Comments on evolution of cooperation evolutionary

arXiv:cond-mat/0310266v1[cond-mat.dis-nn]11Oct2003Commentsonevolutionofooperation:evolutionarystabilityinenhanedDove-HawkmodelPawelSobkowiz∗11thOtober20031AbstratOneofthebestexamplesoftraditionalanalysisofevolutionarystablestrate-gies(ESS)isprovidedbythesoalledDove-Hawkmodel.Inthispaperwepresentseveralenhanementstothemodelaimedatdesribingtheevolutionofooperativebehavior.InadditiontoDovesandHawksweintroduesev-eralgroupsofCooperators,whoatasDoveswithintheirowngroup,butasHawksoutsideit.Thisallowstostudyhowooperatinggroupsmaygrowandahievestabilitywithinanonrepeatingevolutionarygamesframework.Dependingoninitialonditions,thenalstablepopulationmayhaveone,all-enompassingCooperatorpopulationorseveralompetingliques.Af-tertakingintoaountthatCooperatorsbearostsneessarytoreognizemembersofone’sowngroupitispossibletoseepopulationswhereCooper-atorseventuallyloseagainstHawksorpopulationswhereseveralliquesofCooperatorsoexistwithaHawkpopulation.2TheoretialbakgroundThelassialworksbyMaynardSmith[1,2℄introduingandpopularizingtheevolutionarygametheoryandtheoneptofEvolutionaryStableStrategyhaveresultedinwidespreadinterestinsuhmodelingapproahbysoialsientists,biologistsandeonomists.∗e-mailaddress:pawelsobpozta.onet.pl1Withintheevolutionarygamestheoryonedesribesapopulationon-sistingofNTOT‘players’,whointeratamongthemselves.Theformoftheinterationissimpliedto‘ontests’,whiharesingleeventsduringwhihtwoplayersdeidehowtoshareamongthemselvessomevaluableresoure.Forsimpliityitisusuallyassumedthattheresourevalueisonstant(de-notedbyV).For‘peaefully’resolvedonteststhevalueisdividedamongthepartiipantplayers(equallyorunequally).Insituationswhereontestsdevelopsintoaghtovertheresoure,playerspartiipatingintheghtmaysuerinjuries,expendunneessaryresoureset.,whihismodeledbyin-ludingapenaltyost−h(harm).Forsimpliitytheosthisassumedtobeequalforbothplayerspartiipatinginaght.Therearetwomainvariantsofevolutionarygames:Nonrepetitive,singleinstaneontests,inwhihpartiipantshavenomemoryofpreviousen-ountersandthusannotadapttheirstrategiestopastbehaviouroftheiropponents,andRepetitiveGames,wherethememoryallowsompliatedstrategies,individualbehaviorandlearning.WithinthispaperweonentrateonNonrepetitiveGames.Evenforsuhsimpliedsituationplayersmayusedierentstrategies.ToahievethestatusofEvolutionaryStableStrategy,speibehaviourσmusteither:dobetteragainstitselfthananyothervariantstrategyor,ifsomemutantstrategywoulddojustaswellagainstσasσitself,thenσmustdobetteragainstthemutantthanthemutantagainstitself[1℄.Themodelingothepopulationdynamisisbestperformedassumingthattheativitiestakeplaewithinseparateroundsoriterations.Suhiterationsmayorrespond,forexample,toyearlyylesorbiologialgenera-tionsormaynotorrespondtoanydiserniblequantizationofthemodeledsystem,butbeaonvenientsimpliation.Withinsuhiterationsplayersinteratamongthemselvesandompeteforresoures.Betweeniterations,followingtheaverageoutomes(payos)fordierentgroups,theompositionofthetotalplayerpopulationisadjusted.Afterseveral(many)iterationsoneandeterminewhihofthestrategiessueedsandbeomesanESS.ThebestknownofsuhmodelsistheHawk-DovemodelofMaynardSmithandPrie[1℄.Thepopulationonsistsoftwogroups:Doves(D)andHawks(H).Thesegroupsarebestharaterizedbytheirbehaviorinon-tests.WhenaDovemeetsaDove,theyresolvetheontestpeaefully,with-outaght.ThevalueVisthen,statistially,splitinhalfbetweentheontestants.Thustheaverageoutome(payo)foraplayerinaD-DmathisV/2.WhenaDovemeetsaHawk,thelatter‘bullies’theDoveandgetsthewholevalueofV,leavingtheDoveemptyhanded.Ontheotherhand,whentwoHawksmeetaghtensuesandtheaveragepayoisV/2−h.ForV/2−h0(ifonly‘pure’strategiesareallowed)theHawkbeomesanESS,2byvirtueofwinningtheontestswiththeDoves.TheHawksandDovesmodelservesasperfetexamplethatanevolution-arystablestrategyisnotneessarilytheonethatreturnstheebestresultsforthepartiipatingplayers.InapureDovepopulationtheaveragepay-oofV/2isgreaterthaninHawkpopulation.Nevertheless,beausetheoutomeofinter-‘speies’enountersarealwaysfavorabletoHawks,evenasmallHawkpopulationwouldinvadeaDovesoietyandhaveevolution-aryadvantage.InaDovepopulationinvadedbyHawks,theaveragepayodiminishesastheproportionofHawksgrows.TheaimofthispaperistodesribesomeonsequenesoftheextensionoftheHawk-Dovemodelwithinanonrepetitiveparadigm.Theextensionisbaseduponanintrodutionofanewstrategy,alledhere‘Cooperator’.ACooperatorbehaveslikeaDovewhenitompeteswithamemberofitsowngroup.ButwhenaCooperatorenountersaHawkoraDoveitbehavesasaHawk.ToobtainmoregeneralresultsweintrodueseveralseparateCooperatorgroups,denotedbyCi,whihatfriendlywithintheirowngroup,butompetitivelyoutsideit.Inthesimplestase,withjustoneCooperatorgroup,thedynamisofthesystemistrivial.TheCooperatorpayoisalwaysbetterthanthepayoforaHawk.CooperatorsouldeasilyinvadeaDovesoiety.ThussinglegroupCooperatorstrategyisanESS.Itisworthnoting,thatalthoughweplaythegamewithout‘memory’,toallowtheCooperator-likebehaviorsomesortofreognitionofgroupmem-bershipisneessary.Thisanbeahievedthroughsomelabeling,pre-ontestbehaviouret1.Asweareinterestedinexploringfurtherfurth