微觀經(jīng)濟(jì)學(xué)Microeconomics英文課件 29 Game Applications

Lec24GameApplicationsChapter29NashEquilibriumInanyNashequilibrium(NE)eachplayerchoosesa“best〞r(shí)esponsetothechoicesmadebyalloftheotherplayers.AgamemayhavemorethanoneNE.Howcanwelocateeveryoneofagame’sNashequilibria?IfthereismorethanoneNE,canwearguethatoneismorelikelytooccurthananother?BestResponsesThinkofa2×2game;i.e.,agamewithtwoplayers,AandB,eachwithtwoactions.AcanchoosebetweenactionsaA1andaA2.BcanchoosebetweenactionsaB1andaB2.Thereare4possibleactionpairs;

(aA1,aB1),(aA1,aB2),(aA2,aB1),(aA2,aB2).Eachactionpairwillusuallycausedifferentpayoffsfortheplayers.BestResponsesSupposethatA’sandB’spayoffswhenthechosenactionsareaA1andaB1are

UA(aA1,aB1)=6andUB(aA1,aB1)=4.Similarly,supposethat

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.IfBchoosesactionaB1thenA’sbestresponseis??BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.IfBchoosesactionaB1thenA’sbestresponseisactionaA1(because6>4).BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.IfBchoosesactionaB1thenA’sbestresponseisactionaA1(because6>4).IfBchoosesactionaB2thenA’sbestresponseis??BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.IfBchoosesactionaB1thenA’sbestresponseisactionaA1(because6>4).IfBchoosesactionaB2thenA’sbestresponseisactionaA2(because5>3).BestResponsesIfBchoosesaB1thenAchoosesaA1.IfBchoosesaB2thenAchoosesaA2.A’sbest-response“curve〞isthereforeA’sbest

responseaA1aA2aB2aB1B’saction++BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.IfAchoosesactionaA1thenB’sbestresponseis??BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.IfAchoosesactionaA1thenB’sbestresponseisactionaB2(because5>4).BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.IfAchoosesactionaA1thenB’sbestresponseisactionaB2(because5>4).IfAchoosesactionaA2thenB’sbestresponseis??.BestResponsesUA(aA1,aB1)=6andUB(aA1,aB1)=4

UA(aA1,aB2)=3andUB(aA1,aB2)=5

UA(aA2,aB1)=4andUB(aA2,aB1)=3

UA(aA2,aB2)=5andUB(aA2,aB2)=7.IfAchoosesactionaA1thenB’sbestresponseisactionaB2(because5>4).IfAchoosesactionaA2thenB’sbestresponseisactionaB2(because7>3).BestResponsesIfAchoosesaA1thenBchoosesaB2.IfAchoosesaA2thenBchoosesaB2.B’sbest-response“curve〞isthereforeA’sactionaA1aA2aB2aB1B’sbestresponseBestResponsesIfAchoosesaA1thenBchoosesaB2.IfAchoosesaA2thenBchoosesaB2.B’sbest-response“curve〞isthereforeA’sactionaA1aA2aB2aB1B’sbestresponseNoticethataB2isa

strictlydominant

actionforB.BestResponses&NashEquilibriaA’sresponseaA1aA2aB2aB1aA1aA2aB2aB1++A’schoiceB’schoiceB’sresponseHowcantheplayers’best-responsecurvesbe

usedtolocatethegame’sNashequilibria?BABestResponses&NashEquilibriaA’sresponseaA1aA2aB2aB1aA1aA2aB2aB1++A’schoiceB’schoiceB’sresponseHowcantheplayers’best-responsecurvesbe

usedtolocatethegame’sNashequilibria?Put

onecurveontopof

theother.BAHowcantheplayers’best-responsecurvesbe

usedtolocatethegame’sNashequilibria?Put

onecurveontopof

theother.BestResponses&NashEquilibriaA’sresponseaA1aA2aB2aB1aA1aA2aB2aB1++A’schoiceB’schoiceB’sresponseBAHowcantheplayers’best-responsecurvesbe

usedtolocatethegame’sNashequilibria?Put

onecurveontopof

theother.BestResponses&NashEquilibriaA’sresponseaA1aA2aB2aB1++B’sresponseIsthereaNashequilibrium?Howcantheplayers’best-responsecurvesbe

usedtolocatethegame’sNashequilibria?Put

onecurveontopof

theother.BestResponses&NashEquilibriaA’sresponseaA1aA2aB2aB1++IsthereaNashequilibrium?

