The Blessing of Sexuality: Evolution of Altruism with Mating Preference

14 Pages·2016·1.4 MB·English

The Blessing of Sexuality:
Evolution of Altruism with Mating Preference
TanzheTang1,2andHangYe2
1SchoolofMedicine,ZhejiangUniversity,38ZhedaRoad,Hangzhou,310027,China
2CollegeofEconomics,ZhejiangUniversity,866YuhangtangRoad,Hangzhou,310027,China
[email protected]
JournalofArtificialSocietiesandSocialSimulation19(2)2,2016
Doi:10.18564/jasss.3058 Url:http://jasss.soc.surrey.ac.uk/19/2/2.html
Received:17-10-2015 Accepted:24-02-2016 Published:31-03-2016
Abstract: Currentsimulationpracticesinartificialsocietiestypicallyignorethecontributionofsexualityasa
drivingforcefortheevolutionofprosocialbehaviours. Asrecentresearchesinbiologyandgeneticsargued,
sexualattractiveness,viathemethodofsexualselection,canexplainmanyaspectsofthesecond-orderso-
cialdilemma. Thebasichypothesisisthataltruismisasexuallyattractivevirtue. Tointroducethehypothesis
intotheanalysisofhumanaltruism,weemploytheconceptsofaltruisticpunishmentandthebehaviour-based
sexualattractivenesstodevelopagender-basedevolutionarymodelwherematingpreferenceactsasthecom-
pensationtothemalepunishersfromfemalesinthegivenpublicgoodsgame.Inthemodel,theforceofsexual
selectionisexpressedastheeffectofmatingpreferenceonaltruism. Thecomputersimulationindicatesthat
socialcohesioncanbeachievedbytheexistenceofsexualityinanartificialsocietywheretheco-evolutionof
matingpreference,altruisticpunishmentandcooperationexist.Wethenextendthemodelintwoways:(1)we
employthevariablesizepopulationassumptiontotesttheinvasioncapacityofcooperators,and(2)individual
variationinaltruisticinvestmentisintroducedtoreplacetheaveragepopulationpayofffunctioninthebase-
linemodel. Thevariablesizepopulationandindividualvariationininvestmentarefoundtohaveamplifying
effectsontheevolutionofaltruismfromdifferentperspectives.Finally,wediscussthedefinitionofaltruismin
dynamicevolutionarygames,aswellasthegenderdifferencesintheformationofaltruisminprimitivetribes.
Keywords:AltruisticPunishment,MatingPreference,SexualAttractiveness,SocialDilemma
Introduction
1.1 Sexualityhasbeenviewedasthenemesisofmoralityinnearlyeverytraditionalculture,yetinpreviouslitera-
turesinbiologyandgenetics,sexualityisconsideredasoneofthedrivingforcesfortheevolutionofprosocial
behaviours(Miller2007).Selectioninrelationtosexprovidesauniqueperspectivetosolvethepersistentpuz-
zleoftheexistenceofsocialcooperationandaltruisticbehaviour(thestrategythatbenefitsothersandcosts
theplayer)thatisreferredtoasthesocialdilemma(Dawes&Messick2000).Inotherfieldsofvirtues,biological
evolutionarymodelsshowedthatinter-sexualselectioncanexplaintheevolutionofmale-onlycareevenifsex-
ualselectionmaydisfavourparentalcarewhencarecomesatanetlosstofertilizationsuccess(Alonzo2012),
andprovidedtemplatesforthevirtueofaltruism.
1.2 Tosolvethepuzzleofcooperation,researchersproposedthataltruismcanbenefitthefitnessofthecooper-
ator’sgenes(Preston2013). Basedonthisproposal,biologistshaveinvestigatedinwaysofinclusivefitness
(Hamilton1964;Dugatkin2007)andvariousselectionsatdifferentlevels(Alexander1987;Boyd&Richerson
1990;Sober&Wilson1999;McAndrew2002).Particularly,pioneeringworkshaveattemptedtoexplainthemain-
tenanceandevolutionofkinaltruism(Hamilton1964)andreciprocalaltruism(Trivers1971;Ridley1997;Axelrod
&Hamilton1981;Sugiyamaetal.2002)viavariousformsofgametheory,socialselectionmodelsandempirical
studies. Inthefieldofpurealtruism,theintroductionofacostlyaltruisticpunishmentfoundineconomicex-
periments(Fehr&Gächter2002;Gintis2000)pavesthewayfortheemergenceofaltruisminvolvingnogenetic
ordirectrewards.
