Evolutionary-prisoner’s-dilemma-game-on-a-square-l

Evolutionaryprisoner’sdilemmagameonasquarelatticeGyo¨rgySzabo´1andCsabaTo˝ke21ResearchInstituteforMaterialsScience,P.O.Box49,H-1525Budapest,Hungary2Eo¨tvo¨sUniversity,Mu´zeumkrt.6-8,H-1088Budapest,Hungary~Received10October1997!Asimplifiedprisoner’sgameisstudiedonasquarelatticewhentheplayersinteractingwiththeirneighborscanfollowtwostrategies:tocooperate(C)ortodefect(D)unconditionally.Theplayersupdatedinrandomsequencehaveachancetoadoptoneoftheneighboringstrategieswithaprobabilitydependingonthepayoffdifference.UsingMonteCarlosimulationsanddynamicalclustertechniques,westudythedensitycofcooperatorsinthestationarystate.Thissystemexhibitsacontinuoustransitionbetweenthetwoabsorbingstateswhenvaryingthevalueoftemptationtodefect.Inthelimitsc!0and1wehaveobservedcriticaltransitionsbelongingtotheuniversalityclassofdirectedpercolation.@S1063-651X~98!00303-1#PACSnumber~s!:02.50.Le,05.50.1q,05.40.1j,64.60.HtI.INTRODUCTIONTheevolutionaryprisoner’sdilemmagameswereintro-ducedbyAxelrod@1#tostudytheemergenceofcooperationratherthanexploitationamongselfishindividuals.SincethepioneeringworkofAxelrodthisapproachhasbecomeafruitfultoolintheareaofpoliticalandbehaviorsciences,biologyandeconomics@2–4#.Intheprisoner’sdilemma~PD!gameeachoftwoplayershastodecidesimultaneouslywhetheritwishestocooperatewiththeotherortodefect.Therewardsdependentontheirchoicesareexpressedby232payoffmatricesinagreementwiththefourpossibilities.AssumingasymmetricgametheplayersgetrewardsR(P)ifbothchoosetocooperate~de-fect!.Intheremainingtwocasesthedefector’sandcoopera-tor’spayoffareT~temptationtodefect!andS~sucker’spayoff!,respectively.Theelementsofthepayoffmatrixsat-isfythefollowingconditions:T.R.P.Sand2R.T1S.Inthisgamethemutualcooperationleadstothehighesttotal~average!payoff.Thehighestindividualpayoff(T)canonlybereachedagainsttheotherplayerdecreasingtheaveragepayoff.ThesefeaturesmakesthePDgameinterestinginthementionedareas.InearlierstudiesNcontestantsplayedaniteratedround-robinprisoner’sdilemmagame.Thepopulationofcontes-tants,whichapplydifferentalgorithmstochoosebetweendefectionandcooperationintheknowledgeofpreviousde-cisions,wasmodifiedaccordingtoaDarwinianselectionruleroundbyround.Forexample,eliminatingtheworstplayer,thebestonewillhaveanoffspringinheritingtheparent’sstrategy.Inadifferentinterpretation,theworstplayeradoptsthebestalgorithm.Computertournaments~simulations!wereperformedtostudyhowthepopulationofcontestantsvarieswithtime@1#.Evidently,thefinal~station-ary!statedependsontheinitialpopulation.Thesimulationshaveclarifiedtheemergenceofmutualcooperationamongalltheplayersundersomeconditions.Inthesetournamentsthewinner,theso-calledtitfortat~TFT!algorithm,hasacrucialrole.Thisverysimplealgorithmcooperatesinthefirstroundandlateritreciprocatesthepartner’spreviousdecision.Itforcestheplayerstocooperatemutuallyandmaintainsthisstateagainstdefectors.Inadditiontothehomogeneoussystemwithplayersfol-lowingtheTFTalgorithm,thestatewherealltheplayerschoosetodefecthasprovedtobestationarytoo;morepre-cisely,sparecooperatorswillbesuppressedduetotheevo-lutionaryruleinthelarge-Nlimit.Moreprecisely,onlyasufficientlylargeportionofmutualcooperatorscansurviveamongdefectors.Theemergenceofuniformcooperationbe-comeseasierwhen,combiningtheevolutionarygamewithspatialeffects,theplayersinteractmuchmorewiththeirneighborsthanwiththosewhoarefaraway,asitistypicalinrealpopulations.ThespatialeffectspromotethesurvivalofcooperatorsevenifwedonotuseanykindofelaboratestrategiessuchastheTFT.Recently,NowakandMay@5#haveintroducedaspatialevolutionaryPDgame.Inthismodelindividualslocatedonalatticeplaywiththeirneighborsandwiththemselves.Thestrategicalcomplexitiesandmemoriesofpastencountersareneglectedbyconsideringonlytwosimplekindsofindividu-als:thosewhocooperate(C)andthosewhodefect(D)un-conditionally.Theevolutionaryrulewasalsosimplifiedbyusingdiscretetimesteps.Betweentworoundsindividualsadoptthestrategythathasreceivedthehighestpayoffamongitsneighborsincludingthemselves.Thisdeterministicmodelisequivalenttoatwo-statecellularautomatonwherethenextstateatagivenlatticepointisdeterminedbythestatesonthesurroundingpoints.Theoutcomedependsontheinitialcon-figurationandtherescaledpayoffmatrixdescribedbyasingleparameterbcharacterizingthemeasureoftemptationtodefect~seethematrixinSec.II!.Thismodelwithandwithoutself-interactionwasinvestigatedondifferentlatticestructures~square,triangle,andcubic!.Themostexhaustiveanalysisisperformedonasquarelatticetakingintoaccounttheinteractionswiththefirstandsecondneighborsandself-interaction.NowakandMayobservedarichvarietyofspa-tialandtemporaldynamicsdependentonthevalueofb.Forexample,thecooperatorscaninvadetheworldofdefectorsalongstraightborderlines,whiledefectorsgainalongirregu-larboundariesforagivenintervalofb.Furthermore,theaboverulesconservethesymmetriesoftheinitialstateforadequateboundaryconditions.Duetothediscretenatureoftotalpayoff,sharpstepsappearwhenvaryingb.IntroducingstochasticevolutionaryrulesbetweentwoPHYSICALREVIEWEJULY1998VOLUME58,NUMBER1PRE581063-651X/98/58~1!/69~5!/$15.006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