How Altruism Can Prevail Under Natural Selection by Ted Bergstrom and Oded Stark University - PDF document

HowAltruismCanPrevailUnderNaturalSelectionbyTedBergstromandOdedStarkUniversityofMichiganandHarvardUniversityCurrentversion:March22,2002 HowAltruismCanPrevailUnderNaturalSelectionTedBergstromandOdedStarkIntroductionWhyhaveweeconomistsbeenconvincedforsolongthatouroldfriend,homoeco-nomicusmustbeselsh?Nodoubtwendconsiderablesupportforthishypothesisinthebehaviorofourcolleagues.Wemightalsoexpectthatevolutionarypressurestendtoproduceselshbehavior{withthenotableexceptionoftherelationbetweenparentsandospring.Butcanweexpectnaturalselectiontoactinfavorofaltruisticbehaviorinotherrelationships?Evolutionarybiologistshavecreatedatheorythatpredictsaltruisticbehavior,notonlybetweenparentsandchildren,butalsoamongsiblingsandothercloserelatives.1RichardDawkins'expressionofthisviewinTheSelshGene,isthatthereplicatingagentinevolutionisthegeneratherthantheanimal.Ifagenecarriedbyoneanimalislikelytoappearinitsrelatives,thenageneforhelpingone'srelatives,atleastwhenitischeaptodoso,willprosperrelativetogenesfortotallyselshbehavior.Thispaperpresentsaseriesofexamplesinwhichnaturalselectionsustainscooperativebehaviorinsingle-shotprisoners'dilemmagames.Inprisoners'dilemma,cooperational-waysgetsalowerpayoforoneselfandahigherpayoforone'sopponentthandefection.Thereforeitseemsappropriateinthissimplecasetoidentifyaltruismwithplayingcoop-erateinprisoners'dilemma.2Thereasonthatcooperativebehaviortowardsiblingscanbesustainedevenwheredefectionisadominantstrategy,isthatanindividualwhohasageneforcooperatingwithitssiblingshasagoodchanceofbenetingfromthepresence 1See,forexample,WilliamHamilton(1964a,1964b),RichardDawkins(1976),JohnMaynardSmith(1982),andRobertTrivers(1985).2Moresubtlequestionsaboutthenatureofaltruisticpreferencesareleftforotherinvestigations.Eachofushasdonesomeworkofthiskind.SeeB.DouglasBernheimandOdedStark(1988),OdedStark(1989),andTedBergstrom(1988),(1989),(1992).1 ofthesamegeneinitssiblings.Similarreasoningappliestobehaviorthatisimitativeratherthangeneticallyinheritedifthosewhosharecommonrolemodelsaremorelikelytointeractwitheachotherthanwithrandomlyselectedmembersofthepopulation.1.TheGameCreaturesPlayandtheNatureofEquilibriumIndividualswillbeassumedtoplayone-period,two-persongamesofprisoners'dilemmawiththeirsiblingsorneighbors.Ineachgamethatitplays,anindividualcanchooseoneoftwostrategies,cooperateordefect.Thepayosfromthisgamearelistedinthematrixbelow.IftheparameterssatisfytherestrictionSPRT,thendefectwillbeadominantstrategyforeachgame.Forthegametobecalleda\prisoners'dilemma",itshouldalsosatisfytherestrictionthatS+T2R.Prisoners'DilemmaPlayer1Player2CooperateDefect Cooperate R;R S;T Defect T;S P;P Totalpayotoanindividualwillbetheaverageofitspayosintheprisoners'dilemmagamesthatitplays.Wherebehaviorisgeneticallyinherited,weassumethattheexpectednumberofsurvivingospringthatanindividualproduceswillbehigher,thehigheritstotalpayo.Wherebehavioriscopiedfromneighbors,theprobabilitythatanindividual'sbehavioriscopiedwilldependonitspayo.Apopulationcanhaveeitheramonomorphicequilibriumorapolymorphicequilibrium(orpossiblyboth).Inastablemonomorphicequilibrium,onlyonetypeofindividualispresentandifamutantindividualoftheothertypeshouldarise,itmustreproducelessrapidlythannormalindividuals.Inapolymorphicequilibrium,morethanonetype2 ofindividualispresentandeachtypethatispresentreceivesthesameexpectedpayo.Stabilityofpolymorphicequilibriumrequiresthatifonetypehappenstobecomemorecommonthantheequilibriumproportion,itwillhavealowerexpectedpayothantheothertype.2.EvolutionofGeneticallyTransmittedBehaviorSincelittleisknownabouttheenvironmentswhichshapedourgeneticinheritance,theevo-lutionaryhypothesismaynotbeveryinformativeaboutmanyaspectsofourpreferences.Butthefundamentalprocessesofmating,child-rearingandrelationsbetweensiblingsap-peartohavechangedlittleoverthemillennia.Accordingly,wemaylearnagooddealaboutthe\economicsofthefamily"fromalookattheevolutionarytheoryofrelationsamongkin.3AltruisticSororitiesWithoutSexJusttohelpusunderstandthelogicofinheritance,webeginwithatoymodelthatseemsunrealisticforhumans{asexualreproduction.Letusassumethatanyindividualwill,ifshesurvivestoreproductiveage,haveexactlytwochildren,whosegenesarejustlikeherown(exceptinthecaseofraremutations).Asurvivingindividualwithageneforcooperatewillhavetwoospringwithgenesforcooperate.Asurvivingindividualwithagenefordefectwillhavetwoospringlikeherself.Tokeepthepopulationsizeconstant,wemustassumethatonlyhalfoftheindividualswhoarebornwillsurvivetoreproductiveageandthattheprobabilitythatanyindividualwillsurvivetoreproducewillbehigher,thegreaterherpayointhegamethatsheplayswithhersister.Weclaimthattheonlyequilibriumisapopulationconsistingentirelyofcooperators. 