1.3 Acostlypunishmentfromacooperatorisnotanadvantageousstrategyinthegameoflife,givenothercoop-
erators’non-punishmentstrategy(Yeetal.2011). Thisviolationofcausality, whichisalsoreferredtoasthe
JASSS,19(2)2,2016 http://jasss.soc.surrey.ac.uk/19/2/2.html Doi:10.18564/jasss.3058
second-ordersocialdilemma(Hauertetal.2007),waspartiallysolvedbyHauertetal.(2007)viaadynamic
evolutionarymodelwhereplayerscouldoptoutofthecollectiveaction.Thepossibilityofplayingthenonpar-
ticipantstrategyrequirescrucialassumptionsconcerninghumansociety.Otherwise,themodellosesitspower
toprovideacogentmechanismfortheestablishmentofpunishment(Boyd&Mathew2007). Toexpandthe
breadthofthe‘narrowwaytocooperation’proposedbyHauertetal.(2007),variousadjustedmodelandsim-
ulationshavebeenconductedwithouttheassumptionofnonparticipation.Jaffe&Zaballa(2010)proposeda
newformofpunishmentcalledcooperativepunishmentthatwascontingentupontheconcurrentparticipation
ofotherplayers.Withtheexistenceofcooperativepunishment,socialcohesionmightbeachievedincomputer
simulations.Yeetal.(2011)introducedthemoralconceptofsympathyascompensationforthepunisher.InYe’s
modelofpunishmentandsympathy,individualsinthesocietyareclassifiedintothreetypes:cooperator,de-
fectorandpunisher.Theysharethebenefitsfrompublicgoodsequallywhileonlydefectorsdonotcontribute.
Punishersimposeanaltruisticpunishmentondefectorsatacosttothemselves,andcooperatorswhodonot
punishthedefectorsdirectlysympathizewiththepunisherandbearacostduetothesympathy.Theexistence
ofsympathyinthemodeloperatesasastabilizerforthesocialcohesion,wherethepunisherswillpersistently
dominatethepopulation.
1.4 Alltheadjustedmodelsbasedonaltruisticpunishmentarecontributiveandsalient. However, takenalone
theydonotelucidateanunderstandingofhowsexualityshapestheevolutionaryequilibriumofaltruismin
humansociety. Besides,theoreticalbiologistsandeconomistshavebroadlystudiedthebenefit-costanalysis
ofaltruism(Axelrod&Hamilton1981;Axelrod1984)withoutexploringthehumanmotivationforselflesshelp
(Preston2013),whilesexualitypossiblyplaystheroleofmotivationtoinvestinaltruismaccordingtosexual
selectiontheoryandcostlysignallingtheory(Gintisetal.2001;McAndrew2002). Theignoranceofsexuality
inmodelsinvolvingalternationsofgenerationsisusuallyassociatedwiththeimplicitassumptionofasexual
reproduction,whichisatraditionfromtheMoranprocess(Moran1958). Ashasbeenmentionedbeforeand
notedbymany,sexualselectionisakeyfactorintheevolutionofhumanaltruism.Withouttheanalysisofthe
effectofsexualselectioninaltruism,themodelsarenotcompleteforthesimulationsofhumansocieties.
1.5 Inthispaper, weproposeadynamicmodelofpunishmentandsexualselection, wherethesocialcohesion
canbeachievedbythebehaviour-basedsexualattractivenessfrommalesthatwillincreasethefitnessofmale
punishers.SexualattractivenessviasexualselectionisananalogoustothesympathyinYe’smodel(2011).The
fundamentaldifferencebetweenYe’ssympathyandsexualattractivenessinthispaperisthedivisionbetween
asexualandsexualreproduction.Theconceptofsexualattractivenessinthemodelopensupthenovelsources
tostudythegameofpublicgoods,altruismandotherprosocialbehavioursfromtheperspectiveofsexuality.
