BERG SERGUEI NORINE FRANCIS EDWARD SU ROBIN THOMAS AND PAUL WOLLAN Manuscript August 2006 My idea of an agreeable person is a person who agrees with me Benjamin Disraeli 7 1 Introduction When is agreement possible An important aspect of group decis ID: 34948
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2D.BERG,S.NORINE,F.E.SU,R.THOMAS,ANDP.WOLLANApprovalvotinghasnotyetbeenadoptedforpoliticalelectionsintheUnitedStates.However,manyscienticandmathematicalsocieties,suchastheMathematicalAssociationofAmericaandtheAmericanMathematicalSociety,useapprovalvotingfortheirelections.Additionally,countriesotherthantheUnitedStateshaveusedapprovalvotingoranequiv-alentsystem.Fordetails,seeBramsandFishburn[2],whogivemanyreasonswhytheybelieveapprovalvotingisadvantageous.Inwhatfollows,ourstudyofagreeabilitywillhelpusunderstandwhenwecanguaranteeamajorityunderapprovalvoting.Considerthe2003Californiagubernatorialrecallelection,with135candidatesinthemix[6].Wemightimaginethesecandidatespositionedat135pointsonthelineinFigure1.IfeachCaliforniavoterapprovesofcandidateswithinsomerangeofpositions(callthisthevoter'sapprovalset),wemightwonderifandwhentheremightbeapointonthelinecoveredbyamajorityofthevoterapprovalsets,i.e.,aplatformonwhichamajorityofthevotersagree.Inthissetting,wemayassumethateachapprovalsetisaclosedintervalonRandwecallacollectionofvoters,togetherwiththeirapprovalsets,alinearsociety.Callasocietysuper-agreeableifforeverypairofvotersthereissomeplatformtheywouldbothapprove,i.e.,eachpairofapprovalsetshasanon-emptyintersection.Forlinearsocietiesthislocalconditionguaranteesastrongglobalproperty,namely,thatthereisaplatformthateveryvoterapproves!AsweshallseeinTheorem3,thisisaconsequenceofHelly'stheoremaboutintersectionsofconvexsets.Inthisarticle,weconsideravarietyofsimilartheorems.Forinstance,werelaxtheconditionaboveandcallasocietyagreeableifithasatleastthreevotersandamongeverythreevoters,thereissomepairofvoterswhoagreeonsomeplatform.Thenweprovethefollowing:Theorem1(TheAgreeableLinearSocietyTheorem).Inanagreeablelinearsociety,thereisaplatformwhichhastheapprovalofamajorityofvoters,i.e.,awinningplatform.Forexample,Figure2showsapprovalsetsforanagreeablelinearsocietyofsixvoters,andindeedthereareplatformsthatamajorityofvotersapprove.Asanotherapplicationofourtheorem,considerasituationinwhicheachvoter'sapprovalsetisaclosedsubintervalof[0;1]oflengthatleast1=3.ThenTheorem1guaranteesawinningplatform,sinceamonganythreeapprovalsetstheremustbeapairthatintersect.Weconsiderotherdegreesof\agreeability"andproveamoregeneralresultinTheorem8givingalowerboundforthesizeofthepluralityinapprovalvoting.Wealsobrie ystudysocietieswhoseapprovalsetsareconvexsubsetsofRd.Ageneralthemeofthisarticleisthatclassical(andnew)convexitytheoremshaveinter-estingsocialinterpretations,andthesesocialquestionsmotivatethestudyofsetintersec-tionsandperfectgraphs,sincetheyhavenaturalinterpretationsinthisvotingcontext.2.DefinitionsInthissection,wexterminologyandexplainsomeofthebasicconceptsuponwhichourresultsrely.LetussupposethatthesetofpossiblepreferencesismodeledbyasetX,calledthespectrum.Eachelementofthespectrumisaplatform.AssumethatthereisanitesetVofvoters,andeachvotervhasanapprovalsetAvofplatforms.WedeneasocietyStobeatriple(X;V;A)consistingofaspectrumX,asetofvotersV,andacollectionAofapprovalsetsforallthevoters.Ofparticularinteresttouswillbethecaseforalinearsociety,whereXisRandapprovalsetsinAareclosedintervals, 4D.BERG,S.NORINE,F.E.SU,R.THOMAS,ANDP.WOLLAN