3Wendourselvesingoodcompanyinthisheresy.Becker(1976,1981)andHirshleifer(1977,1978),exploregeneticexplanationsforaltruismamongclosefamilymembers.RobertFrank(1985)seeksanevolutionaryexplanationforhumanemotions,andArthurRobson(1992a,1992b)exploresanevolutionaryexplanationofhumanattitudestowardrisk.3 Toseethis,considerapopulationconsistingentirelyofcooperators.Whatwouldhappentoamutantdefectorthatappearedinthispopulation?Sincehersisterisacooperator,themutantgetsapayoofT�R,andsosheismorelikelytosurvivethananyothermemberofthepopulation.Buthergoodfortunewillnotbesustainedbyherdescendants.Herdaughterswillbothinheritherdefectgeneandwillbothdefect.SistersineachgenerationofherdescendantswillalsodefectandgetPR,andhencegraduallydisappearfromthepopulation.Similarly,inapopulationofdefectors,amutantcooperatorwouldfaceadefectingsisterandwouldgetapayoofS,whilethesurroundingdefectorswouldgetP&#x-277;S.Althoughhersurvivalprobabilitywillbelowerthanthepopulationaverage,herdaughtersandtheirdescendantswouldallbecooperators.EachofthemwillreceiveapayoofR&#x-277;Pandtheirnumberswouldgrowrelativetothoseofthedefectors.DiploidSiblingsareSometimesAltruistic(ButNotasOftenasTheirParentsWouldLike)Diploidparentswillnotbesurprisedtodiscoverthatinourownspecies,siblingsarenotalwaysascooperativeasasexualsiblingswouldbe.Wewillshowthatthereisarichmenuofpossibleequilibriawithdiploidsiblings.Dependingontheparametersoftheprisoners'dilemmagame,theremaybeauniquestablemonomorphicequilibriumwithcooperatorsonly,auniquestablemonomorphicequilibriumwithdefectorsonly,ortheremaybetwolocallystablemonomorphicequilibria{onewithcooperatorsonlyandtheotherwithdefectorsonly.Finally,thereareparametervaluesforwhichtherearenostablemonomorphicequilibria,butfortheseparameterstherewillbeastablepolymorphicequilibriumwithsomecooperatorsandsomedefectorsinthepopulation.Weconsideralargepopulationwhichreproducessexuallyandhasdiploidgeneticstruc-ture.Eachindividualplaysasingle-shotgameofprisoners'dilemmawitheachofitssib-lings.Tosimplifyexposition,wewillassumethateachindividualwhosurvivestomateandreproducehasexactlythreeospring.Theprobabilitythatanindividualsurvivestoreproducewillbehigher,thehigherthetotalpayothatitgetsinthegamesitplayswith4 itstwosiblings.4Individualsareabletodistinguishtheirsiblingsfromothermembersofthepopulationandmayusedierentstrategiesingamesplayedwithsiblingsfromthestrategiesusedwithoutsiders.Thestrategythatanyindividualusesinplaywithitssiblingsisdeterminedbythecontentsofasinglegeneticlocus.Thislocuscontainstwogenes,onerandomlyselectedfromeachofitsparents'twogenes.Forthepresentdiscussion,weassumethatmatingismonogamousandrandomwithrespecttothegenescontrollingbehaviortowardsibling.Weassumethattherearetwokindsofgenes,ac(cooperate)geneandad(defect)gene.Thentherewillbethreepossibletypesofindividuals,namelytypecchomozygoteswhocarrytwocgenes,typecdheterozygoteswhocarryonecgeneandonedgene,andtypeddhomozygoteswhocarrytwodgenes.Typecchomozygotesalwaysplaycooperateandtypeddhomozygotesalwaysplaydefect.Ifheterozygotesalwaysdefect,thenthedgeneissaidtobedominantandthecgeneissaidtoberecessive.Ifheterozygotesalwayscooperate,thenthecgeneisdominantandthedgeneisrecessive.5Inthispaper,weconneourattentiontomonomorphicequilibriawhicharestableagainstinvasionbydominantmutantgenes.Thus,wewillconsiderwhetherapopulationconsistingentirelyofcchomozygotescouldbe\invaded"bymutant\dominant"dgenessuchthatcdheterozygotesalwaysplaydefect.Similarly,wewillaskwhetherapopula-tionconsistingentirelyofddhomozygotescouldbeinvadedbydominantcgenes.6Thepossibilityofinvasionbyrecessivemutantsleadstoaninteresting,butratherelaborate 