherearethreecandidates,anditisassumedthatthevoters'preferencesarerepresentedbylinearorderings,theneachlinearorderingcanoccurwithprobability1/6.Underthisassumption,theprobabilityofamajoritycycleoccurringhasbeencalculated(seeGehrlein2006,fordetails).Riker(1982,p.122)hasatableoftherelevantcalculations.Twoobservationsaboutthisdata:First,asthenumberofcandidatesandvotersincreases,theprobabilityofamajoritycyclesincreasestocertainty.Second,forafixednumberofcandidates,theprobabilityofamajoritycyclestillincreases,thoughnotnecessarilytocertainty(thenumberofvotersistheindependentvariablehere).Forexample,iftherearefivecandidatesandsevenvoters,thentheprobabilityofamajoritycycleis21.5percent.Thisprobabilityincreasesto25.1percentasthenumberofvotersincreasestoinfinity(keepingthenumberofcandidatesfixed)andto100percentasthenumberofcandidatesincreasestoinfinity(keepingthenumberofvotersfixed).Primafacie,thisresultsuggeststhatweshouldexpecttoseeinstancesoftheCondorcetandrelatedparadoxesinlargeelections.Ofcourse,thisinterpretationtakesitforgrantedthattheimpartialcultureisarealisticassumption.Manyauthorshavenotedthattheimpartialcultureisasignificantidealizationthatalmostcertainlydoesnotoccurinreallifeelections.Tsetlinetal.(2003)goevenfurtherarguingthattheimpartialcultureisaworstcasescenariointhesensethatanydeviationresultsinlowerprobabilitiesofamajoritycycle(seeRegenwetteretal.2006,foracompletediscussionofthisissue).

Asecondwaytoarguethattheabovetheoreticalobservationsarerobustistofindsupportingempiricalevidence.Forinstance,isthereevidencethatmajoritycycleshaveoccurredinactualelections?WhileRiker(1982)offersanumberofintriguingexamples,themostcomprehensiveanalysisoftheempiricalevidenceformajoritycyclesisprovidedbyMackie(2003,especiallyChapters14and15).Theconclusionisthat,instrikingcontrasttotheprobabilisticanalysisreferencedabove,majoritycyclestypicallyhavenotoccurredinactualelections.However,thisliteraturehasnotreachedaconsensusaboutthisissue(cf.Riker1982):Theproblemisthattheavailabledatatypicallydoesnotincludevoters'opinionsaboutallpairwisecomparisonofcandidates,whichisneededtodetermineifthereisamajoritycycle.So,thisinformationmustbeinferred(forexample,byusingstatisticalmethods)fromthegivendata.

Howdothedifferentvotingmethodscompareinactualelections?Inthisarticle,Ihaveanalyzedvotingmethodsunderhighlyidealizedassumptions.But,intheend,weareinterestedinaverypracticalquestion:Whichmethodshouldagivensocietyadopt?Ofcourse,anyanswertothisquestionwill



dependonmanyfactorsthatgobeyondtheabstractanalysisgivenabove.Aninterestinglineofresearchfocusesonincorporatingempiricalevidenceintothegeneraltheoryofvoting.Evidencecancomeintheformofacomputersimulation,adetailedanalysisofaparticularvotingmethodinreallifeelections(forexample,seeBrams2008,Chapter1,whichanalyzesApprovalvotinginpractice),orasinsituexperimentsinwhichvotersareaskedtofillinadditionalballotsduringanactualelection(Laslier2010,2011).

However,themoststrikingresultsherecanbefoundintheworkofMichaelRegenwetterandhicolleagues.Theyhaveanalyzeddatasetsfromavarietyofelections,showingthatmanyoftheusualvotingmethodsthatareconsideredirreconcilable(e.g.,plurality,BordacountandmethodsthatchoosetheCondorcetwinner)are,infact,inperfectagreement.Thissuggeststhatthetheoreticalliteraturemaypromoteoverlypessimisticviewsaboutthelikelihoodofconsensusamongconsensusmethods(Regenwetteretal.2009,p.840).SeeRegenwetteretal.2006foranintroductiontothemethodsusedintheseanalysesand(Regenwetteretal.2009)forthecurrentstateoftheart.

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