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(c)Therearethreepossibleequilibria:
astablemonomorphicequilibriumofallLowtypes(h=0),astablemonomorphicequilibriumofallHightypes(h=1),andanunstablepolymorphicequilibriumwheretwo-fifthsofthepopulationareHightypes(h=0.4).
S2.Throughouttheanswersforthisexercise,weletxrepresentthepopulationproportionoftheinvadingtype.
(a)InapopulationprimarilyconsistingofAtypeswithonlyasmallproportion(x)ofinvadingTtypes,theA-typefitnessisF(A)=864(1–x)+936x=864+72xandtheT-typefitnessisF(T)=792(1–x)+972x=792+180x.F(A)>
F(T)aslongas864+72x>
792+180x,or72>
108x,orx<
72/108=2/3.Anall-Apopulationcan’tbeinvadedbyTtypesunlesstheTtypesaremorethantwo-thirdsofthepopulation,soasmallnumberofmutantTscannotsuccessfullyinvade.Similarlyforasmallproportion(x)ofinvadingNs:
F(A)=864(1–x)+1,080x=864+216xandF(N)=648(1–x)+972x=648+324x.F(A)>
F(N)aslongas864+216x>
648+324x,or216>
216/108=2.Thisisalwaystruebecausexmustbebetween0and1.Therefore,theAtypesarealwaysfitterthantheNtypes,soanall-Apopulationcan’tbeinvadedbyNs.
(b)InaprimarilyNpopulation,mutantTshavefitnessF(T)=972(1–x)+972x=972andNshavefitnessF(N)=972(1–x)+972x=972.Thefitnessesareequal,soTsandNsdoequallywellinthepopulationandNscannotpreventTsfrominvading.ApopulationofNsinvadedbyTsthusexhibitsneutralstability,whereboththeprimaryandsecondarycriteriaforanESSgiveties.Sinceneithertypeismorefitthantheother,theirproportionsinthepopulationwillpersist,onlyslightlyadjustingasmutationsoccur.
AgainstagroupofmutantAs,theNtypeshavefitnessF(N)=972(1–x)+648x=972–324xandtheAtypeshavefitnessF(A)=1,080(1–x)+864x=1,080–216x.F(N)>
F(A)when972–324x>
1,080-216x,or108+108x
<
0.Thisconditionneverholds,soAscaninvadeanall-Npopulation.Anall-NpopulationisunstablewhenanAmutationispossible.
(c)InaprimarilyTpopulation,mutantAshavefitnessF(A)=936(1–x)+864x=936–72x,andtheT-typefitnessisF(T)=972(1–x)+792x=972–180x.F(T)>
F(A)when972–180x>
936–72x,or36>
36/108=1/3.Anall-Tpopulationcan’tbeinvadedbyAtypesunlesstheAtypesaremorethanone-thirdofthepopulation,soasmallnumberofmutantAscannotsuccessfullyinvade.WhenthemutantsaretypeN,F(T)=972(1–x)+972x=972,andthemutantNsalsohavefitnessF(N)=972(1–x)+972x=972.Again,thefitnessesareequal,soTsandNsdoequallywellinthepopulation,andTscannotpreventNsfrominvading.
Column
A
T
Row
20,20
11,35
35,11
6,6
S3.(a)Thepayofftableisatright.
(b)LetxbethepopulationproportionofTplayers.ThentheAs’expectedyearsinjailare20(1–x)+11x=20–9x,andtheTs’expectedyearsinjailare35(1–x)+6x=35–29x.ThentheTtypeisfitterthantheAtypeiftheformerspendsfeweryearsinjail:
rememberthatpayoffsareyearsinjailhere,sosmallernumbersarebetter.TsarefitterthanAswhen35–29x<
20–9x,or15<
20x,orx>
0.75.Ifmorethan75%ofthepopulationisalreadyT,thenTplayersarefitter,andtheirproportionswillgrow;
wehaveonestableESSwhenthepopulationisallT.Similarly,ifthepopulationstartswithfewerthan75%Ttypes,thentheAtypesarefitter,andtheirproportionwillgrow;
wehaveanotherstableESSwhenthepopulationisallS.Finally,thereisanotherequilibriuminwhichexactly75%ofthepopulationistypeTand25%ofthepopulationistypeA;
thatequilibriumispolymorphicbutunstable.
N
2,50
50,2
(c)Seethepayofftableatright.ThestrategyNdoesnotdowellinthisgame.Inamixedpopulationthatincludesallthreetypes,letxandybethepopulationproportionsofTandN,respectively.ThentheAs’expectedyearsinjailequal20(1–x–y)+11x+2y,theTs’expectedyearsinjailare35(1–x–y)+6x+6y,andtheNs’expectedyearsinjailare50(1–x–y)+6x+6y.TisstrictlyfitterthanNinthispopulation(rememberthatsmallernumbersarebetter)andwilleventuallydominateN.Usingthesameequations,weseethatifthepopulationconsistsinitiallyofonlyNandA,amutantTcaninvadesuccessfullyandtheneventuallydominateN.EvenifthepopulationisinitiallyallN,amutantAwillbeabletoinvade.Inthatcase,forasmallproportionaofAtypes,theNsget50a+6(1–a)yearsinjail,whereastheAsget20a+2(1–a);
theAsarefitterforallvaluesofa,sotheycansuccessfullyinvade.ThisanalysisimpliesthatNcannotbeanESSinthisgame.TheESSherearethesameasthoseinpart(b).
