4NashequilibriumForWSNEthecurrentbestresultwasgivenbyFearnleyetal10and ndsa2 3WSNEwhere000473ItbuildsonanapproachofKontogiannisandSpirakis13which ndsa2 3WSNEinpolynomialtimeusing ID: 455466
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1IntroductionANashequilibriumofabimatrixgameisapairofstrategiesinwhichthesupportsofbothplayersconsistonlyofbestresponses.TheapparenthardnessofcomputinganexactNashequilibrium[6,5]eveninabimatrixgamehasledtoworkoncomputingapproximateNashequilibria,andtwonotionsofapproximateNashequilibriahavebeendeveloped.Therstandmorewidelystudiednotionisofan-approximateNashequilibrium(-Nash).Here,norestrictionisplaceduponthesupports;anystrategycanbeinthesupportsprovidedeachplayerachievesanexpectedpayo thatiswithinofabestresponse.Therefore,-Nashequilibriahaveapracticaldrawback:aplayermightplaceprobabilityonastrategythatisarbitrarilyfarfrombeingabestresponse.Thesecondnotion,denedtorectifythisproblem,iscalledan-wellsupportedapproximateNashequilibrium(-WSNE).Here,thecontentofthesupportsarerestricted,butlessstringentlythaninanexactNashequilibrium.Specically,bothplayerscanonlyplacepositiveprobabilityonstrategiesthathavepayo withinofapurebestresponse.Observethatthelatternotionisastrongerequilibriumconcept:every-WSNEisan-Nash,buttheconverseisnottrue.Approximatewell-supportedequilibriarecentlyplayedanimportantroleinunderstandingthehardnessofcomputingNashequilibria.Theyaremoreusefulinthesecontextsthan-Nashequilibriabecausetheirdenitionismorecombina-torialandmorecloselyresemblesthebestresponseconditionthatcharacterizesexactNashequilibria.Indeed,approximatewell-supportedequilibriawereintro-ducedin[12,6]inthecontextofPPADreductionsthatshowthehardnessofcomputing(approximate)Nashequilibria.TheyweresubsequentlyusedasthenotionofapproximateequilibriumbyChenetal.[5]thatshowedthePPAD-hardnessofcomputinganexactNashequilibriumevenforbimatrixgames.Anotheractiveareaofresearchistoinvestigatethebest(smallest)thatcanbeguaranteedinpolynomialtime.For-Nash,thecurrentbestalgorithm,duetoTsaknakisandSpirakis[18],achievesa0:3393-Nashequilibrium;see[8,7,3]forotheralgorithms.Fortheimportantclassofwin-losegames{gameswithpayo sinf0;1g{[18]givesa1 4-Nashequilibrium.For-WSNE,thecurrentbestresultwasgivenbyFearnleyetal.[10]andndsa(2 3�)-WSNE,where=0:00473.ItbuildsonanapproachofKontogiannisandSpirakis[13],whichndsa2 3-WSNEinpolynomialtimeusinglinearprogramming.ThealgorithmofKontogiannisandSpirakisproducesa1 2-WSNEofwin-losegamesinpolynomialtime,whichisbest-known(themodicationsofFearnleyetal.donotleadtoanimprovedapproximationguaranteeforwin-losegames).2 wecanhopefor{aprobabilisticargument[13]showsthatthereexist-WSNEwithsupportsofcardinalityO(1 2logn),forany�0.72ALowerBoundontheSupportSizeofWellSupportedNashEquilibriaWebeginbyformallydeningbimatrixwin-losegamesandwell-supportedNashequilibria.Abimatrixgameisa2-playergamewithmnpayo matricesAandB;wemayassumethatmn.Thegameiscalledwin-loseifeachmatrixentryisinf0;1g.Apairofmixedstrategiesfp;qgisaNashequilibriumifeverypurerowstrategyinthesupportofpisabestresponsetoqandeverypurecolumnstrategyinthesupportofqisabestresponsetop.Arelaxationofthisconceptisthefollowing.Apairofmixedstrategiesfp;qgisan-wellsupportedNashequilibriumifeverypurestrategyinthesupportofp(resp.q)isan-approximatebestresponsetoq(resp.p).Thatis,foranyrowriinthesupportofpwehaveeiTAqmax`e`TAq�and,foranycolumncjinthesupportofqwehavepTBejmax`pTBe`�:Inthissectionweproveourmainresult.Theorem1.Forany2 