4Arichermodelwouldhavensiblingsplayageneraln-persongameratherthanhaveeachindividualplaytwoseparatetwo-persongameswithitstwosiblings.Suchamodelcouldfocusonquestionsofreturnstoscalewithinfamilies.Theassumptionthatallindividualswhosurvivetomateandreproduceallowsustosidestepthecomplicationsthatwouldarisefromreconcilingtheassumptionofmonogamousrandommatingwiththeassumptionthatthenumberofchildrenwashasdependsononespayosinprisoners'dilemma.Whatshouldwedoifhusbandandwifehavedierentexpectednumberofospring?5Alternatively,onecouldassumethatheterozygotesplayamixedstrategywithsomexedprobabilitiesofcooperationanddefection.6Noticethatwedonottaketheviewthateithercgenesordgenesmustbeintrinsicallydominant.Insteadweaskwhetherinamonomorphicpopulation,ifadominantmutantoftheoppositetypeshouldarise,themutantstrainwouldincreaseasaproportionofthepopulationorwouldultimatelydisappear.5 analysiswhichwillnotbepursuedhere.7Firstletusaskwhenapopulationconsistingonlyofcooperatorswouldberesistanttoinvasionbymutantswitha(dominant)genefordefection.Supposethattheentirepop-ulationconsistsoftypecchomozygotes,allofwhomcooperate.Nowletsomeindividualexperienceamutationwhichchangesoneofitscgenestoadgenewhichisdominantoverthetypecgene.Themutantwillthereforebeacdheterozygoteandwillplaydefect.Inthegamesitplayswithitssiblings,thismutantwillgetahigherpayothannormalmembersofthepopulation,sinceitsnormalsiblingscooperatewhileitdefects.ThereforethemutantreceivesTineachgamewhileordinarymembersofthepopulationreceiveRT.Butinordertondoutwhetherthemutanttypewillinvadethepopulationinthelongrun,wemustfollowthefortunesofitsospringwhoinheritthemutantgene.Whenthemutantcdtypeisrare,itwillalmostcertainlymatewithanormaltypecc.Themutant'sospringwillthereforebeoftypecdwithprobability1/2andoftypeccwithprobability1/2.Anospringofthemutantwhocarriesthemutantgenewillbeoftypecdandwillplaydefect.Withprobability1/2,arandomlychosensiblingofthisindividualwillbeatypecchomozygoteandwithprobability1/2,thatsiblingwillbeanothertypecd.Thereforewithprobability1/2,thisindividualcanexploitacooperativesiblingandreceiveapayoofT,butwithprobability1/2,thesiblingwillalsodefect.Itthenfollowsthattheexpectedpayotoeachheterozygoteospringofthemutantis(T+P)=2.TheospringofnormalcctypeswillreceiveapayoofRinthegamestheyplaywiththeirsiblings.Itfollowsthatwhilethemutantgeneisrare,carriersofthemutantgenewillreproducemorerapidlythannormalindividualsifT+P&#x-354;2RandlessrapidlyifT+P2R.Nowletusaskwhenapopulationconsistingentirelyoftypeddindividualscouldbeinvadedbyamutantcgenewherethemutantgeneisdominantoverthenormalgenes.Asinglemutatinggenewouldrstappearinacdheterozygote.Assumingthecgeneis 7Aheterozygotewitharecessivemutantgenewillactjustlikethenormalpopulationandsotherewillbenoselectioneitherfororagainstrecessivegenesuntil\geneticdrift"producesenoughmutantheterozygotessothattheyoccasionallymate,therebyproducinghomozygoteswhoactdierentlyfromtheremainingpopulation.EquilibriawhichareresistanttoinvasionbothbydominantandbyrecessivemutantsarestudiedbyBergstrom(1992).6 dominant,themutantindividualwouldcooperate.ThemutantindividualwouldreceivethelowpayoS,sinceitplaysitssiblingswhoplaydefect.Butonaverage,itsospringwilldobetterthanSandperhapswilldobetterthanthenormalpopulationofdefectors,allofwhomreceiveP.Whenthemutanttypeisrare,amutantwillalmostcertainlymatewithanormalddtype.Halfofthemutant'sospringwillbecdheterozygotes,whocooperate,andhalfofthemwillbeddhomozygotes,whodefect.Anospringofthemutantwhocarriesthemutantgenewillbeoftypecdandwillplaycooperate.Withprobability1/2,arandomlychosensiblingofthisindividualwillbeatypeddhomozygotewhodefectsandwithprobability1/2,thatsiblingwillbeanothertypecdwhocooperates.Thereforewithprobability1/2,anospringthatcarriesofthemutantgenewillbeexploitedbyitssiblingandgetapayoofS,butwithprobability1/2,itssiblingwillalsocooperateandeachofthemwillreceiveapayoofR.Theexpectedpayotoatypecdospringofthemutantistherefore(S+R)=2.Thispayowillbesmallerthanthepayotonormalddtypesif2P�S�R�0andlargeriftheinequalityisreversed.Aswehaveshown,therewillbeastablemonomorphicequilibriumwithallcooperatorsifT+P�2R�0andtherewillbenosuchequilibriumiftheinequalityisreversed.Therewillbeastablemonomorphicequilibriumwithalldefectorsif2P�S�R�0andnosuchequilibriumiftheinequalityisreversed.Itturnsthatthereareprisoners'dilemmagameswhereeachoftheseinequalitiestakeseithersign.ThepossibilitiesareillustratedinFigure1.InthisgurewehavenormalizedthegametosetS=0andT=1.8Withthisnormalization,therewillbeastablemonomorphicequilibriumwithtypecconlyifR�(P+1)=2�0andtherewillbeastablemonomorphicequilibriumwithtypeddonlyifR2P.ForparametervaluesinRegionCofFigure1thereisastablemonomorphicequilibriumwithcooperatorsonlyandnostableequilibriumwithdefectorsonly.ForparametervaluesinRegionDofFigure1,thereisastablemonomorphicequilibriumwithdefectorsonlyandnostableequilibriumwithcooperators 