BaselineModel
Modelsetup
2.1 WemodelthedynamicevolutionofanisolatedpopulationofaconstantscaleN(wewillmodifythisconstant
populationassumptioninnextsections).Thepopulationiscomposedoffourtypesofpeopleincluding:
• MP:MalePunisher
• MD:MaleDefector
• FC:FemaleCooperator
• FD:FemaleDefector
2.2 Thestrategiesandbehavioursofeachtypearedescribedasfollows:
MP isaltruistic; thatis, hewillinvestI intothealtruisticbehaviour. Theinvestmentincursacostc, which
isassumedtobeproportionaltotheinvestment(i.e.,c = αI). However,MPalsopunishesdefectors
regardlessoftheirgenders.Thestrengthofthepunishmentisδ≥c,whichwillincuracostofγ <δtoMP.
ThepopulationofMPisassumedtobeN .
MP
MD isegoistic,contributesnothingtothealtruisticbehaviourandhencebearsnoaltruisticcost.However,asa
memberofthesociety,hebenefitsfromMPandFC’scontributionjust.MDsufferspunishmentsfromMP.
ThepopulationofMDisN .
MD
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FC isaltruistic. ShealsoinvestsIintothealtruisticbehaviour(weassumethatIisidenticalineveryMPand
FC,andthisassumptionwillbegeneralisedtoindividualvariationinthelatersection). FCalsoadmires
MPwiththestrengthofs(i.e.,thestrengthofsexualattractivenessorMP’ssexualcapital),whilethecost
incurredbythesissetto0.ThepopulationofFCisN
FC
FD isthesameasMDexceptfortheirgenders. Forsimplicity,weassumethatFDwillnotadmireanyMP.The
populationofFDisN .
FD
2.3 EveryindividualequallysharesthebenefitfromthealtruisticcontributionfromMPandFC.
2.4 Itshouldbenotedthatthenamesof‘defector’or‘cooperator’donotnecessarilyimplythattheindividualsare
involvedinatypicalpublicgoodsgamedefinedinthefieldofeconomics.Itwouldbemoreappropriatetore-
nametheindividualsinthesocietyas‘altruist’and‘egoist’.However,topaytributetothepreviouscontribution
bypioneers,weutilizetheirsystemofnamesinthispaper.
2.5 BythedefinitionofbirthratebyKoella(2000),theexpectedpayoffsforeachindividualineachofthefourtypes
inasingleperiodaredefinedas
(cid:16) (cid:17)
P =β 1−e−kI(cid:101) −αI −γ(N +N )+sN (1)
MP MD FD FC
(cid:16) (cid:17)
P =β 1−e−kI(cid:101) −δN (2)
MD MP
(cid:16) (cid:17)
P =β 1−e−kI(cid:101) −αI (3)
FC
(cid:16) (cid:17)
P =β 1−e−kI(cid:101) −δN (4)
FD MP
whereP ,P ,P ∧P arethepayoffsofMP,MD,FCandFD,respectively. Thebenefitfromtheentire
MP MD FC FD
societyiscalculatedbythemeanaltruisticinvestment
(cid:80)
I (N )I
I(cid:101)= i = MP+NFC (5)
N N
whereNisthetotalpopulationofthesociety,N +N +N +N =N.β∧κareconstantparameters.
MP MD FC FD
2.6 Weadopttheexponentialformforthefunctionoffitness(i.e.,f = eωP),wheref denotesthefitnessofan
individual,P denotesthepayoffandωdenotestheselectionstrength(0<ω <1).Theformoffitnesscomes
fromTraulsenetal.(2008).
2.7 WeadoptageneralisedMoranProcessadjustedforsexualityandsexualreproductiontomodeltheevolution
ofthesocietygiventheassumptionsabove. Theprobabilityofdeathdisidenticaltoeachindividualinthe
societyandremainsconstantduringeachperiod.Tokeepthepopulationunchanged,thenumberofnewborn
offspringsissettobeequaltothenumberofthedeadintheperiod.Theprocessofdeterminingthetypeofthe
offspringiscompletedthroughtwosteps: (1)naturerandomlydecideswhethertheoffspringismale/female
withaprobabilityof50%respectively;(2)iftheoffspringisamale,thenhisprobabilityofbeinganMP(MD)is
proportionaltothefitnessofanMP(MD),andiftheoffspringisafemale,thenherprobabilityofbeinganFC
(FD)isproportionaltothefitnessofanFC(FD).Thisprocesspermitsthebalanceofbothgenders.
