How Altruism Can Prevail Under Natural Selection by Ted Bergstrom and Oded Stark University of Michigan and Harvard University Current version March How Altruism Can Prevail Under Natural Selectio - PDF document

How Altruism Can Prevail Under Natural Selection by Ted Bergstrom and Oded Stark University of Michigan and Harvard University Current version March    How Altruism Can Prevail Under Natural Selectio
How Altruism Can Prevail Under Natural Selection by Ted Bergstrom and Oded Stark University of Michigan and Harvard University Current version March    How Altruism Can Prevail Under Natural Selectio

How Altruism Can Prevail Under Natural Selection by Ted Bergstrom and Oded Stark University of Michigan and Harvard University Current version March How Altruism Can Prevail Under Natural Selectio - Description


We might also expect that evolutionary pressures tend to produce sel64257sh behaviorwith the notable exception of the relation between parents and o64256spring But can we expect natural selection to act in favor of altruistic behavior in other relat ID: 35322 Download Pdf

Tags

might also expect

Embed / Share - How Altruism Can Prevail Under Natural Selection by Ted Bergstrom and Oded Stark University of Michigan and Harvard University Current version March How Altruism Can Prevail Under Natural Selectio


Presentation on theme: "How Altruism Can Prevail Under Natural Selection by Ted Bergstrom and Oded Stark University of Michigan and Harvard University Current version March How Altruism Can Prevail Under Natural Selectio"— Presentation transcript