8Thiscanbedonewithoutlossofgenerality,sincethepopulationdynamicsdiscussedinthispaperareinvarianttoanetransformationsofthepayomatrix.7 only.ForparametervaluesinRegionB,therewillbestablemonomorphicequilibriaofbothtypes,andforparametervaluesinRegionA,therewillnotbeastablemonomorphicequilibriumofeithertype.Inorderforthegametobeaprisoners'dilemma,itmustalsobethatR�PandthatR�:5.TheregioninFigure1abovethetwodottedlinessatisestheseconditions.WenotethateachoftheregionsA,B,C,andDcanoccurwithparameterssuitableforprisoners'dilemma.Figure1justiestheclaimsmadeinthetitleofthissection.Forprisoners'dilemmagameswithparametervaluesinRegionC,diploidsiblingswillcooperate,eventhoughitistotheirselshadvantagetodefect.Forprisoners'dilemmagameswithparametervaluesinRegionD,diploidsiblingswillbothdefect,althoughparentswhowishthemtomaximizetheirjointpayowouldpreferthembothtocooperate.ItisinterestingtoconsiderRegionsAandB.ForparametervaluesinRegionB,therearetwostableequilibria{onewithamonomorphicequilibriumofeachkind.ForparametervaluesinRegionA,therearenostablemonomorphicequilibrium.Inordertounderstandthesecases,itisnecessarytoworkoutthedetailedlawsofmotionforthedynamicalsystemthatresultsfromthismodel.ThisisdonebyBergstromandBergstrom(1992),whereitisfoundthatforparametervaluesinRegionA,thereexistsexactlyonestablepolymorphicequilibriumandforparametervaluesinRegionB,thereisoneunstablepolymorphicequilibriumandnostablepolymorphicequilibria.3.WhenChildrenImitatetheirParentsorTeachersHerewestudyamodelinwhichbehaviorisacquiredbyimitation,ratherthangeneti-cally.ThemodeldiscussedhereisavariantofmodelsofculturaltransmissionwhichweredevelopedbyCavalli-SforzaandFeldman(1980),andBoydandRichardson(1985).Weassumethateachindividualhastwosiblingsandplaysagameofprisoners'dilemmawitheachofthem.Wewillalsoassumethattheprobabilitythatanyindividualsurvivestomateandreproduceisproportionaltotheaveragepayothatitreceivesinthegamesitplayswithitssiblings.8 Assumethatwithprobabilityv,achildadoptsthestrategythatwasusedbyarandomlychosenoneofitstwoparentsandwithprobability1�vitadoptsthestrategyusedbyanonparent,randomlyselectedfromtheentirepopulation.Weassumethatmarriageismonogamous,sothatallsiblingssharethesamemotherandfather.Parent-couplescanbeoneofthreepossibletypes;two-cooperatorcouples,\mixedcouples"withonecooperatorandonedefector,andtwo-defectorcouples.Matingissaidtobeassortativeifadultsalwaysmatewithindividualsoftheirowntype.9Letxbethefractionoftheadultpopulationwhoarecooperators.Ifmarriageispurelyrandom,thefractionofallmarriageswhicharemixedcoupleswillbe2x(1�x).Wedeneaparametermwhere0m1insuchawayastoallowmatingpatternsthatliebetweenthepolarcasesofpurelyrandom(m=0)andpurelyassortative(m=1)mating.Inthepopulationatlarge,theproportionofmixedcouplesis2(1�m)x(1�x),theproportionoftwo-cooperatorcouplesisx2+mx(1�x),andtheproportionoftwo-defectorcouplesis(1�x)2+mx(1�x).10Giventheproportionsofcouplesofeachtype,wecandeterminetheproportionsofallsiblingpairsconsisting,respectively,oftwocooperators,onecooperatorandonedefector,andtwodefectors.Thisenablesustodeterminenotonlytheproportionofospringofeachtype,butalsotheexpectedpayostoospringofeachtype,sincewewillknowtheprobabilitythatarandomlychosensiblingofanindividualofeachtypewillbeacooperatororadefector.Withthisinformation,wearebeabletodeterminetherelativegrowthratesofthepopulationofcooperatorsandofdefectors.ThedetailsofthisprocessareworkedoutintheAppendixofthispaper.Thismodelturnsouttohavearemarkablyconvenientmathematicalstructure.Therateofchangeofthenumberofsurvivingindividualsofeachtypeinanygenerationturns 