Yes,(aA2,aB2).Why?B’sresponseHowcantheplayers’best-responsecurvesbe

usedtolocatethegame’sNashequilibria?Put

onecurveontopof

theother.BestResponses&NashEquilibriaA’sresponseaA1aA2aB2aB1++IsthereaNashequilibrium?

Yes,(aA2,aB2).Why?aA2isabestresponsetoaB2.

aB2isabestresponsetoaA2.B’sresponseBestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAaA2istheonlybestresponsetoaB2.

aB2istheonlybestresponsetoaA2.Hereisthestrategic

formofthegame.BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAHereisthestrategic

formofthegame.aA2istheonlybestresponsetoaB2.

aB2istheonlybestresponsetoaA2.Istherea2ndNash

eqm.?BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAIstherea2ndNash

eqm.?No,because

aB2isastrictly

dominantaction

forPlayerB.aA2istheonlybestresponsetoaB2.

aB2istheonlybestresponsetoaA2.Hereisthestrategic

formofthegame.BestResponses&NashEquilibriaNowallowbothplayerstorandomize(i.e.,mix)

overtheiractions.6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerABestResponses&NashEquilibriaNowallowbothplayerstorandomize(i.e.,mix)

overtheiractions.6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.A

choosesactionaA1.B1istheprob.B

choosesactionaB1.BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.A

choosesactionaA1.B1istheprob.B

choosesactionaB1.GivenB1,what

valueofA1isbest

forA?BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAEVA(aA1)=6B1+3(1-B1)=3+3B1.A1istheprob.A

choosesactionaA1.B1istheprob.B

choosesactionaB1.GivenB1,what

valueofA1isbest

forA?BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.A

choosesactionaA1.B1istheprob.B

choosesactionaB1.GivenB1,what

valueofA1isbest

forA?EVA(aA1)=6B1+3(1-B1)=3+3B1.

EVA(aA2)=4B1+5(1-B1)=5-B1.BestResponses&NashEquilibriaA1istheprob.AchoosesactionaA1.B1istheprob.BchoosesactionaB1.GivenB1,whatvalueofA1isbestforA?EVA(aA1)=3+3B1.

EVA(aA2)=5-B1.

3+3B15-B1asB1??>

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<BestResponses&NashEquilibriaA1istheprob.AchoosesactionaA1.B1istheprob.BchoosesactionaB1.GivenB1,whatvalueofA1isbestforA?EVA(aA1)=3+3B1.

EVA(aA2)=5-B1.

3+3B15-B1asB1?.>

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<BestResponses&NashEquilibriaA1istheprob.AchoosesactionaA1.B1istheprob.BchoosesactionaB1.GivenB1,whatvalueofA1isbestforA?EVA(aA1)=3+3B1.

EVA(aA2)=5-B1.

3+3B15-B1asB1?.

A’sbestresponseis: aA1ifB1>? aA2ifB1<? aA1oraA2ifB1=?>

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<BestResponses&NashEquilibriaA1istheprob.AchoosesactionaA1.B1istheprob.BchoosesactionaB1.GivenB1,whatvalueofA1isbestforA?EVA(aA1)=3+3B1.

EVA(aA2)=5-B1.

3+3B15-B1asB1?.

A’sbestresponseis: aA1(i.e.A1=1)ifB1>? aA2(i.e.A1=0)ifB1<? aA1oraA2(i.e.0A1

1)ifB1=?>

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<BestResponses&NashEquilibria0A11B10A’sbestresponse?A’sbestresponseis:aA1(i.e.A1=1)ifB1>?aA2(i.e.A1=0)ifB1<?aA1oraA2(i.e.0A1

1)if

B1=?1BestResponses&NashEquilibria0A11B10A’sbestresponse?1A’sbestresponseis:aA1(i.e.A1=1)ifB1>?aA2(i.e.A1=0)ifB1<?aA1oraA2(i.e.0A1

1)if

B1=?BestResponses&NashEquilibria0A11B10A’sbestresponse?1A’sbestresponseis:aA1(i.e.A1=1)ifB1>?

aA2(i.e.A1=0)ifB1<?aA1oraA2(i.e.0A1

1)if

B1=?BestResponses&NashEquilibria0A11B10A’sbestresponse?1ThisisA’sbestresponse

curvewhenplayersare

allowedtomixovertheir

actions.A’sbestresponseis:aA1(i.e.A1=1)ifB1>?