Figure3.ThesocietyofFigure2withagreementnumber4andagreementsetmarked.ForRd-convexsocieties,workconcerningsetintersectionscanbeappliedtotheagreementnumberproblem.ThemostwellknowntheoreminthisareaisHelly'stheorem.ThistheoremwasprovenbyHellyin1913,buttheresultwasnotpublisheduntil1921,byRadon[14].Theorem2(Helly).GiventconvexsetsinRdwheredt,ifeveryd+1ofthemintersectatacommonpoint,thentheyallintersectatacommonpoint.Helly'stheoremhasaniceinterpretationforRd-convexsocieties,especiallywhend=1,wheretheHellyconditionforapprovalsetsisequivalenttotheconditionforalinearsuper-agreeablesociety:Theorem3.Foreveryd1,a(d+1;d+1)-agreeableRd-convexsocietymustcontainatleastoneplatformthatisacceptabletoallvoters.Inparticular,whend=1andtheapprovalsetsofeverypairofvotersintersect,wehave:Corollary4(TheSuper-AgreeableSocietyTheorem).Alinearsuper-agreeablesocietymustcontainatleastoneplatformthatisacceptabletoallvoters.WeprovideasimpleproofoftheSuper-AgreeableSocietyTheorem(equivalently,Helly'stheoremindimension1)asitwillbeneededlater.AproofofHelly'stheoremforgeneraldmaybefoundin[12].Proof.Sinceeachvoteragreesonatleastoneplatformwitheveryothervoter,weseethatthesetsAimustbenon-empty.Thus,eachAiisanon-emptyclosedintervalin[0;1].Letx=maxifminfp2Aiggandy=minjfmaxfp2Ajgg.Weclaimthatxy.Why?Letibethevoterwhoseapprovalsetminimumismaximal,andletjbethevoterwhoseapprovalsetmaximumisminimal.Sincetheapprovalsetsofiandjintersect,theonlywaythiscouldholdisifxy.Therefore,everyapprovalsetcontainsthenon-emptyinterval[x;y];hencethereisaplatformcommontoallapprovalsets.BesidesHelly'stheorem,anotherfamoustheoremaboutsetintersectionsistheKKMlemma[10],whichisconcernedwithsetintersectionsonsimplices.Thereisavariantofthistheoremfortrees(e.g.,see[13])thatgeneralizesbothHelly'stheoremandtheKKMlemma,andsincealineisatree,Theorem4alsofollowsasaconsequence.Hereisanexampledemonstratingthattheconvexityassumptionisessential.Letn2beanintegerandletthespectrumofasocietySconsistofall2-elementsubsetsof 8D.BERG,S.NORINE,F.E.SU,R.THOMAS,ANDP.WOLLAN6.Rd-convexSocietiesInthissectionweproveahigherdimensionalanalogueofTheorem8bygivingalowerboundontheagreementproportionofa(k;m)-agreeableRd-convexsociety.Weneedadi erentmethodthanourmethodford=1,becauseford2,neitherFact1norFact2holds.WeusethefollowinggeneralizationofHelly'stheorem,duetoKalai[11].Theorem9(TheFractionalHelly'sTheorem).Letd1andnd+1beintegers,let 2[0;1]bearealnumber,andlet=1�(1� )1=(d+1).LetF1;F2;:::;FnbeconvexsetsinRdandassumethatforatleast �nd+1ofthe(d+1)-elementindexsetsIf1;2;:::;ngwehaveTi2IFi6=;.ThenthereexistsapointinRdcontainedinatleastnofthesetsF1;F2;:::;Fn.ThefollowingisthepromisedanalogueofTheorem8.Theorem10.Letd1,k2andmkbeintegers.Thenevery(k;m)-agreeableRd-convexsocietyhasagreementproportionatleast1�1��kd+1�md+11=(d+1).Proof.LetSbea(k;m)-agreeableRd-convexsociety,andletA1;A2;:::;Anbeitsvoterapprovalsets.LetuscallasetIf1;2;:::;nggoodifjIj=d+1andTi2IAi6=;.ByTheorem9itsucestoshowthatthereareatleast�kd+1�nd+1�md+1goodsets.Wewilldothisbycountingintwodi erentwaysthenumberNofallpairs(I;J),whereIJf1;2;:::;ng,IisgoodandjJj=m.Letgbethenumberofgoodsets.Sinceeverygoodsetisofsized+1andextendstoanm-elementsubsetoff1;2;:::;ngin�n�d�1m�d�1ways,wehaveN=g�n�d�1m�d�1.Ontheotherhand,everym-elementsetJf1;2;:::;ngincludesatleastonek-elementsetKwithTi2KAi6=;(becauseSis(k;m)-agreeable),andKinturnincludes�kd+1goodsets.ThusN�kd+1�nm,andhenceg�kd+1�nd+1�md+1,asdesired.Ford=1,Theorem10givesaworseboundthanTheorem8,andhenceford2,theboundismostlikelynotbestpossible.However,apossibleimprovementmustuseadi erentmethod,becausetheboundinTheorem9isbestpossible.AboxinRdistheCartesianproductofdclosedintervals,andwesaythatasocietyisad-boxsocietyifeachofitsapprovalsetsisaboxinRd.ItfollowsfromTheorem3thatd-boxsocietiessatisfytheconclusionofFact1(namely,thatthecliquenumberequalstheagreementnumber),andhencetheiragreementgraphscapturealltheessentialinformationaboutthesociety.Unfortunately,agreementgraphsofd-boxsocietiesare,ingeneral,notperfect.Forinstance,thereisa2-boxsocietywhoseagreementgraphisthecycleonvevertices.SeeFigure7.Forkm2k�2,thefollowingtheoremandcorollarywillresolvetheagreementproportionproblemforall(k;m)-agreeablesocietiessatisfyingtheconclusionofFact1,andhenceforall(k;m)-agreeabled-boxsocietieswhered1.Theorem11.Letm;k2beintegerswithkm2k�2,andletGbeagraphonnmverticessuchthateverysubsetofV(G)ofsizemincludesacliqueofsizek.Then!