HowAltruismCanPrevailUnderNaturalSelectionbyTedBergstromandOdedStarkUniversityofMichiganandHarvardUniversityCurrentversion:March22,2002 HowAltruismCanPrevailUnderNaturalSelectionTedBergstromandOdedStarkIntroductionWhyhaveweeconomistsbeenconvincedforsolongthatouroldfriend,homoeco-nomicusmustbesel sh?Nodoubtwe ndconsiderablesupportforthishypothesisinthebehaviorofourcolleagues.Wemightalsoexpectthatevolutionarypressurestendtoproducesel shbehavior{withthenotableexceptionoftherelationbetweenparentsando spring.Butcanweexpectnaturalselectiontoactinfavorofaltruisticbehaviorinotherrelationships?Evolutionarybiologistshavecreatedatheorythatpredictsaltruisticbehavior,notonlybetweenparentsandchildren,butalsoamongsiblingsandothercloserelatives.1RichardDawkins'expressionofthisviewinTheSel shGene,isthatthereplicatingagentinevolutionisthegeneratherthantheanimal.Ifagenecarriedbyoneanimalislikelytoappearinitsrelatives,thenageneforhelpingone'srelatives,atleastwhenitischeaptodoso,willprosperrelativetogenesfortotallysel shbehavior.Thispaperpresentsaseriesofexamplesinwhichnaturalselectionsustainscooperativebehaviorinsingle-shotprisoners'dilemmagames.Inprisoners'dilemma,cooperational-waysgetsalowerpayo foroneselfandahigherpayo forone'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.Thepayo sfromthisgamearelistedinthematrixbelow.IftheparameterssatisfytherestrictionSPRT,thendefectwillbeadominantstrategyforeachgame.Forthegametobecalleda\prisoners'dilemma",itshouldalsosatisfytherestrictionthatS+T2R.Prisoners'DilemmaPlayer1Player2CooperateDefect Cooperate R;R S;T Defect T;S P;P Totalpayo toanindividualwillbetheaverageofitspayo sintheprisoners'dilemmagamesthatitplays.Wherebehaviorisgeneticallyinherited,weassumethattheexpectednumberofsurvivingo springthatanindividualproduceswillbehigher,thehigheritstotalpayo .Wherebehavioriscopiedfromneighbors,theprobabilitythatanindividual'sbehavioriscopiedwilldependonitspayo .Apopulationcanhaveeitheramonomorphicequilibriumorapolymorphicequilibrium(orpossiblyboth).Inastablemonomorphicequilibrium,onlyonetypeofindividualispresentandifamutantindividualoftheothertypeshouldarise,itmustreproducelessrapidlythannormalindividuals.Inapolymorphicequilibrium,morethanonetype2 ofindividualispresentandeachtypethatispresentreceivesthesameexpectedpayo .Stabilityofpolymorphicequilibriumrequiresthatifonetypehappenstobecomemorecommonthantheequilibriumproportion,itwillhavealowerexpectedpayo thantheothertype.2.EvolutionofGeneticallyTransmittedBehaviorSincelittleisknownabouttheenvironmentswhichshapedourgeneticinheritance,theevo-lutionaryhypothesismaynotbeveryinformativeaboutmanyaspectsofourpreferences.Butthefundamentalprocessesofmating,child-rearingandrelationsbetweensiblingsap-peartohavechangedlittleoverthemillennia.Accordingly,wemaylearnagooddealaboutthe\economicsofthefamily"fromalookattheevolutionarytheoryofrelationsamongkin.3AltruisticSororitiesWithoutSexJusttohelpusunderstandthelogicofinheritance,webeginwithatoymodelthatseemsunrealisticforhumans{asexualreproduction.Letusassumethatanyindividualwill,ifshesurvivestoreproductiveage,haveexactlytwochildren,whosegenesarejustlikeherown(exceptinthecaseofraremutations).Asurvivingindividualwithageneforcooperatewillhavetwoo springwithgenesforcooperate.Asurvivingindividualwithagenefordefectwillhavetwoo springlikeherself.Tokeepthepopulationsizeconstant,wemustassumethatonlyhalfoftheindividualswhoarebornwillsurvivetoreproductiveageandthattheprobabilitythatanyindividualwillsurvivetoreproducewillbehigher,thegreaterherpayo inthegamethatsheplayswithhersister.Weclaimthattheonlyequilibriumisapopulationconsistingentirelyofcooperators. 