aA2(i.e.A1=0)ifB1<?

aA1oraA2(i.e.0A11)if

B1=?BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.A

choosesactionaA1.B1istheprob.B

choosesactionaB1.GivenA1,what

valueofB1isbest

forB?BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.A

choosesactionaA1.B1istheprob.B

choosesactionaB1.GivenA1,what

valueofB1isbest

forB?EVB(aB1)=4A1+3(1-A1)=3+A1.BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.A

choosesactionaA1.B1istheprob.B

choosesactionaB1.GivenA1,what

valueofB1isbest

forB?EVB(aB1)=4A1+3(1-A1)=3+A1.

EVB(aB2)=5A1+7(1-A1)=7-2A1.BestResponses&NashEquilibriaA1istheprob.AchoosesactionaA1.B1istheprob.BchoosesactionaB1.GivenA1,whatvalueofB1isbestforB?EVB(aB1)=3+A1.

EVB(aB2)=7-2A1.

3+A17-2A1asA1??>

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<BestResponses&NashEquilibriaA1istheprob.AchoosesactionaA1.B1istheprob.BchoosesactionaB1.GivenA1,whatvalueofB1isbestforB?EVB(aB1)=3+A1.

EVB(aB2)=7-2A1.

3+A1<7-2A1forall0A1

1.BestResponses&NashEquilibriaA1istheprob.AchoosesactionaA1.B1istheprob.BchoosesactionaB1.GivenB1,whatvalueofA1isbestforA?EVB(aB1)=3+A1.

EVB(aB2)=7-2A1.

3+A1<7-2A1forall0A1

1.

B’sbestresponseis: aB2always(i.e.B1=0always).BestResponses&NashEquilibria0A11B10B’sbestresponse?B’sbestresponseis aB2always(i.e.B1=0always).1ThisisB’sbestresponse

curvewhenplayersare

allowedtomixovertheir

actions.BestResponses&NashEquilibria0A11B10B’sbestresponse?10A11B10A’sbestresponse?1BABestResponses&NashEquilibria0A11B10B’sbestresponse?10A11B10A’sbestresponse?1BAIsthereaNashequilibrium?BestResponses&NashEquilibria0A11B10B’sbestresponse?10A11B10A’sbestresponse?1BAIsthereaNashequilibrium?BestResponses&NashEquilibria0A11B10B’sbestresponse?1A’sbestresponseIsthereaNashequilibrium?BestResponses&NashEquilibria0A11B10B’sbestresponse?1A’sbestresponseIsthereaNashequilibrium?Yes.Justone.(A1,B1)=(0,0);i.e.AchoosesaA2only

&BchoosesaB2only.BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerALet’schangethegame.3,1BestResponses&NashEquilibria6,43,55,74,3aA1aA2aB1aB2PlayerBPlayerAHereisanew

2×2game.3,1BestResponses&NashEquilibria6,45,74,3aA1aA2aB1aB2PlayerBPlayerAHereisanew

2×2game.Again

letA1betheprob.

thatAchoosesaA1

andletB1bethe

prob.thatBchooses

aB1.WhataretheNE

ofthisgame?NoticethatPlayerBnolongerhasastrictlydominantaction.3,1BestResponses&NashEquilibria6,45,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.thatA

choosesaA1.

B1istheprob.thatB

choosesaB1.EVA(aA1)=??

EVA(aA2)=??3,1BestResponses&NashEquilibria6,45,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.thatA

choosesaA1.

B1istheprob.thatB

choosesaB1.EVA(aA1)=6B1+3(1-B1)=3+3B1.

EVA(aA2)=??

3,1BestResponses&NashEquilibria6,45,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.thatA

choosesaA1.

B1istheprob.thatB

choosesaB1.EVA(aA1)=6B1+3(1-B1)=3+3B1.

EVA(aA2)=4B1+5(1-B1)=5-B1.3,1BestResponses&NashEquilibria6,45,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.thatA

choosesaA1.

B1istheprob.thatB

choosesaB1.EVA(aA1)=6B1+3(1-B1)=3+3B1.

EVA(aA2)=4B1+5(1-B1)=5-B1.

3+3B15-B1asB1?.>

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=

<BestResponses&NashEquilibriaEVA(aA1)=6B1+3(1-B1)=3+3B1.

EVA(aA2)=4B1+5(1-B1)=5-B1.