3,thereexistwin-losegamesforwhichevery-well-supportedNashequilibriumhassupportsofcardinality (3p logn).Toprovethisresult,werstformulateourwin-losegamesgraphically.Thiscanbedoneinastraight-forwardmanner.Simplyobservethatwemayrepresenta2-playerwin-losegamebyadirectedbipartitegraphG=(R[C;E).Thereisavertexforeachrowandavertexforeachcolumn.Thereisanarc(ri;cj)2Eifandonlyif(B)ij=bij=1;similarlythereisanarc(cj;ri)2Eifandonlyif(A)ij=aij=1.Consequently,wearesearchingforagraphwhosecorrespondinggamehasnohighqualitywell-supportedNashequilibriumwithsmallsupports.Weshow 7ForNashequilibria,Althofer[1]andLiptonandYoung[15]independentlyprovedsimilarresultsforzero-sumgames,andLiptonatal.[16]laterprovedsimilarresultsforgeneral-sumbimatrixandmulti-playergames.4 theexistenceofsuchagraphprobabilistically.TheConstruction.LetT=(V;E)bearandomtournamentonNnodes.NowcreatefromTanauxiliarybipartitegraphG(T)=(R[C;A)correspondingtoa2-playerwin-losegameasfollows.Theauxiliarygraphhasavertex-bipartitionR[CwherethereisavertexofRforeachnodeofTandthereisavertexofCforeachsetofkdistinctnodesofT.(Observethat,forclaritywewillrefertonodesinthetournamentTandverticesinthebipartitegraphG.)TherearetwotypesofarcinG(T):thoseorientedfromRtoCandthoseorientedfromCtoR.Forarcsoftheformertype,eachvertexX2Cwillhavein-degreeexactlyk.Specically,letXcorrespondtothek-tuplefv1;:::;vkgwherevi2V(T),forall1ik.Thentherearearcs(vi;X)inGforall1ik.NextconsiderthelattertypeofarcinG.Foreachnodeu2Rthereisanarc(X;u)inGifandonlyifudominatesX=fv1;:::;vkginthetournamentT,thatisif(u;vi)arearcsinTforall1ik.Thiscompletestheconstructionoftheauxiliarygraph(game)G.WesaythatasetofverticesW=fw1;:::;wtgiscoveredifthereexistsavertexysuchthat(wj;y)2A,forall1jt.Furthermore,abipartitegraphisk-coveredifeverycollectionofkverticesthatlieonthesamesideofthebipartitioniscovered.NowwithpositiveprobabilitytheauxiliarygraphG(T)isk-covered.Lemma1.Forallsucientlylargenandk3p logn,thereexistsatourna-mentTwhoseauxiliarybipartitegraphG(T)isk-covered.Proof.Observethatthepayo matricesthatcorrespondtoG(T)havem=Nrowsandn=�Nkcolumns.Furthermore,byconstruction,anysetofkverticesinRiscovered.Thus,rstwemustverifythatanysetofkverticesinCisalsocovered.SoconsideracollectionX=fX1;:::;XkgofkverticesinC.SinceeachXi2Ccorrespondstoak-tupleofnodesofT,weseethatXcorrespondstoacollectionofatmostk2nodesinT.Thus,foranynodeu=2[iXi,wehavethatuhasanarcinTtoeverynodein[iXiwithprobabilityatleast2�k2.Thuswithprobabilityatmost(1�1 2k2)N�k2thesubsetXofCnotcoveredinG(T).Applyingtheunionboundwehavethatthereexiststhedesiredtournamentifnk1�1 2k2N�k21(1)5 Nowsetk=log1 3n.Thereforelogn1 k=log2 3n=k2.Inaddition,becausen=�Nk,wehavethatNk en1 k.Hence,N�k2�n1 k.