3We ndourselvesingoodcompanyinthisheresy.Becker(1976,1981)andHirshleifer(1977,1978),exploregeneticexplanationsforaltruismamongclosefamilymembers.RobertFrank(1985)seeksanevolutionaryexplanationforhumanemotions,andArthurRobson(1992a,1992b)exploresanevolutionaryexplanationofhumanattitudestowardrisk.3 Toseethis,considerapopulationconsistingentirelyofcooperators.Whatwouldhappentoamutantdefectorthatappearedinthispopulation?Sincehersisterisacooperator,themutantgetsapayo ofT�R,andsosheismorelikelytosurvivethananyothermemberofthepopulation.Buthergoodfortunewillnotbesustainedbyherdescendants.Herdaughterswillbothinheritherdefectgeneandwillbothdefect.SistersineachgenerationofherdescendantswillalsodefectandgetPR,andhencegraduallydisappearfromthepopulation.Similarly,inapopulationofdefectors,amutantcooperatorwouldfaceadefectingsisterandwouldgetapayo ofS,whilethesurroundingdefectorswouldgetP&#x-277;S.Althoughhersurvivalprobabilitywillbelowerthanthepopulationaverage,herdaughtersandtheirdescendantswouldallbecooperators.Eachofthemwillreceiveapayo ofR&#x-277;Pandtheirnumberswouldgrowrelativetothoseofthedefectors.DiploidSiblingsareSometimesAltruistic(ButNotasOftenasTheirParentsWouldLike)Diploidparentswillnotbesurprisedtodiscoverthatinourownspecies,siblingsarenotalwaysascooperativeasasexualsiblingswouldbe.Wewillshowthatthereisarichmenuofpossibleequilibriawithdiploidsiblings.Dependingontheparametersoftheprisoners'dilemmagame,theremaybeauniquestablemonomorphicequilibriumwithcooperatorsonly,auniquestablemonomorphicequilibriumwithdefectorsonly,ortheremaybetwolocallystablemonomorphicequilibria{onewithcooperatorsonlyandtheotherwithdefectorsonly.Finally,thereareparametervaluesforwhichtherearenostablemonomorphicequilibria,butfortheseparameterstherewillbeastablepolymorphicequilibriumwithsomecooperatorsandsomedefectorsinthepopulation.Weconsideralargepopulationwhichreproducessexuallyandhasdiploidgeneticstruc-ture.Eachindividualplaysasingle-shotgameofprisoners'dilemmawitheachofitssib-lings.Tosimplifyexposition,wewillassumethateachindividualwhosurvivestomateandreproducehasexactlythreeo spring.Theprobabilitythatanindividualsurvivestoreproducewillbehigher,thehigherthetotalpayo thatitgetsinthegamesitplayswith4 itstwosiblings.4Individualsareabletodistinguishtheirsiblingsfromothermembersofthepopulationandmayusedi erentstrategiesingamesplayedwithsiblingsfromthestrategiesusedwithoutsiders.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.Theassumptionthatallindividualswhosurvivetomateandreproduceallowsustosidestepthecomplicationsthatwouldarisefromreconcilingtheassumptionofmonogamousrandommatingwiththeassumptionthatthenumberofchildrenwashasdependsononespayo sinprisoners'dilemma.Whatshouldwedoifhusbandandwifehavedi erentexpectednumberofo spring?5Alternatively,onecouldassumethatheterozygotesplayamixedstrategywithsome xedprobabilitiesofcooperationanddefection.6Noticethatwedonottaketheviewthateithercgenesordgenesmustbeintrinsicallydominant.Insteadweaskwhetherinamonomorphicpopulation,ifadominantmutantoftheoppositetypeshouldarise,themutantstrainwouldincreaseasaproportionofthepopulationorwouldultimatelydisappear.5 analysiswhichwillnotbepursuedhere.7Firstletusaskwhenapopulationconsistingonlyofcooperatorswouldberesistanttoinvasionbymutantswitha(dominant)genefordefection.Supposethattheentirepop-ulationconsistsoftypecchomozygotes,allofwhomcooperate.Nowletsomeindividualexperienceamutationwhichchangesoneofitscgenestoadgenewhichisdominantoverthetypecgene.Themutantwillthereforebeacdheterozygoteandwillplaydefect.Inthegamesitplayswithitssiblings,thismutantwillgetahigherpayo thannormalmembersofthepopulation,sinceitsnormalsiblingscooperatewhileitdefects.ThereforethemutantreceivesTineachgamewhileordinarymembersofthepopulationreceiveRT.Butinorderto ndoutwhetherthemutanttypewillinvadethepopulationinthelongrun,wemustfollowthefortunesofitso springwhoinheritthemutantgene.Whenthemutantcdtypeisrare,itwillalmostcertainlymatewithanormaltypecc.Themutant'so springwillthereforebeoftypecdwithprobability1/2andoftypeccwithprobability1/2.Ano springofthemutantwhocarriesthemutantgenewillbeoftypecdandwillplaydefect.Withprobability1/2,arandomlychosensiblingofthisindividualwillbeatypecchomozygoteandwithprobability1/2,thatsiblingwillbeanothertypecd.Thereforewithprobability1/2,thisindividualcanexploitacooperativesiblingandreceiveapayo ofT,butwithprobability1/2,thesiblingwillalsodefect.Itthenfollowsthattheexpectedpayo toeachheterozygoteo springofthemutantis(T+P)=2.Theo springofnormalcctypeswillreceiveapayo ofRinthegamestheyplaywiththeirsiblings.Itfollowsthatwhilethemutantgeneisrare,carriersofthemutantgenewillreproducemorerapidlythannormalindividualsifT+P&#x-354;2RandlessrapidlyifT+P2R.Nowletusaskwhenapopulationconsistingentirelyoftypeddindividualscouldbeinvadedbyamutantcgenewherethemutantgeneisdominantoverthenormalgenes.Asinglemutatinggenewould