3+3B15-B1asB1?.>

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<0A11B10A’sbestresponse?1BestResponses&NashEquilibriaEVA(aA1)=6B1+3(1-B1)=3+3B1.

EVA(aA2)=4B1+5(1-B1)=5-B1.

3+3B15-B1asB1?.>

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<0A11B10A’sbestresponse?13,1BestResponses&NashEquilibria6,45,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.thatA

choosesaA1.

B1istheprob.thatB

choosesaB1.EVB(aB1)=??

EVB(aB2)=??3,1BestResponses&NashEquilibria6,45,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.thatA

choosesaA1.

B1istheprob.thatB

choosesaB1.EVB(aB1)=4A1+3(1-A1)=3+A1.

EVB(aB2)=??

3,1BestResponses&NashEquilibria6,45,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.thatA

choosesaA1.

B1istheprob.thatB

choosesaB1.EVB(aB1)=4A1+3(1-A1)=4+A1.

EVB(aB2)=A1+7(1-A1)=7-6A1.3,1BestResponses&NashEquilibria6,45,74,3aA1aA2aB1aB2PlayerBPlayerAA1istheprob.thatA

choosesaA1.

B1istheprob.thatB

choosesaB1.EVB(aB1)=4A1+3(1-A1)=3+A1.

EVB(aB2)=A1+7(1-A1)=7-6A1.

3+A17-6A1asA1.>

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<47/BestResponses&NashEquilibriaEVB(aB1)=4A1+3(1-A1)=3+A1.

EVB(aB2)=A1+7(1-A1)=7-6A1.

3+A17-6A1asA1.>

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<47/0A11B10147/B’sbestresponseBestResponses&NashEquilibriaEVB(aB1)=4A1+3(1-A1)=3+A1.

EVB(aB2)=A1+7(1-A1)=7-6A1.

3+A17-6A1asA1.>

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=

<47/0A11B10147/B’sbestresponseBestResponses&NashEquilibria0A11B10B’sbestresponse10A11B10A’sbestresponse?1BA47/BestResponses&NashEquilibria0A11B10B’sbestresponse10A11B10A’sbestresponse?1BA47/IsthereaNashequilibrium?BestResponses&NashEquilibria0A11B10B’sbestresponse1BA47/0A11B10A’sbestresponse?1IsthereaNashequilibrium?BestResponses&NashEquilibria0A11B10B’sbestresponse147/A’sbestresponse?IsthereaNashequilibrium?BestResponses&NashEquilibria0A11B10B’sbestresponse147/A’sbestresponse?IsthereaNashequilibrium?Yes.3ofthem.BestResponses&NashEquilibria0A11B10B’sbestresponse147/A’sbestresponse?IsthereaNashequilibrium?Yes.3ofthem.(A1,B1)=(0,0)BestResponses&NashEquilibria0A11B10B’sbestresponse147/A’sbestresponse?IsthereaNashequilibrium?Yes.3ofthem.(A1,B1)=(0,0)

(A1,B1)=(1,1)IsthereaNashequilibrium?BestResponses&NashEquilibria0A11B10B’sbestresponse147/A’sbestresponse?IsthereaNashequilibrium?Yes.3ofthem.(A1,B1)=(0,0)

(A1,B1)=(1,1)(A1,B1)=(,)?47/IsthereaNashequilibrium?SomeImportantTypesofGamesGamesofcoordinationGamesofcompetitionGamesofcoexistenceGamesofcommitmentBargaininggamesCoordinationGamesSimultaneousplaygamesinwhichthepayoffstotheplayersarelargestwhentheycoordinatetheiractions.Famousexamplesare:TheBattleoftheSexesGameThePrisoner’sDilemmaGameAssuranceGamesChickenCoordinationGames;TheBattleoftheSexesSissypreferswatchingballettowatchingmudwrestling.Jockpreferswatchingmudwrestlingtowatchingballet.Bothpreferwatchingsomethingtogethertobeingapart.CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.Whataretheplayers’

best-response

functions?CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.Whataretheplayers’

best-response

functions?EVS(B)=8JB+(1-JB)=1+7JB.CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.Whataretheplayers’

best-response

functions?EVS(B)=8JB+(1-JB)=1+7JB.EVS(MW)=2JB+4(1-JB)=4-2JB.EVS(B)=8JB+(1-JB)=1+7JB.EVS(MW)=2JB+4(1-JB)=4-2JB.1+7JB4-2JBasJB.13/CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.Whataretheplayers’

best-response

functions?>

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<EVS(B)=8JB+(1-JB)=1+7JB.EVS(MW)=2JB+4(1-JB)=4-2JB.1+7JB4-2JBasJB.13/CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.>