9Ifmatingrequiresmutualconsent,iftypesarecostlesslyrecognizableandsearchcostsarenegigible,thiswouldbeanaturaloutcome,sincecooperatorscanexpectmoreospringiftheymatewithothercooperatorsthaniftheymatewithdefectorssolongas2R�S+P.10Cavelli-SforzaandFeldmanattributethisparameterizationofassortativematingtoSewallWright(1921).Theseproportionswouldbeachievedifcoupleswererstrandomlymatchedandthenthefractionmofthemixedcoupleswerebrokenupandthefreedindividualspairedwithpersonsoftheirowntype.9 outtobealinearfunctionofthefractionofthepopulationintheparentgenerationwhoarecooperators.Thereforethedierencebetweenthegrowthratesofthenumberofindividualsofthetwotypesisalsoalinearfunctionofthefractionofthepopulationwhoarecooperators.11Inparticular,thedierencebetweenthegrowthratesofthepopulationofcooperatorsandthepopulationofdefectorsisexpressedbyD(x)=A+Bx,whereA=v2(1+m)(R�S)�2(P�S)andwhereA+B=v2(1+m)(T�P)�2(T�R):Dependingontheparametervalues,v,m,S,P,R,andT,thedynamicsofthissystemfallsintooneofthefollowingfourqualitativelydistinctcases.Casei.IfA�0andA+B�0,thentheonlystableequilibriumisamonomorphicequilibriuminwhichtheentirepopulationconsistsofcooperators.ThissituationisillustratedinFigure2a.Caseii.IfA�0andA+B0,thentherearetwostableequilibria,oneinwhichtheentirepopulationconsistsofcooperatorsandanotherinwhichtheentirepopulationconsistsofdefectors.Thereisalsoanunstablepolymorphicequilibriuminwhichtheproportionofcooperatorsis�A=B.ThissituationisillustratedinFigure2b.Caseiii.IfA0andA+B&#x]TJ/;ñ 9;&#x.963;&#x Tf ;!.9;i 0;&#x Td[;0,thentherearetwounstablemonomorphicequi-libria,oneinwhichtheentirepopulationconsistsofcooperatorsandanotherinwhichtheentirepopulationconsistsofdefectors.Theonlystableequilibriumisapolymor-phicequilibriuminwhichtheproportionofcooperatorsis�A=B.ThissituationisillustratedinFigure2c.Caseiv.IfA0andA+B0,thentheonlyequilibriumisamonomorphicequi-libriuminwhichtheentirepopulationconsistsofdefectors.Thissituationisillustrated 11Thislinearitydoesnotholdinmodelswithdiploidsiblings.Withdiploidinheritance,theequationforthedierenceinexpectedgrowthratesistypicallyquadraticorcubic.SeeBergstromandBergstrom,1992.10 inFigure2d.Itisinterestingtolookatsomespecialcases.Supposethatthereisperfectlyassortativemating,m=1,andthatchildrenalwaysimitatetheirparents,v=1.Thenthemodelisformallythesameasthemodelofasexualreproductiondiscussedabove.Inthiscase,A=A+B=2(R�P).Foreveryprisoners'dilemmagame,R�P,sotheonlyequilibriumforprisoners'dilemmawouldbeapopulationconsistingofcooperatorsonly.Anothersimplespecialcaseiswherev=1andthereisrandommating,sothatm=0.Inthiscase,A=R+S�2PandA+B=2R�T�P.Inthiscase,theparametervaluescorrespondingtoeachofthefourcasesareexactlythesameasthosecharacterizingthefourpossiblecasesforadiploidpopulation,asdisplayedinFigure1.Ifv=0,thenA=2(S�P)0andA+B=2(R�T)0.Inthiscase,foraprisoners'dilemmagame,theonlyequilibriumisapopulationconsistingonlyofdefectors.Noticethattheparametersmandvin uenceequilibriumonlythroughtheirin uenceontheexpression,(1+m)v2.Anincreasein(1+m)v2willincreasebothAandBforgivenpayoparameters,S,P,R,andT.Thismeansthatthelargeris(1+m)v2,thelargeristhesetofpayoparametersforwhichthereisamonomorphicequilibriumwithallcooperatorsandthesmallerthesetofpayoparametersforwhichthereisanequilibriumwithalldefectors.Thatistosay,themorelikelychildrenaretoimitatetheirparents,andthemorelikelytheirparentsaretobethesameaseachother,themorelikelycooperativebehavioristoprevail.4.WhendoesProvincialismPromoteCooperation?Intheevolutionaryexamplesthatwelookedat,thereisagoodchancethatonewillplayagamewithanopponentwhosebehaviorisinheritedfromthesameparentasonesownbehavior.Wecanexpectsimilareectsinspatialmodels,whereneighborsinteractingamesofprisoners'dilemmaandwherebehaviorcomesfromimitationofrelativelysuccessfulneighbors.ThiseecthasbeendocumentedinaseriesofcomputersimulationscarriedoutbyNowakandMay(1992),carriedoutonatwo-dimensionalgrid.NowakandMayshownotonlythatcooperativebehaviorcanbesustained,butalsothatagreat11 varietyofcyclesandwavescanoccur.Herewewillshowthatcooperationcanbesustainedininterestingwayseveninaone-dimensionalmodelthatissimpleenoughtobestudiedwithapadandpaper.Imaginearoadwhichrunsaroundalake.Alongthisroadliveseveralfarmers,eachofwhomhasoneneighboronhisleftandoneonhisright.Eachfarmerplaysagameofprisoners'dilemmawithhisneighborsandhistotalincomeisthesumofhispayosfromthesegames.Thefarmers'sonsgrowup,observingtheactionsoftheirfathersandtheirneighbors.Whenthefathersdie,theirsonstakeoverthefarmsanddecidewhethertobecooperatorsordefectors.Thesonschoosetheirstrategiesafterobservingtheactionsbyandthepayosreceivedbytheirfathersandtheirneighbors.Forthisdiscussion,letusconsiderprisoners'dilemmagamesforwhich2PS+Rand2R&#x-277;T+P.