(Notethat,sinceNkthisimpliesthatG(T)isdened.)Consequently,nk1�1 2k2N�k2nk1�1 2k2n1 knke�1 2k2n1 knke�1 ek2log2n1 kThus,takinglogarithms,weseethatInequality(1)holdsifek2log2klognn1 k(2)Butn1 k=ek2,soInequality(2)clearlyholdsforlargen.Theresultfollows.ApropertyoftheauxiliarygraphG(T)thatwillbeveryusefultousisthatitcontainsnocycleswithlessthansixvertices.Lemma2.TheauxiliarygraphG(T)containsnodigonsandno4-cycles.Proof.SupposeG(T)containsadigonfw;Xg.Thearc(w;X)impliesthatX=fx1;:::;xk�1;wg.Ontheother-hand,thearc(X;w)impliesthatwdominatesXinTand,thus,w=2X.SupposeG(T)containsa4-cyclefw;X;z;YgwherewandzareinRandwhereX=fx1;:::;xk�1;wgandY=fy1;:::;yk�1;zgareinC.ThenzmustdominateXinTandwmustdominateYinT.ButthenwehaveadigoninTas(w;z)and(z;w)mustbearcsinT.ThiscontradictsthefactthatTisatournament.Lemmas1and2arealreadysucienttoproveamajordistinctionbetweenapproximate-Nashequilibriaandwell-supportedNashequilibria.Recallthattherealwaysexist1 2-Nashequilibriawithsupportsofcardinalityatmosttwo[8].Insharpcontrast,forsupportsofcardinalityatmosttwo,noconstantap-proximationguaranteecanbeobtainedfor-well-supportedNashequilibria.Theorem2.Forany�0,thereexistwin-losegamesforwhichnopairofstrategyvectorswithsupportsizesatmosttwoisa(1�)-well-supportedNashequilibrium.Proof.Taketheauxiliarywin-losegameG(T)fromLemma1forthecasek=2.Nowconsideranypairofstrategyvectorsp1andp2withsupportsofcardinality6 Supposep1andp2forma-well-supportedequilibriumforsome2 3.Thenitmustbethecasethateachri2S1hasexpectedpayo atleast1��1 3againstp2.Similarly,eachcj2S2hasexpectedpayo atleast1��1 3againstp1.Butthiscannothappen.ConsiderthesubgraphofG(T)inducedbyS1[S2whereeachri2S1hasweightwi=p1(ri)andeachcj2S2hasweightwj=p2(cj).WeconvertthisintoanunweightedgraphHbymakingLwvcopiesofeachvertexv,forsomelargeintegerL.NowHisanLLbipartitegraphwithminimumin-degree(1�)L�1 3L.Thus,byTheorem3,Hcontainsa4-cycle.Thisisacontradiction,byLemma2.utWeremarkthatthe2 3inTheorem1cannotbeimprovedusingthisprooftechnique.Specicallytheminimumin-degreerequirementof1 3kinTheorem3istight.Toseethis,takeadirected6-cycleCandreplaceeachvertexinCby1 3kcopies.ThuseacharcinCnowcorrespondstoacompletek 3k 3bipartitegraphwithallarcorientationsinthesamedirection.ThegraphHcreatedinthisfashionisbipartitewithallin-degrees(andallout-degrees)equalto1 3k.ClearlytheminimumlengthofadirectedcycleinHissix.3ConclusionAnoutstandingopenproblemiswhetheranyconstantapproximationguaranteebetterthan1isachievablewithconstantcardinalitysupports.Wehaveshownthatsupportsofcardinalitytwocannotachievethis;apositiveresolutionofConjecture1wouldsucetoshowthatsupportsofcardinalitythreecan.How-ever,Conjecture1seemsahardgraphproblemanditiscertainlyconceivablethatitisfalse.8Ifso,thatwouldleadtotheintriguingpossibilityofaverymajorstructuraldi erencebetween-Nashand-WSNE;namely,thatforany�0,thereexistwin-losegamesforwhichnopairofstrategieswithconstantcardinalitysupportsisa(1�)-well-supportedNashequilibrium.Theexistenceofsmallsupport-WSNEclearlyimpliestheexistenceofofpolynomialtimeapproximationalgorithmstondsuchequilibria.Obtainingbetterapproximationguaranteesusingmorecomplexalgorithmsisalsoanin-terestingquestion.Asdiscussed,thebestknownpolynomial-timeapproxima-tionalgorithmforwell-supportedequilibriainwin-losegamesndsa1 2-wellsupportedequilibrium[13]bysolvingalinearprogram(LP).Forgameswith 8Forexample,theconjectureresemblesaquestionabouttheexistenceofk-existentiallycompletetriangle-freegraphsfork�3referredtoin[9],whichtheauthorsconsidertobewideopen.9