Proceedings of the TwentySecond International Joint Conference on Artificial Intelligence gameInparticularwefocusonthenaturalcasewherepotentialcoalitionsofplayersarefullyconnected132thatisth ID: 470738
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ConsiderateEquilibriumMartinHoeferMichalPennMariaPolukarovAlexanderSkopalikBertholdVDept.ofComputerScience,RWTHAachenUniversity,Germany,mhoefer,voeckingFacultyofIndustrialEngineeringandManagement,Technion,Israel,mpenn@ie.technion.ac.il Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence game.Inparticular,wefocusonthenaturalcasewherepo-tentialcoalitionsofplayersarefullyconnected„thatis,thesetoffeasiblecoalitionscorrespondstothesetofallpossiblecliquesinthegraph.Crucially,besidestheabilityofcooperation,thepresenceofsociallinksmayalsoaffectthestrategicinterestsofplayersinthegame.Inthisspirit,socialcontextgames[2]werepro-posedtomodelscenarioswhereaplayersutilitycandependonthepayoffsofotherplayers.Forexample,aplayermaybeinterestedinrankinghispayoffashighaspossiblecompar-ingtotheotherspayoffs[4],oraplayermaycareaboutthetotalpayoffofasubsetofhisfriendsŽ,asincoalitionalcon-gestiongames[8,11].Asocialcontextgameisthende“nedbysomeunderlyinggame,thesocialcontextgivenbysometopologicalorgraph-theoreticstructureofneighborhood,andaggregationfunctionscapturingtheeffectsofutilitychangesintheunderlyinggameonplayerincentives.In[2],RSGsareconsideredastheunderlyinggames,andfournaturalsocialcontextsarestudied.However,unlikeforpartitionequilib-rium,thisworkdealsonlywithunilateraldeviations.Thispaperstudiestheinterplaybetweensocialstructureandtheoutcomeofmulti-agentinteractioninyetanotherway.Insteadofrelatingaplayersutilitytothepayoffsofotherpar-ticipants,weconsidertheeffectstheirmayhaveonit.Inpresenceofsocialconnections,theseeffectsintroducead-ditionalincentivesfortheplayersandmayhavecrucialin”u-enceonthedecisionstheymakeinthegame.Forexample,insocialnetworkssuchasFaceBookorLinkedIn,linksamongtheagentsrepresentfriendships,professionalpartnerships,orevenfamilyrelations.Inothercontextsofinterest,agentsmaybetiedbybusinesscontracts,technologicaldependen-ciesorcommunicationlines.Insuchscenarios,itisnaturaltoexpectthatanagentwillbehaveinaconsiderateŽmannerandavoidtakingactionsthatmayharmhisneighborsinthenetwork.Thismotivatesthestudyofconsiderationinstrate-gicgames,whichisinthemainfocusofourwork.Asfarasweareaware,thispaperisthe“rsttoaddressthisissue.Thesolutionconceptsnaturallycorrespondingtoconsider-atebehaviorextendthenotionsofNE,SEandSSEtocon-siderdecisions(eithergrouporindividual)thatdonotdeteri-orateanyneighboringplayers.Focusingonthenaturalcasewherecoalitionsofplayersthatexecuteastrategychangemustbefullyconnected,wede“netheconsiderateequilib-tobeastateinwhich(1)nocoalitonformedbyacliqueinthesocialnetworkcandeviatesothattheutilityofatleastonememberofthecoalitionstrictlyimprovesand(2)noneoftheplayersneighboringthecliquegetsworse.Weobservethatpartitionequilibriumevolvesasaspecialcaseofconsiderateequilibriumwhenthesocialnetworkiscomposedofasetofdisjointcliques.Indeed,onemay“ndthattherestrictionofcoalitionaldeviationsinpartitionequi-libriumessentiallypostulatestwostructuralproperties:(1)coalitionsofplayersthatexecuteastrategychangehavetobecloseŽtoeachother,and(2)theirdecisionmuststrictlyben-e“tatleastoneofthembutnotstrictlydeteriorateanyotherplayerclosetothem.Thenotionofclosenessisde“nedinbothcasessimplyasbeinginthesamepartitionset.However,while[2,5]areinitialstepsinrelatingthesocialstructuretotheoutcomeofagame,theyarequiterestrictiveinthatonlyparticularsocialcontextsand“xedcoalitionalstructures(par-titions)areconsidered.Inaddition,theygenerallyignorethephenomenonofconsideratebehaviorwhichispresentinourwork.Similarargumentsapplyw.r.t.[7],where“xedcoali-tionstructuresinloadbalancingandcongestiongamesarestudied.HerecoalitionsactassinglesplittableŽplayersthatstrivetominimizethemakespanorthesumofcostsoftheagentsinthecoalition.Weexploretheconceptofconsideratebehaviorintheprominentclassofresourceselectiongames.InanRSG,eachplayerchoosesoneofa“nitesetofresources,anditscostisgivenbyadelayfunctiondependingonthenumberofplayerschoosingthesameresource.RSGsareafundamentalsettingincomputerscience,operationsresearchandeconomics,duetotheirpracticalapplicability(e.g.,inelectroniccommerceandcommunicationnetworks)andplausibleanalyticalprop-erties.Inparticular,forstrictlyincreasingdelayfunctions,SEalwaysexist[9,10],butSSEisnotguaranteed[5].Thelatterfacthasbeenprominentlyutilizedtodemonstratethepoweroflimitedcoalitionaldeviations[1,5].1.1OurResultsWeshowthatregardlessofthesocialnetworktopology,allRSGswithstrictlyincreasingdelayfunctionspossessacon-siderateequilibrium.OurproofinSection3isconstructiveandyieldsanef“cientalgorithmforcomputingsuchanequi-librium.Importantly,thecomputedconsiderateequilibriumisalsoastandardNEforagivenRSG,thusshowingthatthereexistsastatethatisstableagainstsel“shandconsid-eratebehaviorsimultaneously.Observethatthenumberofcliquesmightbeexponentialinthenumberofplayers,whichmakesnon-trivialeventhecomputationofasingleimprov-ingmove.Wesolvethisproblembyshowingthat,inanNE,everypro“tabledeviationofacliqueiswitnessedbyamoveofasingleplayerthatdecreasesasuitablyde“nedpotentialfunction.Inaddition,ourproofisfundamentallydifferentandsigni“cantlysimplerthantheexistenceproofforthespecialcaseofpartitionequilibriumin[1].InSection4,westudyconvergencepropertiesofconsid-eratedynamics.Letusremarkthatthepotentialfunctionargumentusedinourexistenceproofdoesnotimplythatthesequentialdynamicsde“nedbydeviationsofcliquesisacyclic,sincethesingleplayermovesconsideredintheexis-tenceproofdonotnecessarilycorrespondtoallowedimprov-ingmoves.Indeed,weshowthatevenforidentical,strictlyincreasingdelaystherearein“nitesequencesofimprovingmovesofcliques.Thisisincontrasttothedynamicscor-respondingtopartitionequilibrium,forwhichweshowthe“niteimprovementpropertyinthissetting.2PreliminariesandInitialResultsstrategicgameisatuple,whereisthesetofplayers,ansisastrategyspaceofplayerofthegameisavectorofstrategies.Forconvenience,weusetodenote,i.e.,reducedbythesingleen-tryofplayer.Similarly,forastateweusetodenotethestrategychoicesofacoalitionforthe