2.8 Additionally,weassumethatanyindividualofthepopulationcouldrandomlyswitchtoanothertypewithin
thesamegenderwithasmallprobabilityµ,whichisreferredtoasthemutationrate.
Simulationresults
2.9 Theoutcomeofthesimulationisnotable.Figure1showsthatwhenpunishmentandsexualattractivenessexist
(Figure1C),beingaltruisticisadvantageous,whichmeansthatMPandFCarethedominanttypesinthesoci-
ety,whereasMDandFDareatthelimitofnear-extinctionaftersomeinitialperiods.However,inFigure1A(no
punishmentandnoattractiveness)and1B(punishmentwithoutattractiveness),thestatuswheretheegoists
becomedominantoverthealtruists(especiallyinthemalesociety)isasuperlativeexampleofthesecond-order
socialdilemmainasocietywithnosexualattractiveness.Thedifferenceinresultsbetweenthesimulationsin
Figure1A,BandCindicatestheeffectofsexualityasthedrivingforcetoovercomethesocialdilemmaofcostly
punishment. Theapparentabsenceofsexualattractivenessandsexualselectionintheevolutionarydynam-
icsraisestheextinctionofpunishersinFigure1B.Meanwhile,thesexualattractivenessofamalepunishertoa
JASSS,19(2)2,2016 http://jasss.soc.surrey.ac.uk/19/2/2.html Doi:10.18564/jasss.3058
Figure1:Figure1SimulationResultofthePunishment-SexualityGame:(A)Nopunishmentandnosexualattrac-
tiveness. Theparametersforthissimulationsare: N=400,N =N =N =N =100,κ=1,β=1,α=0.5,γ=
MP MD FC FD
0,ω=0.5,δ=0,I=1,µ=0.001,d=0.1,s=0,period=5,000.(B)Punishmentwithoutsexualattractiveness.The
parametersforthissimulationareN=400,N =N =N =N =100,κ=1,β=1,α=0.5,γ=0.4,ω=0.5,δ=
MP MD FC FD
0.6,I=1,µ=0.001,d=0.1,s=0,period=5,000.(C)Punishmentandsexualattractiveness.N=400,N =N =
MP MD
N =N =100,κ=1,β=1,α=0.5,γ=0.4,ω=0.5,δ=0.6,I=1,µ=0.001,d=0.1,s=1,period=5,000.
FC FD
femalecooperatorshapestheevolutionaryequilibriumwherethevirtueofaltruismcanescapetheDarwinian
Demonofecologicalselection.
Robustnesscheck
2.10 Weperformrobustnesschecksonvariousparametersandinitialsettingstotestthepowerofourmodel. We
selectseveralimportantparameterstoreporttheirrobusttestresults. Theconclusionwithstandstherobust
testontheinitialproportionofthefourtypesinthesociety. Figure2illustratestheresultsoftherobusttest
withnoMPandFCatperiod0withthreedifferenttreatmentsasinFigure1.
2.11 Itisnecessarytoclarifythattheexacttimingoftheturningpointwherealtruiststakepowerinthesocietyfrom
theegoistsisinconsistentinrepeatedcomputersimulationsduetotherandomnessofthemodeldynamics.
Figure3indicatesthevarioustimingsoftheturningpointsfromthreerepeatedcomputersimulationsthatre-
portdifferentevolutionarypathsandroutines.
2.12 Wenowconducttherobusttestonthestrengthofsexualattractiveness,s.
2.13 Insimulationswheres>0.2,MPandFCtakedominanceoverMDandFD;inthesimulationwheres=0.1,MD
takesdominanceoverMPwhileFCandFDcoexist.Theparametersinthesesimulationsarethesameasthose
inFigure1C,withtheexceptionofthevalueofs. Itshouldbenotedthattheresultsfors = 1ands = 0have
beenreportedinprevioussimulations.
2.14 Figure5teststhevalueofmutationrateµ,whichis0.001inthebaselinemodel.
2.15 Figure5showsthatmutationplaystheroleof‘revolution’,tryingtoredistributethepopulation.Asmallermu-
tationratecanwidenthegapbetweencooperatorsanddefectors.
2.16 Figure6showstherobusttestforperiods.Theresultswithholdlongperiods.
2.17 Additionally,robusttestsonotherparametersareconductedtoensurethestabilityofthesimulationresultsin
thissection.
Discussionofthebaselinemodel
2.18 InYe’ssimulationmodel, humansympathycanbeexpressedbycare, praise, solace, orevenadorationand
admiration(2011).However,themoralsenseofsympathyproposedanddiscussedbymoralphilosophersand
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Figure2:SimulationResultofthePunishment-SexualityGamewith100%Defectorsinitially.(A)Nopunishment
andnosexualattractiveness. Theparametersforthissimulationsare: N=400,N =N =0,N =N =200,κ
MP FC MD FD
=1,β =1,α=0.5,γ =0,ω =0.5,δ =0,I =1,µ=0.001,d=0.1,s=0,period=5,000. (B)Punishmentwithout
attractiveness.TheparametersforthissimulationareN=400,N =N =0,N =N =200,κ=1,β=1,α=0.5,
MP FC MD FD
γ=0.4,ω=0.5,δ=0.6,I=1,µ=0.001,d=0.1,s=0,period=5,000.(C)Punishmentandsexualattractiveness.N
=400,N =N =0,N =N =200,κ=1,β=1,α=0.5,γ=0.4,ω=0.5,δ=0.6,I=1,µ=0.001,d=0.1,s=1,period
MP FC MD FD
=5,000.
Figure3:SimulationResultsofthePunishment-SexualityGamewith100%Defectors:(A)Thetimingdoesnot
appearbefore50periods.(B)Thetimingappearsbetween50and100periods.(C)Thetimingappearsapproxi-
matelyat50periods.TheparametersinallthreesimulationsareidenticaltothesimulationinFigure2Cexcept
forthenumbersofperiods.
moralpsychologistscannotcompensatethepunisherdirectlybyincreasingthepunisher’sfitness. Matepref-
erence(orequivalentlysexualattractivenessfromtheoppositeperspective)servesasaparticularformofsym-
pathythatcanincreasethefitnessoftheindividualtowhomthepreferenceisgiven.
2.19 Thesimulationresultsdescribingtheevolutionarypathofeachtypeofindividualprovidecriticalevidencefor
sexualattractivenessasastrongstabilizertogenerateasocialequilibriumwhereprosocialstrategy(i.e.,be-
ingaltruistic)isadvantageousinanartificialhumansociety. Theintroductionofsexualitytothetraditional
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Figure4: SimulationResultsofthePunishment-SexualityGamewithDifferentStrengthsofSexualAttractive-
ness.TheparametersinthesesimulationsarethesameasthoseinFigure1C,withtheexceptionofthevalueof
s.
Figure5:SimulationResultsofthePunishment-SexualityGamewithDifferentmutationrate. Theparameters
inthesesimulationsarethesameasthoseinFigure1C,withtheexceptionofthevalueofµ.Themutationrate
µhasbeenlabeledoneachfigure.
altruisticpunishmentgameisbeyondthescopeofthepreviousliteraturesonsocialevolutionviamethodsof
computersimulation.Wefoundthattheintroductionofsexualityandsexualattractivenesschangedthesocial
equilibriumfromthesoledominanceofpunishers(Yeetal.2011)tothecoexistenceofboth(male)punishers
and(female)cooperators,whichfitsrealitybetter.
ExtensiontoVariableSizePopulation
3.1 AstrictconstraintinMoranprocessistheconstantpopulationsize,whichgreatlysimplifiesthecalculations
fortheoreticalmodellingandservesasacommondefaultinthefieldofevolutionarysimulations. However,
theassumptionoffixedpopulationwillbefulfilledonlyinmodelswithhardresourcelimitations(Claussen&
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Figure6:SimulationResultsofthePunishment-SexualityGamewithDifferentPeriods.Theparametersinthese
simulationsarethesameasthoseinFigure1C,withtheexceptionofthevalueofperiod.