(G)n�m+k.Beforeweembarkonaproofletusmakeafewcomments.Firstofall,theboundn�m+kisbestpossible,asshownbythegraphconsistingofacliqueofsizen�m+kandm�kisolatedvertices.Second,theconclusion!(G)n�m+kimpliesthateverysubsetofV(G)ofsizemincludesacliqueofsizek,andsothetwostatementsareequivalentunderthe 10D.BERG,S.NORINE,F.E.SU,R.THOMAS,ANDP.WOLLAN!(G)!(Gnfx;yg)n�2�(m�2)+k�1=n�m+k�1.Wemayassumeinthelaststatementthatequalityholdsthroughout,becauseotherwiseGsatisestheconclusionofthetheorem.LetGdenotethecomplementofG;thatis,thegraphwithvertexsetV(G)andedgesetconsistingofpreciselythosepairsofdistinctverticesofGthatarenotadjacentinG.LetusnoticethatasetQisacliqueinGifandonlyifV(G)�QisavertexcoverinG.ThusthesizeofaminimumvertexcoverinGism�k+1.Since2(m�k+1)mn,byLemma12,thegraphGhasaninducedsubgraphHonexactlymverticeswithnovertexcoverofsizem�korsmaller.Byhypothesis,thegraphHhasacliqueQofsizek,butV(H)�QisavertexcoverinHofsizem�k,acontradiction.Corollary13.Letd1andm;k2beintegerswithkm2k�2,andletSbea(k;m)-agreeabled-boxsocietywithnvoters.ThentheagreementnumberofSisatleastn�m+k,andthisboundisbestpossible.Proof.TheagreementgraphGofSsatisesthehypothesisofTheorem11,andhenceithasacliqueofsizeatleastn�m+kbythattheorem.Sinced-boxsocietiessatisfytheconclusionofFact1,therstassertionfollows.Theboundisbestpossible,becausethegraphconsistingofacliqueofsizen�m+kandm�kisolatedverticesisanintervalgraph.7.SpeculationandOpenQuestionsAswehaveseen,setintersectiontheoremscanprovideausefulframeworktomodelandunderstandtherelationshipsbetweenpreferencesetsinmanysocialcontexts.Additionally,recentresultsindiscretegeometryhavesocialinterpretations.Thepiercingnumber[9]ofapprovalsetscanbeinterpretedastheminimumnumberofplatformsthatarenecessarysuchthateveryonehassomeplatformofwhichheorsheapproves.Setintersectiontheoremsonotherspaces(suchastreesandcycles)arederivedin[13]andsocialapplicationsareexplored,includinganapprovalvotinginterpretationwhenthesocietyhasacircularpoliticalspectrum.Wesuggestseveraldirectionswhichthereadermaywishtoexplore.ThemostnaturalproblemseemstobetodeterminetheagreementproportionforRd-convexandd-box(k;m)-agreeablesocieties.Thesmallestcasewherewedonotknowtheanswerisd=2,k=2,andm=3.RajneeshHegde(privatecommunication)foundanexampleofa(2;3)-agreeable2-boxsocietywithagreementproportion3=8.Additionally,wemustexamineourinitialassumptions.Forinstance,weassumedthatvotersplacecandidatesalongalinearspectruminexactlythesameorder,eventhoughvotersmayordercandidatesalongaspectrumdi erently.Also,whileconvexityseemstobearationalassumptioninthelinearcase,inmultipledimensions,additionalconsiderationsmayneedtobemade.TheoriginalconceptofanagreementgraphcouldbeappliedtoRd-convexsocietiestokeeptrackofmoreinformation.Forinstance,twovotersmightnotagreeoneveryaxis,meaningthattheirapprovalsetsdon'tintersect,butitmightbethecasethatmanyoftheprojectionsoftheirapprovalsetsdo.Inthiscase,onemaywishtoconsideranagreementgraphwithweightededges.Finally,wemightwonderabouttheagreementparameterskandmforvariousissueswhicha ectuspersonally.Forinstance,asocietyconsideringoutlawingmurderwouldprobablybemuchmoreagreeablethanthatsamesocietyconsideringtaxreform.Notonlydotheissuesmatter,however,butalsothesocieties.Groupsofsimilarpeopleseemlikely