(ForexampleS=0,P=1=4,R=3=4,T=1.)Avarietyofinterestingpatternsemerge.Thenatureofequilibriumwilldependonthedetailsofneighborsinteract,whichneighborsareobservedbythesons,andhowthesonschoosetherebehavior.Considerrstthecasewhereeachfarmerplaysprisoners'dilemmawithhistwoim-mediateneighborsandwherethesonsimitatethebehavioroftheirfatheroroneofhisneighbors,dependingonwhoreceivesthehighestpayo.Thissetupleadstoarelativelytranquiloutcomeinwhichtherearemanypossiblestablecongurations.Infact,anyar-rangementofdefectorsandcooperatorswhichconsistsofclustersof3ormorecooperatorsandclustersof2ormoredefectorswillbestable.12Cooperationinclusterssmallerthantwowilldisappear.13Anisolatedcooperatorwillgetahigherpayothaneitherofhis 12Consideraclusterof3ormorecooperatorswhichabutsaclusterof2ormoredefectors.Thesonsoffarmersintheinteriorofcooperatorclustercanseeonlytheirfathersandtwocooperativeneighbors,sotheywillcooperate.Thesonofacooperatorontheboundaryofaclusterseesacooperatorneighborwhoreceives2R,hisfatherwhoreceivesS+R,andadefectorneighborwhoreceivesapayoofT+P.ByassumptionthelargestofthesepayosisT+P,sohecooperatesashisfatherdid.Thesonofadefectorintheinteriorofadefectorclusterseesonlydefectandwilldefect.ThesonofadefectorontheboundaryofaclusterseesacooperatorneighborwhoreceivesS+R,hisdefectingfatherwhoreceivesT+P,andadefectorneighborwhoreceives2P.Sinceforaprisoners'dilemmagameSPRT,hisfatherwillhaveahigherpayothanthecooperatorneighbor,sothesonchoosestodefect,justashisfatherdid.13Anisolatedcooperatorgets2SandhisneighborsgetatleastT+P&#x-339;2S.ApairofcooperatorssurroundedbydefectorswilleachgetR+S,whiletheadjacentdefectorswilleachgetatleastT+P&#x-339;R+S.Soineachcasethecooperators'sonswilldefect.12 neighborsandsowillbeimitatedbyhissonandbythesonsofbothofhisneighbors.Somethingmoreexcitinghappensifwechangetheprecedingmodelsothatthesonspaynoattentiontotheirfathers,butimitatetheirfathers'mostprosperousneighbor.Inthiscase,weseesomeremarkableculturalpatternswhichseemsto\pickuptheirfeetandwalkdowntheroad."Forexample,supposethatsomewherealongtheroadthereisagroupingofvefarmersconsistingofacooperatorwithadefectoronhisright,followedbyastringofthreecooperatorstotherightofthedefector,makingapatternCDCCC.Supposethatallotherfarmersontheroadaredefectors.Itisnothardtoshowthatwiththisconguration,everysonalongtheroadwilladoptthebehaviorofhisfather'sneighborontheleft.ThismeansthatthebehaviorclusterCDCCCmovesonefarmtotherightineachgeneration.14Anobserverwhowatchedthebehavioroftheresidentofasinglefarmoveralongperiodoftimewouldseecycles,inwhichaspellofdefectionswouldbeinterruptedbyacooperation,thenadefection,thenthreecooperationsandthenareturntodefection.Othersimilarpatternswhichwalkdowntheroadcanbeconstructedfromanyblockofthreeormorecooperatorsfollowedbya\tail"ofanarbitrarynumberofalternatingcooperatorsanddefectors.Anotherexampleofinterestisthecasewhereeachfarmerobservestwoneighborstohisleftandtwoneighborstohisrightandplaysprisoners'dilemmawithallfourofthem.Eachsoncopiesthemostprosperousfarmerfromthesetwhichincludeshisfatherandhisfather'sfournearestneighbors.Forthiscase,also,thereareequilibriainwhichcooperationissustained.Allsuchequilibriahaveblocksofcooperatorspunctuatedbypatternsofdefectionofoneoftwokinds:1)Stablepairsofdefectorssurroundedbycooperators.2)\Blinkers",whichcycleinthefollowingway.Atonestage,thereisasingledefector,surroundedbycooperators.Thisdefectordoesbetterthananyofhisneighborsandisimitatedbythesonsofallthefarmerswhocanseehim,makingaclusterofvedefectorsinthenextgeneration.Thesonsoftheoutertwoofthesevedefectorsthencooperate|leavingaclusterofthreedefectors.Thesonsoftheoutertwoofthesethreedefectorswill 14Similarphenomenaoccuringinthewell-knowncellularautomatongameof\Life"areknownas\glid-ers".NowakandMayalsondglidersintheirtwo-dimensionalsimulations.13 