Traulsen2005),e.g. latticemodelswithviscouspopulations(Koella2000). Byassumingconstantpopulation,
theinterplaybetweenreproductivestrategiesandpopulationdynamicsisneglectedandonlytheeffectofaltru-
ismoneachindividual’sfitnessistakenintoaccount.Evenatcapacity,thebehaviourofaltruismcanincrease
thepopulationsizeduetocooperators’contributiontothepublicgoods. Thedependencyofpopulationsize
oncooperatorshappenswhenindividualsinanisolatedsocietycompeteforlimitedresources: egoismleads
totheso-calledcommontragedyandaltruismleadstoabundantpublicgoodsduetothescaleofeconomics
(Houchmandzadeh2015). Simulationshaveledtotheoutcomethatanincreasingpopulationhasadiminish-
ingeffecttoproduceahighermeanaltruisticlevelinstructuredsocietieswithaspatialdistributionofagents
(Shutters&Hales2015). InHouchmandzadeh’sasexualmodel(2015),wherepopulationsizeisdesignedtobe
afunctionoftheproportionofcooperators,clarifiedthatadeleteriousmutant(i.e.,altruism)maypossiblyin-
vadethecommunitythanthewildtype(i.e.,defectors). Basedonthisapproach,itisreasonabletosuppose
thatthevariablesizepopulation(asafunctionofaltruism)isanamplificationfactorforthesocialcohesion
madebyaltruisticpunishmentandsexualselectionaswehavedemonstratedintheprecedingsections.
3.2 Inthissection,weapplythevariablesizepopulationconditioninsteadoftheoriginalconstantpopulationcon-
strainttotestthecontributionofbothsexualselectionandpopulationdynamics.
Variablesizepopulationassumption
3.3 Consideranartificialsocietywithfourtypesofindividualsasdescribedbefore.PopulationsizeNisnowanin-
creasingfunctionoftheproportionofcooperatorsinthepopulation(denotedasx),followingHouchmandzadeh
(2015)assumption.Ineachgeneration,thepopulationreachesitscapacityfortheparticularpopulationstruc-
ture(i.e.thedistributionofeachtype),thatis,nomoreindividualscansurviveundertheresourcelimitations.
Tosimplifythemodel,weadoptthelinearformforN(x)(Houchmandzadeh2015):
N(x)=(1−x)N +xN (6)
D C
(N +N )
x= MP FC (7)
N
whereN andN (N < N )arethecarryingcapacitiesofthesocietywhencomposedofcooperatorsand
C D C D
defectors,respectively.xistheproportionofcooperatorsasdefinedin(6).Thedynamicsprocessfollowsthe
formulaintheBaselineModelsectionexceptforthevariablesizepopulationassumption.
Simulationresults
3.4 Noremarkabledifferencesinxfromthebaselinemodelcanbedetectedwhentheinitialpopulationdistribution
wassettobeevenlydistributedbetweenfourtypes. Toextracttheinvasioncapacityofcooperatorsunder
JASSS,19(2)2,2016 http://jasss.soc.surrey.ac.uk/19/2/2.html Doi:10.18564/jasss.3058
Figure7: SimulationResultsforVariableSizePopulationAssumptions. (A1-C1)Simulationresultsofconstant
populationmodels. (A2-C2)SimulationresultsofvariablesizepopulationmodelswhereN = 1,000and
C
N = 200. (A3-C3)SimulationresultsofvariablesizepopulationmodelswhenN = 2,000andN = 200.
D C D
TreatmentsandnumericalsimulationresultsareshowninTable1.
Figure Punishment1 Attractiveness Pop.Dynamics2 xatperiod5,000 Finalpopulation
A1 (0,0) s=0 (200,200) 0.4416 200
B1 (0.4,0.6) s=0 (200,200) 0.2601 200
C1 (0.4,0.6) s=1 (200,200) 0.9865 200
A2 (0,0) s=0 (1000,200) 0.4392 551
B2 (0.4,0.6) s=0 (1000,200) 0.3208 442
C2 (0.4,0.6) s=1 (2000,200) 0.9904 992
A3 (0,0) s=0 (2000,200) 0.4396 991
A3 (0.4,0.3) s=0 (2000,200) 0.3501 830
C3 (0.4,0.6) s=1 (2000,200) 0.9909 1983
Table1:TreatmentsandnumericalsimulationresultsofFigure73.
variablesizepopulationassumption,weplotacomparisonofsimulationresultsfordifferenttreatments(Figure
7).
3.5 TheresultsofA1,A2andA3indicatethatthedifferencesinxarenotsignificant,andthedifferencesinxinC1,
C2andC3arenotsignificantaswell.NowwecomparetheplotsforFCandFDinB1,B2andB3toseetheroleof
variablesizepopulationtreatmentinthechangeofinvasioncapacity.ThevalueofxincreasesfromB1toB3.In
B1,theplotsofFCoverlapwithFD,thatis,FCandFDsharenearlythesamepopulationfrequency(about20%).
