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SolutionstoChapter13Exercises

SOLVEDEXERCISES

S1.（a）Thepayofftableforthetwotypesoftravelersis:

High

Low

High

100,100

30,70

Low

70,30

50,50

（b）Thegraphis:

（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>108x,orx<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>108x,orx<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

A

20,20

11,35

T

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.

Column

A

T

N

Row

A

20,20

11,35

2,50

T

35,11

6,6

6,6

N

50,2

6,6

6,6

（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

Theater

（prop.x）

1,1

0,0

Movie

（prop.1–x）

0,0

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:

High

Mixer

Low

High

100,100

58,82

30,70

Mixer

82,58

58,58

42,58

Low

70,30

58,42

50,50

（c）ConsiderthecasewhenasingleHightypeattemptstoinvadeapopulationentirelycomposedofMixertypes.Inthissituationweneedonlytheupperleftcornerofthepayofftableinpart（b）:

High

Mixer

High

100,100

58,82

Mixer

82,58

58,58

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）:

Mixer

Low

Mixer

58,58

42,58

Low

58,42

50,50

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）>0wheneverm<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