cooperate,leavinganisolateddefector.Thenthecycleresumes.Itwouldbenicetohavegeneraltheoremsthatwouldallowustoclassifyspatialgamesofthistypeandtopredictthepatternsofoutcomesforbroadclassesofgames.Atthispoint,allwehaveareexampleswhichshowthatcooperationcanbesustainedandthatregularcyclesofalternatingcooperationanddefectionarepossible.5.MaximizersandImitatorsThereisastrikingformalsimilaritybetweenageneticmodelofbehaviortowardssiblingsandamodelinwhichsomeindividualsareimitatorsandothersarerationalmaximizerswhotakeintoaccountthebehaviorofimitators.Inthediploidgeneticmodeldiscussedabove,successfulgenesmust\takeaccountof"thefactthatanindividualwithagenefortreatinghissiblinginagivenwaywill,withprobability1/2,befacedwithasiblingwhotreatshissiblinginthesameway.IthasbeensuggestedbyDonaldCoxandOdedStark(1992)thatevenselshpeoplewouldbekindtotheiragedparentsbecausemuchhumanbehavioris\imprinted"duringchildhood.Thatis,childrenobservehowtheirparentsbehaveandlateradoptthesebehaviorswithoutknowingwhy.Supposethatanadultcouplebelievethattheirbehaviortowardtheirparentswillbeimprintedontheirchildren,sothatwhentheyareold,theirchildrenwilltreatthemastheytreatedtheirparents.Then,eveniftheywereentirelyselsh,theywouldtreattheirparentsastheywouldliketobetreatedwhentheyareold.Butitwouldbeveryoddtoassumethattheparentsinthemiddlegenerationare\freetochoose",rationallyaccordingtotheirself-interest,whilethebehavioroftheirchildrenispredeterminedbyimprinting.Tomakethisstoryinternallyconsistent,weallowthepossibilitythatanyindividualmaybeeitheranimitatororamaximizer,withsomeprobabilitybetween0and1.Parentscannottellwhetherayoungchildisgoingtobeanimitatororamaximizer.Imitatorchildrenwilltreatagedparentsexactlyastheirparentstreatedtheirownparents.Maximizerswillchoosetheirbehaviortomaximizetheirself-interest,butwiththeawarenessthattheiractionsmaybeimitatedbytheirchildren.Tosimplifytheformaltreatment,letusstudythecaseofsingle-parentfamilieswith14 onemotherandonedaughter.LetusassumethatmaximizersseektomaximizeavonNeumann-MorgensternutilityfunctionU(x;y),wherexisthemaximizer'sactionstowardhermotherandyistheactionofherdaughtertowardherwhensheisold.Ifthemaximizerwerecertainthatherdaughterwouldbeanimitator,shewouldchoosethe\Kantian"xthatmaximizesU(x;x).Butifshebelievesthatherdaughtermaybeamaximizerratherthananimitator,thenshewillnotbesogeneroustohermother.Ifaparentchoosesactionxtowardhermother,thenanimitatingdaughterwillchooseactionxtowardher,butamaximizingdaughterwillchooseanactionywhichisindependentofhermother'schoiceofx.Letusassumeastationaryenvironmentsuchthattheplanningproblemfacedbyeachgenerationisthesameasthatfacedbyitssuccessor.Supposethattheprobabilitythatachildisanimitatorisandsupposethattheactiontakenbyamaximizingchildtowardhermotherisy.ThenamotherwhochoosesactionxtowardherparentwillhaveanexpectedutilityofU(x;x)+(1�)U(x;y):Letx(y)bethechoiceofxthatmaximizestheaboveexpression.Sincetheenvironmentisstationary,iftheparent'sdaughterisamaximizer,shewillfacethesamemaximizationproblemashermother.Thereforeheractionytowardhermotherwillbethesameastheactionx(y)oftheparenttowardhermother.Itfollowsthatinanygeneration,amaximizingparentwillchoosexsothatthevalueofxthatmaximizesU(x;x)+(1�)U(x;x)isx.WheretheutilityfunctionUisdierentiable,therst-ordernecessaryconditionformaximizersisfoundbycalculatingthederivativeofU(x;x)+(1�)U(x;x)withrespecttox.Thisrst-orderconditionisU1(x;x)+U2(x;x)=0,whereUi(x;y)isthepartialderivativeofUwithrespecttoitsithargument.Inequilibrium,accordingtothiscondition,maximizerswillchoosexsothatthemarginalcost�U1(x;x)ofkindnessestotheirparentsisequaltotimesthemarginalbenetsofkindnessreceivedfromtheirchildren.15 6.ConclusionWehaveseenseveralenvironmentsinwhichanindividualwillcertainlyreceiveahigherpayofromdefectingthanfromcoooperatingandwhere\copies"ofanindividualaremorelikelytoappear,thehigherthehigherherpayo.Eveninsuchunpromisingsoil,weseethatcooperationcanpersistand uorish.Thereasonisthatbothgeneticinheritanceandculturalinheritancearebluntinstruments.Withgeneticinheritanceageneforbehaviorthatisinheritedbyoneindividualislikelytoappearinitssiblings.Similarly,inmanyenvironments,culturalnormsarelikelytosimultaneouslyin uencebothplayersinthegamesinwhichtheyinteract.16 Appendix{MathematicsofCulturalEvolutionPairsofindividualscanbeofthreetypes.Atype1pairconsistsoftwocooperators,atype2pairconsistsofonecooperatorandonedefector,andatype3pairconsistsoftwodefectors.Ifthefractionofcooperatorsinthepopulationisx,andtheassortativematingparameterism,thenthefractionsofparentpairsoftheithtypeisgivenbytheithentryinthecolumnvector~p(x)=�x2+mx(1�x);2(1�m)x(1�x);(1�x)2+m(1�x)
