Evolutionary stability in quantum games

arXiv:0706.1413v2[quant-ph]12Jun2007EvolutionarystabilityinquantumgamesAzharIqbal∗andTaksuCheonKochiUniversityofTechnologyTosaYamada,Kochi782-8502,Japan.February1,2008Contents1Introduction22Evolutionarygametheoryandevolutionarystability32.1Populationsettingofevolutionarygametheory..........53Quantumgames64Evolutionarystabilityinquantumgames104.1EvolutionarystabilityinEWLscheme...............114.1.1Evolutionarystabilityandentanglement..........134.2EvolutionarystabilityinMWquantizationscheme........154.2.12×2asymmetricgames...................164.2.22×2symmetricgames....................184.2.32×2×2symmetricgames.................224.2.4Rock-Scissors-Papergame..................265ConcludingRemarks32AbstractInevolutionarygametheoryanEvolutionarilyStableStrategy(ESS)isarefinementoftheNashequilibriumconceptthatissometimesalsorec-ognizedasevolutionarystability.Itisagame-theoreticmodel,wellknowntomathematicalbiologists,thatwasfoundquiteusefulintheunderstand-ingofevolutionarydynamicsofapopulation.Thischapterpresentsananalysisofevolutionarystabilityintheemergingfieldofquantumgames.∗Permanentaddress:CentreforAdvancedMathematicsandPhysics,NationalUniver-sityofSciences&Technology,CampusofCollegeofElectricalandMechanicalEngineering,PeshawarRoad,Rawalpindi,Pakistan.11IntroductionGamessuchaschess,warfareandpoliticshavebeenplayedthroughouthistory.Wheneverindividualsmeetwhohaveconflictingdesires,andpriorities,thengamesarelikelytobeplayed.AnalysisandunderstandingofgameshasexistedforlongtimesbuttheemergenceofgametheoryasaformalstudyofgamesiswidelybelievedtohavetakenplacewhenNeumannandMorgenstern[1]pub-lishedtheirpioneeringbook“TheTheoryofGamesandEconomicBehaviour”in1944.Gametheory[2]isnowanestablisheddisciplineofmathematicsthatisavastsubjecthavingarichhistoryandcontent.Roughlyspeaking,gametheoryistheanalysisoftheactionsmadebyrationalplayerswhentheseactionsarestrategicallyinterdependent.The1970ssawgametheorybeingsuccessfullyappliedtoproblemsofevo-lutionarybiologyandanewbranchofgametheory,recognizedasevolutionarygametheory[3,4,5],cameintoexistence.Theconceptofutilityfromeco-nomicswasgivenaninterpretationintermsofDarwinianfitness.Originally,evolutionarygametheoryconsideredanimalconflictsoccurringinmacro-world.Inrecentyears,researchinbiology[6]suggestedthatnaturealsoplaysclassi-calgamesatmicro-level.Bacterialinfectionsbyvirusesareclassicalgame-likesituationswherenatureprefersdominantstrategies.Ingametheory[1,2]onefindsmanyexampleswheremultipleNashequilibria(NE)[7,8]emergeassolutionsofagame.Toselectone(orpossiblymore)outoftheserequiressomerefinementoftheequilibriumconcept[9].Arefinementisarule/criterionthatdescribesthecriteriontopreferone(insomecasesmorethanone)equilibriumoutofmany.Numerousrefinementsarefoundingametheory,forexample,perfectequilibrium(usedforextensive-andnormal-formgames),sequentialequilibrium(afundamentalnon-cooperativesolutioncon-ceptforextensive-formgames),andcorrelatedequilibrium(usedformodellingcommunicationamongplayers).Duringrecentyearsquantumgametheory[10,11,12]hasemergedasanewresearchfieldwithinquantuminformationandcomputation[13].Asignificantportionofresearchinquantumgamesdealswiththequestionaskinghowquan-tizationofagameaffects/changestheexistence/locationofaNE.Thisquestionhasbeenaddressedinanumberofpublications[14]inthisareaandnowitseemsthatitisgenerallyagreedthatquantizationofagameindeedaffects/changestheexistence/locationofaNE.Inthischapterwearguethat,likeaskinghowquantizationofagameaf-fects/changestheexistence/locationofaNE,anequallyimportantquestionforquantumgamesistoaskhowquantizationofagamecanaffectarefinementoftheNEconcept.WenoticethataparticularrefinementoftheNE,knownasanEvolutionarilyStableStrategy(ESS),iscentraltoevolutionarygametheory.Whilefocussingonathisrefinement,wemotivatethosequantumgamesinwhichaNEpersists1inbothofitsclassicalandquantumversionswhileits1BysayingthataNEpersistsinboththeclassicalandquantumversionofagamewemeanthatthereexistsaNEconsistingofquantumstrategiesthatrewardsboththeplayersexactlythesamethecorrespondingNEdoesintheclassicalversionofthegame.2propertyofbeinganESSsurvivesineitherclassicaloritsquantumversion,butnotintheboth.Wearguethat,thequantumgamesofferingsuchsituations,alongwiththeirquantizationprocedures,canjustifiablybesaidtoextendtheboundaryofinvestigationsinquantumgamesfromexistence/locationofNEtoexistence/locationofone(ormore)ofitsrefinements.2EvolutionarygametheoryandevolutionarystabilityTherootsofevolutionarygametheory[5]canbetracedtothepuzzleoftheapproximateequalityofthesexratioinmammals.In1930Fisher([15,16])noticedthatifindividualfitnessisdefinedintermsoftheexpectednumberofgrandchildren,thenitbecomesdependentonhowmalesandfemalesaredis-tributedinapopulation.Fishershowedthattheevolutionarydynamicsthenleadstothesexratiobecomingfixedatequalnumbersofmalesandfemales.AlthoughFisher’sargumentcanberecastingame-theoreticlanguagebutorig-inallyitwasnotpresentedinthoseterms.Perhapsitwasduetothefactthatuntilthattimemoderngametheoryhadnotyetemergedasaformalstudyofgames.Moderngametheorywasused,forthefi