complement,andwewrite.Theinstate.Forastateacoalitionissaidtohaveanimprovingmoveifthereissuchthatforeveryplayer.Inparticular,theimprovingmoveisunilateralstatehasaweakimprovingmoveifthereissuchthatforeveryatleastone(purestrat-egy)Nashequilibrium(NE)[14]isastatethathasnounilat-eralimprovingmoves,astrongequilibrium(SE)[3]isastatethathasnoimprovingmoves,andasuper-strongequilibrium[5]isastatethathasnoweakimprovingmoves.Tomodelconsideratebehavior,weadjustthede“nitionofimprovingmoves.Inparticular,thereisanundirected,un-weightedgraphN,Eoverthesetofplayers.Fora,considerthei,jDe“nition1(ConsiderateImprovingMoves)Astateconsiderateimprovingmoveforacoalitionifthereissuchthatforallforall.Foraconsiderateimprovingmovewehave.AstateweakconsiderateimprovingmoveforacoalitioniftheresuchthatforallforatleastoneNotethatevery(weak/unilateral)considerateimprovingmoveisalsoa(weak/unilateral)improvingmovebutnotviceversa.Tode“necoalitionalequilibria,letus,forthetimebeing,alsoassumethatthereisasetsystemoffeasiblecoali-considerateNashequilibrium(CNE)isathathasnounilateralconsiderateimprovingmoves.A(super)strongconsiderateequilibrium((S)SCE)isastatethathasno(weak)considerateimprovingmoveforacoali-.NotethatforCNEweimplicitlyassumethesetofallsingletonsetsforall.EveryNEisaCNE,andevery(S)SEisa(S)SCE.TheconverseonlyholdsforCNEandNEif.Ingeneral,(S)SCEare(S)SEonlyif.Inthisway,thepresenceofso-cialtiesandanon-trivialsetoffeasiblecoalitionsweakenthestructuralrequirementsfortheexistenceofequilibrium.Intherestofthepaper,wemakethenaturalassumptionthatthesetoffeasiblecoalitionscorrespondstothesetofcliquesin.Inouranalysis,wefocusonweakimprovingmovesandstudysuperstrongconsiderateequilibriaaswebelievethatthissolutionconceptismostinterestingnotonlyfromatechnicalpointofviewbutalsoanaturalandconvinc-ingmodelfortheinteractionofcoalitionalstructuresinthepresenceofasocialnetwork.De“nition2(ConsiderateEquilibria)considerateequi-librium(CE)isastatethathasnoweakconsiderateim-provingmoveforacoalitioncorrespondingtoacliqueinThenotionofCEnicelygeneralizespartitionequilibrium.Inparticular,apartitionequilibriumisaCEifthesocialnetworkispartitionedintoisolatedcliques.Notethatwedonotex-plicitelyassumethatthesetoffeasiblecoalitionsisrestrictedcliques.Ifthegraphispartitionedintoisolatedcliques,however,thisrathertechnicalassumptionmadeinthede“nitionofpartitionequilibriumisanaturalconsequenceoftheassumptionthatthecoalitionsbehaveconsideratelyto-wardstheirneighbors.Inthisway,sinceweakimprovingmovesdonotdecreasetheutilityofneighboringplayers,onecanassumew.l.o.g.thatallmembersofapartitionsetpartic-ipateinacoalition.Weapplytheconceptofconsiderationtoresourceselectiongames(RSG)„abasicclassofpotentialgames[12,15].InanRSG,thereisasetofresources,andforevery.Forastatewedenotebythenumberofplayersthatpick.Foreachresource,thereisadelayfunction.Throughoutthepaperweassumethatalldelayfunctionsarenon-negativeandstrictlyincreasing.Inastatewith,playerInthispaper,wefocusonRSGswithstrictlyincreasingdelays.Inthiscase,itisknownthatNEexist[15],canbecomputedinpolynomialtime[6],andareequivalenttoSE[9].Moreover,thegamespossessa(strong)[9,12],i.e.,everysequenceofunilateralimprov-ingmoveshas“nitelengthandendsinaNE/SE.Trivially,byrestrictionofimprovingmoves,thesameholdsalsoforCNEandSCE.Interestingly,however,SSEarenotguaran-teedtoexisteveninsimplestgames.Incontrast,weprovebelowthatallRSGswithstrictlyincreasingdelayspossessconsiderateequilibria.However,evenforidenticalresources,weshowthattherearein“nitesequencesofweakconsiderateimprovingmovesofcoalitionsbeingcliquesin.ThisisincontrasttoaspecialcasewhereisadisjointsetofcliquesandCEreducestopartitionequilibrium;inthiscase,thereexistsapotentialfunctionforweak(considerate)improvingmovesingameswithidenticalresources.3ExistenceandComputationThissectioncontainsourmaintheoremshowingtheexistenceofCEinRSGswithstrictlyincreasingdelayfunctions.TheexistenceproofisconstructiveandyieldsapolynomialtimealgorithmcomputingastatethatisbothaCEandastandardNE,thusshowingthatthetwoequilibriumconceptsintersect.Theorem1ForanyRSGwithstrictlyincreasingdelayfunc-tionsandanyassociatedsocialnetwork,thereexistsatleastonestatethatisanNEandaCE.Moreover,thereisapolynomialtimealgorithmcomputingsuchastate.Proof:WedescribeaprocessthatstartsinaNashequilib-riumandconvergestoaCE.Thisprocessconsistsofmove-mentsofsingleplayers.Everystrategypro“leinthisse-quenceisastandardNashequilibrium.Considerastate.Letdenotethemaximaldelayofaplayerin.NotethatinaNashequilibrium,eachusedhaseitherdelay+1).Intheformercase,wecallresourceresource,inthelattercase,wecallitaifadditionally+1)=.Letdenotethesetofneighborsofplayerthatareonresource AlsobeingreferredtoassimplecongestiongamescongestiongamesparallellinkgamesConsideragamewith,and Wearenowreadytodescribetheprocess:1.ComputeaNashequilibrium2.Ifthereisaplayerplacedonahighresourceandthereisalowresourcethenset,andrepeatthisstep.3.Ifthereisaplayerplacedonahighresourcethereisalowresourcethensetandcontinuewithstep2.4.OutputNotethateachstateproducedbythisprocessisaNashequi-librium.Duringthisprocess,thefollowingpotentialfunctiondecreasesstrictlyfromsteptostep,whereweuseasashorthandfortheneighborsofonthesamere-sourceandassume.Onecaneasilymod-ifythedelayfunctionssuchthatwithoutchang-ingtheplayerspreferenceswhichimpliesthattheprocessterminatesafterpolynomiallymanysteps.ToprovethatthisprocessresultsinaCE,weshowthatifastateisaNEandthereexistsweakconsiderateimprovingmove,thenthereisalsoamoveofasingleplayerasdescribedabove.denotethesetofhighandlowresourcesin,respectively.Letbethesetofresourcesthatarehighbutnolongerhighin,andletbethesetofresourcesthatarelowinandbecomehighin.By.Letbethesetofplayersonresourcesof,andletbethesetofplayersonresourcesofLemma2Duringthemove,allplayersinaremov-ingfromresourcesintoresourcesoutsideof.Inturn,playersaremovingfromresourcesintothere-sourcesin.Finally,atleastplayersareleavingtowardsresourcesoutsideofProof:isaweakconsiderateimprovingmove,allplayersinmustmovefromresourcesintoresourcesoutsideofastheirdelaywouldincrease,otherwise.Theseplayerscanonlybereplacedbyplayersfromasotherplay-erswouldhaveanincreaseddelayafterthemove,otherwise.Inturn,altogetherplayersneedtomovefromsothattheresourcesofbecomehighresourcesaf-terthemove.Furthermore,weobservethatthenumberofplayersonresourcesindoesnotchangeduringtheconsideredmove,andtherearenoplayersenteringfromoutsideofassuchplayerswouldhaveanincreaseddelay,otherwise.Asaconsequence,theremustbeatleastplayersthatareleavingtowardsinordertohaveplayersthatmovefromThisprovesLemma2.Thelemmaimplies=maxdenotethemaximumnumberofneighborsthataplayerofhasonhisresource.The+1)Notethatnoplayerofhasaneighborthathaschosenaresourcefromandisnotin.Otherwise,thisneigh-borsdelaywouldincreaseduringthemovesothatwouldnotbeaconsideratemove.Therefore,wecanset,wherethechoiceofisirrelevant.Thede“nitionofimmediatelyimpliesLetusderivesomemorehelpfulequationsregardingthedif-ferentkindsofresources.Foreachresourcethatdecreasesitsloadduringtheimprovingmove,thereisatleastoneresourcethatincreasesitsloadbyonebecausethenumberofplayersoneachlowresourcecanonlyincreasebyone.ThisgivesCombiningtheEquations(2),(1),and(3)gives+1)+1)Now,wedistinguishbetweenthefollowingtwocases.Case1:.Inthiscase,wecansetargmax=argmin,whichsatis“estheconditionsofstep2oftheprocess.Case2:.Inthiscase,Equation5yields,which,coupledwithEquation4,implies.SubstitutingthisequalitybackintotheEquation5giveswhichimplies=min.De“ne=max=min.NowEquations2and3yield,whichincombinationwithEqua-tion1givesOnaverage,theresourcesinplayersfrominstateandtheresourcesplayersfrom.Weclaimthatthisimpliesthateachresourceinexactlyplayersfrom;andeachresourceinexactlyplayersfromandnoadditonalneighborofoneofthem.Toseethis,letdenotearesourcefromholdingamaximumnumberofplayersfromandletnotearesourcefromholdingaminimumnumberofplay-ersfrom.Letbeaplayerassignedto.Asisaconsideratemove,doesnothaveneighborsoutsideof.Thus,iftheclaimabovewouldnothold,wouldhaveeitheratleastneighborsonorstrictlylessneighborson,whichwouldim-andthuscontradictourassumption.Asa,forevery,andNow,Lemma2impliesthateachoftheresourcesinisleftbyitsplayersfromandeachoftheresourcesinisleftbyitsplayersfromWemakeafewfurtherobservations:Thede“nitionofimpliesthatthenumberofplayersonaresourcefromdoesnotdecreaseduringtheconsideredmove.Besides,thisnumbercannotincreaseduetoaweakimprovingmove.Nextconsideraresource.Thede“nitionof thatthenumberofplayersoncannotincreaseduringaweakimprovingmove.Nowsupposethenumberofplayersonwoulddecrease.Thenthereisaleavingplayer,whomovestoeitheroranotherresourcein,asitsdelaywouldincrease,otherwise.Inthelattercase,adifferentplayermustmakeroomfor.Byfollowingthisplayer,wecaniterativelyconstructachainofmovingplayersuntil“nallythereisaplayerthatmovestoaresourcein.Thus,togetherwiththeplayersleavingtheresourcesinthereareatleastthatneedtomigratetoaresourcewithadelayoflessthan(afterthemove).However,theresourcesinhaveonlyacapacityfortakingmanyofsuchplayers.Hence,thenumberofplayersonanyresourceoutsideofdoesnotchangeduringthismove.Now,takeoneoftheplayersfrom.Duringtheconsid-eredmove,thisplayermigratestoanotherresourcehavingadelaystrictlylessthan(afterthemove).Ifthisresourcedoesnotbelongtothenanotherplayerneedstoleavethisresourceinordertocompensateforthearrivingplayer.Fol-lowingthatplayer,weiterativelyconstructachainofmovingplayers,leadingfromaresourceintoaresourceinInthismanner,wecandecomposethesetofmovingplayersintoacollectionofmanychainseachofwhichleadsfrom.Asweareconsideringaweakimprovingmove,thedelaysineachofthesechainsdoesnotincreaseandthereisatleastonesuchchainleadingfromaresourcetoare-.Wechooseanarbitraryplayerassignedtoresource.Wehaveshownabovethatforthisplayerholds.Thus,playersatis“estheconditioninstep3ofourprocess,whichcompletesouranalysisforCase2.Thisshowsthat,whentheprocessterminates,therearenoweakconsiderateimprovingmoves.Therefore,theresultingstateisaCE.4ConvergenceNextweshowthatthedynamicsofweakconsiderateim-provingmovesbygeneralcliquesdoesnothavetheimprovementproperty,i.e.,thedynamicscorrespondingtoCEmightcycle(Theorem3).Ourconstructionworksevenforresourceswithidenticaldelays.Thisseparatesconsid-erateequilibriumfrompartitionequilibriumas,inthesamesetting,thedynamicscorrespondingtopartitionequilibriumadmitsthe“niteimprovementproperty(Proposition4).Theorem3TherearesymmetricRSGswithstrictlyincreas-ingandidenticaldelays,forwhichtherearein“nitese-quencesofweakconsiderateimprovingmovesbycoalitionsthatarecliquesinProof:Fortheproofweconstructagamewithamodularstructure.Ourgameconsistsofanumberofsmallergames,referredtoasblocks.Eachblockconsistsof14playersand5resources,andbyitselfitisacyclic.However,bycreatingsocialtiesacrossblocks,wecreatelargercliquesthatareabletoperformresetsŽinoneblockwhilemakingimprovementsinotherblocks.Byacarefulschedulingofsuchresetmovesweconstructanin“nitesequenceofmoves.Moreformally,wehave19blocks,andineach,wehave14players.Thereare8playersinvolvedinoursequence,while6additionaldummyŽplayersnevermove.Thedummyplayersaresingletonnodesinthesocialnetworkandareonlyrequiredto,inessence,simulatenon-identicalre-sourcesbyincreasingsomeofthedelaystolargervalues.Thesocialgraphconsistsofinternallinkswithineachblockandinter-blockconnectionsasfollows.Foreachblock,thereare.Inaddition,foreach,...,therearetwointer-blockcliques,wheretheexponentismeanttocyclethroughthenumbers1to19,i.e.,aboveThe95resourcesaredenotedby.Thedelayfunctionsareidentical.Notethatingeneral,ourexampledoesnotrequirelineardelays,itsuf“cestoensureLetusconsiderasingleblockandasequenceofsixstateswithinthisblockdepictedinFig.1.Notethat ri1ri2ri3 ri4ri5 CiBi PiQi EiDiFi xx xGix xx CiDi PiQi EiBiFi xx xGix xx CiDi PiQi EiBiFi xx xGix xx CiBi PiQi EiDiFi xx xGix xx DiBi PiQi\b EiCiFi xx xGix xx DiBi PiQi\t EiCiFi xx xGix xx CiBi PiQi EiDiFi xx xGix Figure1:Sequenceofsixstateswithinablockthatareat-tainedduringanin“nitesequenceofweakconsiderateim-provingmoves.resentsaweakconsiderateimprovingmoveforperformsthemove,andstrictlyimproves.Sim-ilarly,isaweakconsiderateimprovingmovefor,and.Theareresets,inwhichacyclicswitchisperformedandnoplayerwithintheblockstrictlyimproves.Itsuf“cestoshowthatthesestepscanbeimplementedwithimprovingmovesbyinter-blockcliques. Considerthe“rstreset,inwhichswapplaces,andforsimplicityassumew.l.o.g.thatThisswapisexecutedinthreemoves,wherewe“rstswap,thenswapand“nallyswapouttobringbackin.Thiscyclicswitchistheresultofthefollowingsequenceofweakconsiderateimprovingmoves:(1)coalitionpliesadeviationwhereexchangetheirplaces,movesawayfrominblock9asscribes;(2)coalitionimprovesbyswapping,andmovingawayinblock10;(3)“nally,swapwithcoali-movesawayfrominblock11.Inthe“naldynamics,wewillusethesemovesalsotosimultaneouslyperformswapsintheotherblocks3,4,6,and7.Thesecondresetswapcanbedoneinsimilarfashionbyacircularswapinvolvingandus-ingtheplayersofblocks=12.Notethatouredgesarecarefullydesignednottogenerateanyun-desiredconnections.Inparticular,relyonthemovementoftoexecutetheirswaps.Dur-ingtheseswaps,aredeteriorated.Noneofthedeterioratedplayersareattachedtoplayersintherespec-tiveimprovingcoalitions,i.e.,noneofareneighborswith,noneofareneighborswith,andnoneofareneighborswith.Inaddition,formakingtheswitchbetweenweusethemovementof.Notethatnoneoftheplayersrequiredtoexecutetheswitchesareneighborswithrespectively.Anin“nitesequenceofweakconsiderateimprovingmovescannow,forexample,beobtainedfromastartingstateasfollows.Weindicateforeachblockinwhichstateitisinitialized.Here,andindicatetheintermediatestatesofthecorrespondingcircularresettingswaps. 