0:Assumethatachildimitatesarandomlychosenparentwithprobabilityvandarandomlychosenmemberofthepopulationatlargewithprobability1�v.Theprobabilitythatarandomlychosenpairofospringfromatypeiparent-pairisatypejsiblingpairisgivenbytheijthentryofthefollowingmatrix,M(x)=0@(v+(1�v)x)2(v 2+(1�v)x)2(1�v)2x22(v+(1�v)x)(1�v)(1�x)2(v 2+(1�v)x)(1�v 2�(1�v)x)2(1�v)xv(1�x)(1�v)2(1�x)2(1�v 2+(1�v)x)2(v+(1�v)(1�x))21A:Giventhatthefractionxofthenthgenerationarecooperators,theprobabilitythatarandomlychosenpairofsiblingsfromthen+1stgenerationareoftypeiisgivenbytheithentryofthecolumnvector~s(x)=M(x)~p(x):Calculationshowsthat~s(x)=(s1(x);s2(x);s3(x)),wheres1(x)=xv2(1+m) 2(1�x)+x;s2(x)=2x(1�x)v2(1+m) 2�1;s3(x)=(1�x)v2(1+m) 2x+1�x):Cooperatorsintype1siblingpairswillgetpayosofRandcooperatorsintype2siblingpairswillgetpayosofS.Defectorsintype2siblingpairswillgetpayosofTanddefectorsintype3siblingpairswillgetpayosofP.17 Theprobabilitythatanyindividualsurvivestoreproduceisassumedtobeproportionaltotheaveragepayothatitreceivesinthegamesitplayswithitssiblings.Thismeansthatthetotalnumberofsurvivingcooperatorsinthesecondgenerationwillbeproportionalto2s1(x)R+s2(x)Sandthetotalnumberofospringofcompetitorsinthesecondgenerationwillbeproportionaltos2(x)T+2s3(x)P.Wherexistheproportionofthematingpopula-tioningenerationn,theratioofthenumberofsurvivingcooperatorsingenerationn+1tothenumberofcooperatorsingenerationnmustbec(x)=(2s1(x)R+s2(x))S=xandthecorrespondingratiofordefectorsmustbeproportionaltod=(s2(x)T+2s3(x)P)=(1�x)forsomecommonfactorofproportionality�0.Examiningtheaboveexpressionsfors1(x),s2(x),ands3(x),weseethatc(x)andd(x)arebothlinearexpressionsinx.Infact,thedierencebetweenthetwogrowthratesisjustD(x),whereD(x)isasdenedinthetextofthepaper.18 ReferencesBecker,Gary(1976)\Altruism,Egoism,andFitness:EconomicsandSociobiology,"Jour-nalofEconomicLiterature,14,.Becker,Gary(1981)ATreatiseontheFamily.Cambridge,Ma.:HarvardUniversityPress.Bergstrom,CarlandBergstrom,Ted(1992)\TheEvolutionofAltruismamongDiploidSiblingswhoPlayGamesLikePrisoners'Dilemma,"UniversityofMichiganWorkingPaper.Bergstrom,Ted(1989)\LoveandSpaghetti,TheOpportunityCostofVirtue,"JournalofEconomicPerspectives,,.Bergstrom,Ted(1992)\OntheEvolutionofAltruisticEthicalRulesforSiblings,"Uni-versityofMichiganworkingpaper.Bergstrom,Ted(1988)\SystemsofBenevolentUtilityInterdependence,"UniversityofMichiganworkingpaper.Bernheim,B.DouglasandStark,Oded(1988)\AltruismwithintheFamilyReconsidered:DoNiceGuysFinishLast?,"AmericanEconomicReview,,1034-1045.Boyd,RobertandRicherson,P.(1985)CultureandtheEvolutionaryProcess.Chicago:UnivesityofChicagoPress.Cavalli-Sforza,LuigiandFeldman,Marcus(1981)CulturalTransmissionandEvolution.Princeton,N.J.:PrincetonUniversityPress.Cox,DonaldandStark,Oded(1992)\IntergenerationalTransfersandtheGenerationEect,"HarvardUniversityworkingpaper.Dawkins,Richard(1976)TheSelshGene.NewYork:OxfordUniversityPress.Frank,Robert(1988)PassionswithinReason.NewYork:Norton.19 Hamilton,WilliamD.(1964a)\TheGeneticalEvolutionofSocialBehavior.I,"JournalofTheoreticalBiology,7,1-16.Hamilton,WilliamD.(1964b)\TheGeneticalEvolutionofSocialBehavior.II.,"JournalofTheoreticalBiology,7,17-52.Hirshleifer,Jack(1977)\EconomicsfromaBiologicalViewpoint,"JournalofLawandEconomics,,1-52.Hirshleifer,Jack(1978)\NaturalEconomyVersusPoliticalEconomy,"JournalofSocialandBiologicalStructures,1,319-337.MaynardSmith,John(1982)EvolutionandtheTheoryofGames.NewYork:CambridgeUniversityPress.Nowak,MartinandMay,Robert(1992)\EvolutionaryGamesandSpatialChaos,"Nature,359,826-829.Robson,Arthur(1992a)\Status,theDistributionofWealth,PrivateandSocialAttitudesTowardRisk,"Econometrica,60,837-858.Robson,Arthur(1992b)\TheBiologicalBasisofExpectedUtility,KnightianUncertainty,andtheEllsbergParadox,"WorkingPaper,EconomicsDepartment,UniversityofWesternOntario,LondonOntario.Stark,Oded(1989)\AltruismandtheQualityofLife,"AmericanEconomicReview,79,86-90.(inpress)Stark,Oded(1992)\NonmarketTransfersandAltruism,"EuropeanEconomicReview,,.Trivers,Robert(1985)SocialEvolution.MenloPark,Ca.:Benjamin/Cummings.Wright,Sewall(1921)\SystemsofMating,"Genetics,6,111-178.20