S4.Thepayofftableisshownatright.
Male
Theater
(prop.y)
Movie
(prop.1–y)
Female
(prop.x)
1,1
0,0
(prop.1–x)
2,2
FitnessoffemaleTindividual=1y+0(1–
y)=y.
FitnessoffemaleMindividual=0y+2(1–y)=2(1–
y).
Soforfemales,thetheatertypeisfitterthanthemovietype(andthepopulationproportionxoftheatertypesincreases)ify>
2(1–
y)orify>
2/3.Similarly,formales,thetheatertypeisfitterthanthemovietype(andthepopulationproportionyofTheatertypesincreases)ifx>
2(1–x),orifx>
2/3.
Thediagrambelowshowsthedynamicsofthegame.TherearetwoESS:
(0,0)and(1,
1).
S5.(a)Themostobviousmixturetoplayisthemixed-strategyequilibriumofthetwo-playerversionofthegame:
report$100(i.e.,actlikeaHightype)withprobability0.4.
(b)TocalculateexpectedpayoffsforpairsthatincludeaMixertype,rememberthattheMixerreports$100withprobability0.4and$50withprobability$40.SowhenaMixerplaysagainstaHightype,thereisa40%chancethatbothreport$100anda60%chancethatthereisonereportof$00andonereportof$50.Theexpectedpayoffofthepairing(Mixer,High)isthus0.4*(100,100)+0.6*(70,30)=(82,58).Similarly,theexpectedpayoffofthepairing(Mixer,Low)is0.4*(30,70)+0.6*(50,50)=(42,58).
TheexpectedpayoffwhentwoMixertypesmeetisalittlemorecomplicated,sincetherearefourcasestoconsider.Withprobability0.4*0.4,bothMixersreport$100.Withprobability0.4*0.6,thefirstreports$100andthesecond$50.Withthesameprobability,thefirstreports$50andthesecond$100.Finally,withprobability0.6*0.6,theybothreport$50.Theexpectedpayoffforthepairing(Mixer,Mixer)isthus0.4*0.4*(100,100)+0.4*0.6*(30,70)+0.6*0.4*(70,30)+0.6*0.6*(50,50)=(58,58).
Thethree-by-threetableofexpectedpayoffswiththeMixertypeisthen:
Mixer
58,82
82,58
58,58
42,58
58,42
(c)ConsiderthecasewhenasingleHightypeattemptstoinvadeapopulationentirelycomposedofMixertypes.Inthissituationweneedonlytheupperleftcornerofthepayofftableinpart(b):
Intermsofm—theproportionofMixertypesinthepopulation—thefitnessoftheHightypeis100(1–m)+58m,whereasthefitnessoftheMixertypeis82(1–m)+58m.Whenm=1,itistruethatthefitnessesareequal:
100(0)+58
(1)=82(0)+58
(1).However,wheneverm<
1,theHightypehasahigherfitness,because100(1–m)+58m>
82(1–m)+58m18(1–m)>
0wheneverm<
1.AsingleHightypewillbemorefitthantheMixertypesintherestofthepopulation,sotheHightypewillsuccessfullyinvade.
NowconsiderthecasewhenasingleLowtypeattemptstoinvadeapopulationentirelycomposedofMixertypes.Nowweneedonlythelowerrightcornerofthepayofftableinpart(b):
Intermsofm—theproportionofMixertypesinthepopulation—thefitnessoftheLowtypeis58m+50(1–m),whereasthefitnessoftheMixertypeis58m+42(1–m).Whenm=1,thefitnessesofthetwotypesareequal:
58
(1)+50(0)=58
(1)+42(0).However,wheneverm<
1,theLowtypehasahigherfitness,because58m+50(1–m)>
58m+42(1–m)8(1–m)>
1.AsingleLowtypewillbemorefitthantheMixertypesintherestofthepopulation,sotheLowtypewillsuccessfullyinvade.
SinceeitheraHightypeoraLowtypecouldsuccessfullyinvadeapopulationofMixertypes,theMixerphenotypeisnotanESSofthisgame.
S6.(a)Thefitnessofthesolartype,FS,is2s+3(1–s)=3–s,andthefitnessofthefossilfueltype,FFF,is4s+2(1–s)=2+2s.Graphingthefitnesscurvesofthesolartypeandthefossilfueltypewithrespecttos,wehave:
Therearethreepossibleequilibria:
anunstablemonomorphicequilibriumwhereeveryoneusesfossilfuels(s=0),anunstablemonomorphicequilibriumwhereeveryoneusessolar(s=1),andastablepolymorphicequilibriumwhereone-thirdofthepopulationusessolar(s=1/3).Themonomorphicequilibriaexistaspossibilitiesbecauseintheabsenceofanyalternativetypestheloneexistingtypewillsimplyreproduceinperpetuity.
(b)Withthechangeinthe(solar,solar)payoff,FS=ys+3(1–s)=3+(y–3)s.ThereisnochangeinFFF.Atapolymorphicequilibrium,thefitnessofthesolartypeisequaltothefitnessofthefossilfueltype,so3+(y–3)s
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