Whenthevariablesizepopulationtreatmentisemployed(B2andB3),theplotsofFCandFDseparateandFC
gainsthemajorityoffemales.TheadvantageofFCisenhancedinB3comparedtoB2duetoalargerdifference
betweenN andN .Thus,inthesocietywithaltruisticpunishmentbutwithoutsexualattractiveness,variable
C D
sizepopulationtreatmentoffersamorecooperativefemalesociety(i.e.,FCdominatesFD).Inotherwords,FC
invariablesizepopulationmodelhasahigherinvasioncapacitythaninconstantpopulationmodelwiththis
particularsetting. TheresultcoincidespartiallywithHouchmandzadeh’stheoreticaloutcomesinanasexual
society.
JASSS,19(2)2,2016 http://jasss.soc.surrey.ac.uk/19/2/2.html Doi:10.18564/jasss.3058
Figure PopulationN ale/N emale Period Punishment(γ,δ) Attractiveness
m f
A 200/200 5,000 (0,0) 0
B 200/200 5,000 (0.4,0.6) 0
C 200/200 5,000 (0.4,0.6) 1
D 200/200 5,000 (0.4,0.6) 5
Table2:ModelsettingsinFigure8.
3.6 Overall,thesesimulationresultsimplythatthetreatmentofvariablesizepopulationhasanamplifyingeffect
ontheinvasionofFCinthecaseofaltruisticpunishmentmodel. Varyingpopulationsizeservesasapartial
substituteforattractivenessinthealtruisticevolutionoffemales. Themechanismofvariablesizepopulation
inaltruisticevolutionisnotclear,butitisclearthatvaryingpopulationsizebenefitsthecooperativesocietyas
awhole:adefector-dominatedsocietyisdesignedtohavelesspopulationsizeinthenextperiodasapunish-
ment.Theincreaseinpopulationinducedbycooperatorscanovercomethealtruisticcost(Houchmandzadeh
2015)andpromotetherationalconsumptionoflimitedresources.
ExtensiontotheIndividualVariation
4.1 Variationsinanindividual’ssocialenvironmentandstatevariables,suchascolour,sizeandage,caninfluence
theevolutionofaltruism,buttheinteractionbetweenindividualvariationandcooperationhaverarelybeen
studiedinanevolutionaryway(Barta2016).Therefore,animportantproblemintheevolutionarymodelsbased
ontheaveragepopulationpayofffunctions,asthesimulationswehavedoneintheprevioussections,ishowto
modeltheindividualvariationinthesociety.Thestudyofpendulinetitshasbeendonebasedontheindividual
variationtoresolvetheconflictoverparentalcare(VanDijketal.2011).Inthebaselinesimulationmodels,we
classifythesocietyintofourtypes(i.e.,MP,MD,FC,FD)forthesakeofsimplicity,butitisreasonabletoquestion
theaccuracyofthemodelsinceawell-fittedsimulationrequiresmorethanfourtypesofindividuals.Webelieve
thatintherealworld,ifaltruisticinvestmentcanbescaled,thevalueofIfollowsacertaindistributionandthus
therearenostrictstereotypesaspunishers,cooperatorsordefectors.
4.2 Intheextendedmodelsinthissection,thesocietyiscomposedofmalesandfemalesinvestingIintoaltruism.
Itakesvaluefrom0to1, witharandommutationchangingtheIinstepsof±0.1(thesettingsareadopted
fromKoella’s(2000))withaconstantandidenticalmutationrateµ. Initially,altruisticinvestmentsinmales
andfemalesbothfollowanidenticalnormaldistributionwithameanof0.5andstandarddeviationof0.1.
4.3 Thestrategiesandbehavioursofeachindividualareassumedtosatisfy:
1. FormaleinvestingI∗intoaltruism,hewillpunishindividualswhoseI <I∗regardlessofhis/hergender.
Inturn,hewillsufferfrommaleswhoseI >I∗.
2. ForfemalesinvestingI∗∗intoaltruism,shewillsufferfrommaleswhoseI > I∗∗.Shewillfavourmales
whoseI >I∗∗.