1 2 3 4 5 6 7 8 9 \t2 \t1 \t \t \t \t \t \t \t 10 11 12 13 14 15 16 17 18 19\b 2 1 Inthe“rststep,wecansimultaneouslyadvanceblocks1-3,\tusingmovementof,whichadvancesblock10to.Inthenextstepweadvanceblocks12-14from,usingmovementofwhichadvancesblock18to.Next,wemaketwointernalswitchesinblocks11fromand19from.Inthisway,wehaveshiftedthestatesequencebyoneblock,whichimpliesthatwecanrepeatthissequenceendlessly.contrast,observethatifthegraphisasetofdisjointcliques,thenforgameswithidenticalandstrictlyincreasingdelayfunctionswecaneasilyconstructapotentialfunctionthatim-pliesacyclicitywithrespecttoweak(considerate)improvingmoves.Proposition4IneverysymmetricRSGwithstrictlyincreas-ing,identicaldelaysfunctions,everysequenceofweakim-provingmovesofallowedpartitionsetsis“niteandendsinapartitionequilibrium.Notethatinthiscasewecanassumew.l.o.g.thatforall.Also,eachweakimprovingmovedecreasesthesumofcostsofallplayersinthepartitionset.Thereby,theresultsof[7]forlineardelaysdirectlyimplythe“niteimprovementproperty.Referencesences1]ElliotAnshelevich,BugraCaskurlu,andAmeyaHate.Parti-tionequilibriumalwaysexistsinresourceselectiongames.InProc.3rdIntl.Symp.AlgorithmicGameTheory(SAGT),2010.Toappear..2]ItaiAshlagi,PiotrKrysta,andMosheTennenholtz.Socialcon-textgames.InProc.4thIntl.WorkshopInternet&NetworkEconomics(WINE),pages675…683,2008.2008.3]RobertAumann.Acceptablepointsingeneralcooperativen-persongames.InContributionstotheTheoryofGamesIVvolume40ofAnnalsofMathematicsStudy,pages287…324.PrincetonUniversityPress,1959.1959.4]FelixBrandt,FelixFischer,andYoavShoham.Onstrictlycompetitivemulti-playergames.InProc.21stConf.Arti“cialIntelligence(AAAI),pages605…612,2006.2006.5]MichalFeldmanandMosheTennenholtz.Partitionequilib-rium.InProc.2ndIntl.Symp.AlgorithmicGameTheory(SAGT),pages48…59,2009.2009.6]DimitrisFotakis,SpyrosKontogiannis,EliasKoutsoupias,MariosMavronicolas,andPaulSpirakis.ThestructureandcomplexityofNashequilibriaforasel“shroutinggame.oret.Comput.Sci.,410(36):3305…3326,2009.2009.7]DimitrisFotakis,SpyrosKontogiannis,andPaulSpirakis.Atomiccongestiongamesamongcoalitions.ACMTrans.Al-,4(4),2008.2008.8]AraHayrapetyan,EvaTardos,andTomWexler.Theeffectofcollusionincongestiongames.InProc.38thSymp.TheoryofComputing(STOC),pages89…98,2006.2006.9]RonHolzmanandNissanLaw-Yone.Strongequilibriumincongestiongames.GamesEconom.Behav.,21(1-2):85…101,21(1-2):85…101,10]RonHolzmanandNissanLaw-Yone.Networkstructureandstrongequilibriuminrouteselectiongames.Math.SocialSci.46(2):193…205,2003.2003.11]SergeyKuniavskyandRannSmorodinsky.Coalitionalcon-gestiongames.Mastersthesis,Technion,Haifa,Israel,2007.2007.12]DovMondererandLloydShapley.Potentialgames.Econom.Behav.,14:1124…1143,1996.1996.13]RobertMyerson.GameTheory:AnalysisofCon”ict.HarvardUniversityPress,6thedition,2004.2004.14]JohnNash.Non-cooperativegames.AnnalsofMathematics54(2):286…295,1951.1951.15]RobertRosenthal.Aclassofgamespossessingpure-strategyNashequilibria.Intl.J.GameTheory,2:65…67,1973.1973.16]OlaRozenfeld.Strongequilibriumincongestiongames.Mas-tersthesis,Technion,Haifa,Israel,2007.