rstappearinacdheterozygote.Assumingthecgeneis 7Aheterozygotewitharecessivemutantgenewillactjustlikethenormalpopulationandsotherewillbenoselectioneitherfororagainstrecessivegenesuntil\geneticdrift"producesenoughmutantheterozygotessothattheyoccasionallymate,therebyproducinghomozygoteswhoactdi erentlyfromtheremainingpopulation.EquilibriawhichareresistanttoinvasionbothbydominantandbyrecessivemutantsarestudiedbyBergstrom(1992).6 dominant,themutantindividualwouldcooperate.Themutantindividualwouldreceivethelowpayo S,sinceitplaysitssiblingswhoplaydefect.Butonaverage,itso springwilldobetterthanSandperhapswilldobetterthanthenormalpopulationofdefectors,allofwhomreceiveP.Whenthemutanttypeisrare,amutantwillalmostcertainlymatewithanormalddtype.Halfofthemutant'so springwillbecdheterozygotes,whocooperate,andhalfofthemwillbeddhomozygotes,whodefect.Ano springofthemutantwhocarriesthemutantgenewillbeoftypecdandwillplaycooperate.Withprobability1/2,arandomlychosensiblingofthisindividualwillbeatypeddhomozygotewhodefectsandwithprobability1/2,thatsiblingwillbeanothertypecdwhocooperates.Thereforewithprobability1/2,ano springthatcarriesofthemutantgenewillbeexploitedbyitssiblingandgetapayo ofS,butwithprobability1/2,itssiblingwillalsocooperateandeachofthemwillreceiveapayo ofR.Theexpectedpayo toatypecdo springofthemutantistherefore(S+R)=2.Thispayo willbesmallerthanthepayo tonormalddtypesif2P�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,sincethepopulationdynamicsdiscussedinthispaperareinvarianttoanetransformationsofthepayo matrix.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,althoughparentswhowishthemtomaximizetheirjointpayo wouldpreferthembothtocooperate.ItisinterestingtoconsiderRegionsAandB.ForparametervaluesinRegionB,therearetwostableequilibria{onewithamonomorphicequilibriumofeachkind.ForparametervaluesinRegionA,therearenostablemonomorphicequilibrium.Inordertounderstandthesecases,itisnecessarytoworkoutthedetailedlawsofmotionforthedynamicalsystemthatresultsfromthismodel.ThisisdonebyBergstromandBergstrom(1992),whereitisfoundthatforparametervaluesinRegionA,thereexistsexactlyonestablepolymorphicequilibriumandforparametervaluesinRegionB,thereisoneunstablepolymorphicequilibriumandnostablepolymorphicequilibria.3.WhenChildrenImitatetheirParentsorTeachersHerewestudyamodelinwhichbehaviorisacquiredbyimitation,ratherthangeneti-cally.ThemodeldiscussedhereisavariantofmodelsofculturaltransmissionwhichweredevelopedbyCavalli-SforzaandFeldman(1980),andBoydandRichardson(1985).Weassumethateachindividualhastwosiblingsandplaysagameofprisoners'dilemmawitheachofthem.Wewillalsoassumethattheprobabilitythatanyindividualsurvivestomateandreproduceisproportionaltotheaveragepayo thatitreceivesinthegamesitplayswithitssiblings.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.Thisenablesustodeterminenotonlytheproportionofo springofeachtype,butalsotheexpectedpayo stoo springofeachtype,sincewewillknowtheprobabilitythatarandomlychosensiblingofanindividualofeachtypewillbeacooperatororadefector.Withthisinformation,wearebeabletodeterminetherelativegrowthratesofthepopulationofcooperatorsandofdefectors.ThedetailsofthisprocessareworkedoutintheAppendixofthispaper.Thismodelturnsouttohavearemarkablyconvenientmathematicalstructure.Therateofchangeofthenumberofsurvivingindividualsofeachtypeinanygenerationturns 9Ifmatingrequiresmutualconsent,iftypesarecostlesslyrecognizableandsearchcostsarenegigible,thiswouldbeanaturaloutcome,sincecooperatorscanexpectmoreo springiftheymatewithothercooperatorsthaniftheymatewithdefectorssolongas2R�S+P.10Cavelli-SforzaandFeldmanattributethisparameterizationofassortativematingtoSewallWright(1921).Theseproportionswouldbeachievedifcoupleswere rstrandomlymatchedandthenthefractionmofthemixedcoupleswerebrokenupandthefreedindividualspairedwithpersonsoftheirowntype.9 outtobealinearfunctionofthefractionofthepopulationintheparentgenerationwhoarecooperators.Thereforethedi erencebetweenthegrowthratesofthenumberofindividualsofthetwotypesisalsoalinearfunctionofthefractionofthepopulationwhoarecooperators.11Inparticular,thedi erencebetweenthegrowthratesofthepopulationofcooperatorsandthepopulationofdefectorsisexpressedbyD(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,theequationforthedi erenceinexpectedgrowthratesistypicallyquadraticorcubic.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)v2willincreasebothAandBforgivenpayo parameters,S,P,R,andT.Thismeansthatthelargeris(1+m)v2,thelargeristhesetofpayo parametersforwhichthereisamonomorphicequilibriumwithallcooperatorsandthesmallerthesetofpayo parametersforwhichthereisanequilibriumwithalldefectors.Thatistosay,themorelikelychildrenaretoimitatetheirparents,andthemorelikelytheirparentsaretobethesameaseachother,themorelikelycooperativebehavioristoprevail.4.WhendoesProvincialismPromoteCooperation?Intheevolutionaryexamplesthatwelookedat,thereisagoodchancethatonewillplayagamewithanopponentwhosebehaviorisinheritedfromthesameparentasonesownbehavior.Wecanexpectsimilare ectsinspatialmodels,whereneighborsinteractingamesofprisoners'dilemmaandwherebehaviorcomesfromimitationofrelativelysuccessfulneighbors.Thise ecthasbeendocumentedinaseriesofcomputersimulationscarriedoutbyNowakandMay(1992),carriedoutonatwo-dimensionalgrid.NowakandMayshownotonlythatcooperativebehaviorcanbesustained,butalsothatagreat11 varietyofcyclesandwavescanoccur.Herewewillshowthatcooperationcanbesustainedininterestingwayseveninaone-dimensionalmodelthatissimpleenoughtobestudiedwithapadandpaper.Imaginearoadwhichrunsaroundalake.Alongthisroadliveseveralfarmers,eachofwhomhasoneneighboronhisleftandoneonhisright.Eachfarmerplaysagameofprisoners'dilemmawithhisneighborsandhistotalincomeisthesumofhispayo sfromthesegames.Thefarmers'sonsgrowup,observingtheactionsoftheirfathersandtheirneighbors.Whenthefathersdie,theirsonstakeoverthefarmsanddecidewhethertobecooperatorsordefectors.Thesonschoosetheirstrategiesafterobservingtheactionsbyandthepayo sreceivedbytheirfathersandtheirneighbors.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.13Anisolatedcooperatorwillgetahigherpayo thaneitherofhis 12Consideraclusterof3ormorecooperatorswhichabutsaclusterof2ormoredefectors.Thesonsoffarmersintheinteriorofcooperatorclustercanseeonlytheirfathersandtwocooperativeneighbors,sotheywillcooperate.Thesonofacooperatorontheboundaryofaclusterseesacooperatorneighborwhoreceives2R,hisfatherwhoreceivesS+R,andadefectorneighborwhoreceivesapayo ofT+P.Byassumptionthelargestofthesepayo sisT+P,sohecooperatesashisfatherdid.Thesonofadefectorintheinteriorofadefectorclusterseesonlydefectandwilldefect.ThesonofadefectorontheboundaryofaclusterseesacooperatorneighborwhoreceivesS+R,hisdefectingfatherwhoreceivesT+P,andadefectorneighborwhoreceives2P.Sinceforaprisoners'dilemmagameSPRT,hisfatherwillhaveahigherpayo thanthecooperatorneighbor,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.WheretheutilityfunctionUisdi erentiable,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.ConclusionWehaveseenseveralenvironmentsinwhichanindividualwillcertainlyreceiveahigherpayo fromdefectingthanfromcoooperatingandwhere\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.Theprobabilitythatarandomlychosenpairofo springfromatypeiparent-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):Cooperatorsintype1siblingpairswillgetpayo sofRandcooperatorsintype2siblingpairswillgetpayo sofS.Defectorsintype2siblingpairswillgetpayo sofTanddefectorsintype3siblingpairswillgetpayo sofP.17 Theprobabilitythatanyindividualsurvivestoreproduceisassumedtobeproportionaltotheaveragepayo thatitreceivesinthegamesitplayswithitssiblings.Thismeansthatthetotalnumberofsurvivingcooperatorsinthesecondgenerationwillbeproportionalto2s1(x)R+s2(x)Sandthetotalnumberofo springofcompetitorsinthesecondgenerationwillbeproportionaltos2(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,thedi erencebetweenthetwogrowthratesisjustD(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)\IntergenerationalTransfersandtheGenerationE ect,"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

Shom More....
trish-goza
By: trish-goza
Views: 200
Type: Public

Download Section

Please download the presentation from below link :


Download Pdf - The PPT/PDF document "How Altruism Can Prevail Under Natural S..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Try DocSlides online tool for compressing your PDF Files Try Now