4.4 Tosummary,amalewillpunishindividualswithalowerlevelofaltruismthanhim,andafemalewillfavour
maleswithmorealtruisticthanher(i.e.,amalefavouredbyafemalewillobtainaunitofsexualattractiveness
s).Notethatonlymalespunish,onlyfemalesshowfavourtomalesandonlymalesreceivefemale’sfavourand
thusincreasetheirsexualattractiveness.
4.5 Thedynamicprocessisthesameasinprevioussectionsexceptthesecondstepofbirth.Inthissection,ifthe
offspringisamale,hisprobabilityofhavingI =I∗is
(cid:88) (cid:88)
prob(I∗|male)= f(I∗|male)/ f(I|male) (8)
wheref(I∗|male)representsthefitnessofamalewithI =I∗.
4.6 Forthesimulationresults,wereportthelinegraphsoftheaveragevalueofIofmalesandfemalesrespectively.
Figure8examinestheeffectofpunishmentandattractivenessonthemeanvalueofI.ModelsettingsinFigure
8arelistedinTable2.
4.7 TheplotsinFigure8Bshowthataltruisticpunishmentfavourstheevolutionofhighaltruisticlevelfemales(
I > 0.7anddisfavourstheevolutionofhighaltruisticlevelmales(I < 0.4. Nearlynofluctuationsareob-
servedintheplotofmalesinB.InFigure8Cand8D,weemploybothpunishmentandattractiveness(s=1and5
respectively).PlotsinbothCandDreachthelevelwhereI =1regardlessofgenders.
JASSS,19(2)2,2016 http://jasss.soc.surrey.ac.uk/19/2/2.html Doi:10.18564/jasss.3058
Figure8:MeanValueofIin5,000Periods.
4.8 Toclarify, notethatthemeanvalueofIisameasureofthesocialaltruisticlevel. Acooperativesocietywill
expressahighmeanvalueofI.Figure8A-8Donceagainshowtheroleofpunishmentandattractivenessinthe
altruisticevolution.
4.9 Finally,wereportthefinalsection(atperiod5,000)oftheinvestmentdistributioninFigure9.
4.10 Giventhenormaldistributioninitially,individual’sinvestmentsaredecentralisedfromthemidpointI = 0.5,
themeanvalueofthenormaldistribution)inA.InB,mostmalesinvestabout0.4,andmostfemalesinvest
between0.4and1,withameanvaluelargerthan0.7. InCandD,theplotsofmalescompletelyoverlapwith
females,andmorethan200males(females)invest1,makingthemsaintsoftheirgeneration.
4.11 Inall,theresultsinthissectionarequitesimilartothebaselinemodel. Thetreatmentofindividualvariation
appearstobeasignificantamplifierfortheevolutionofaltruism. Moreover,asuperlativecohesionofsociety
willbeestablishedwhereeveryoneinvests1whenbothpunishmentandsexualattractivenessareemployed.
Surprisingly,thesocietiesinCandDareevenmorecooperativethanthecohesionachievedinthebaseline
model.
4.12 Theabovesimulationresultssuggestthattheevolutionofaltruismwithindividualvariationassumptionisa
self-reinforcingprocess. Thealtruisminthesocietymaybereinforcedbecausethepunishmentmechanism
runssynchronouslywiththeevolutionofaltruism. Punishmentandsexualattractivenesspermittheadvan-
tageofcooperators,andthesocietystructuremay,inturn,enhancethepunishmentandattractiveness. For
example,insimulationinFigure8B,amalewithI =0.8suffersnearlynopunishmentandreceivesthehighest
honourofattractivenessatearlyperiod(i.e.,periodlessthan10),butatlaterperiod(say1,000ormore)hewill
sufferpunishmentfromnearlyeverymalesandreceivenearlynoadmirationifhisinvestmentremains0.8be-
causethemajorityinvest1now.Thecaseforfemalesissomehowsimilar.Ourconclusionisquiteoptimisticin
thesensethatoursocietyislikelytocoordinateonhigh-levelaltruismequilibrium.
Discussion
5.1 Inthebaselinemodelandtheextendedmodels,theroleofsexualattractiveness,variablesizepopulationand
individualvariationintheevolutionofaltruismarepresentinsimulationresults. Inthispaper,weprovidea
JASSS,19(2)2,2016 http://jasss.soc.surrey.ac.uk/19/2/2.html Doi:10.18564/jasss.3058