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<SBJB110013/EVS(B)=8JB+(1-JB)=1+7JB.EVS(MW)=2JB+4(1-JB)=4-2JB.1+7JB4-2JBasJB.13/CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.>

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=

<SBJB110013/SissyCoordinationGames;TheBattleoftheSexesSBJB110013/SissySBJB110013/JockCoordinationGames;TheBattleoftheSexesSBJB110013/SissySBJB110013/JockThegame’sNEare??CoordinationGames;TheBattleoftheSexesSBJB110013/SissySBJB110013/JockThegame’sNEare??CoordinationGames;TheBattleoftheSexesSBJB110013/SissyJockThegame’sNEare??13/CoordinationGames;TheBattleoftheSexesSBJB110013/SissyJockThegame’sNEare:(JB,SB)=(0,0);i.e.,(MW,MW)13/CoordinationGames;TheBattleoftheSexesSBJB110013/SissyJockThegame’sNEare:(JB,SB)=(0,0);i.e.,(MW,MW)

(JB,SB)=(1,1);i.e.,(B,B)13/CoordinationGames;TheBattleoftheSexesSBJB110013/SissyJockThegame’sNEare:(JB,SB)=(0,0);i.e.,(MW,MW)

(JB,SB)=(1,1);i.e.,(B,B)

(JB,SB)=(,);i.e.,both

watchtheballetwithprob.1/9,bothwatchthemud

wrestlingwithprob.4/9,andwithprob.4/9they

watchdifferentevents.13/13/13/CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.ForSissytheexpectedvalueoftheNE(JB,SB)=(,)is

8×+1×+2×+4×=<4and8.19/29/29/49/103/13/13/CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.ForSissytheexpectedvalueoftheNE(JB,SB)=(,)is

8×+1×+2×+4×=<4and8.19/29/29/49/103/13/13/ForJocktheexpectedvalueoftheNE(JB,SB)=(,)is

4×+2×+1×+8×=;4<<8.19/29/29/49/143/13/13//143CoordinationGames;TheBattleoftheSexes1,28,44,82,1BMWBMWJockSissySBistheprob.that

Sissychoosesballet.

JBistheprob.that

Jockchoosesballet.ForSissytheexpectedvalueoftheNE(JB,SB)=(,)is

8×+1×+2×+4×=<4and8.19/29/29/49/103/13/13/ForJocktheexpectedvalueoftheNE(JB,SB)=(,)is

4×+2×+1×+8×=;4<<8.19/29/29/49/143/13/13//143CoordinationGames;ThePrisoner’sDilemmaAsimultaneousplaygameinwhicheachplayerhasastrictlydominantaction.TheonlyNE,therefore,isthechoicebyeachplayerofherstrictlydominantaction.YetbothplayerscanachievestrictlylargerpayoffsthanintheNEbycoordinatingwitheachotheronanotherpairofactions.CoordinationGames;ThePrisoner’sDilemmaTimandTomareinpolicecustody.Eachcanconfess(C)toacrimeorstaysilent(S).Confessionbybothresultsin5yearseachinjail.Silencebybothresultsin2yearseachinjail.IfTimconfessesandTomstayssilentthenTimgetsnopenaltyandTomgets10yearsinjail(andconversely).CoordinationGames;ThePrisoner’sDilemma-10,0-2,-2-5,-50,-10ConfessSilentTomTimSilentConfessForTim,ConfessstrictlydominatesSilent.CoordinationGames;ThePrisoner’sDilemmaConfessSilentTomTimSilentConfessForTim,ConfessstrictlydominatesSilent.

ForTom,ConfessstrictlydominatesSilent.-10,0-2,-2-5,-50,-10CoordinationGames;ThePrisoner’sDilemmaConfessSilentTomTimSilentConfessForTim,ConfessstrictlydominatesSilent.

ForTom,ConfessstrictlydominatesSilent.

TheonlyNEis(Confess,Confess).-10,0-2,-2-5,-50,-10CoordinationGames;ThePrisoner’sDilemmaConfessSilentTomTimSilentConfessForTim,ConfessstrictlydominatesSilent.

ForTom,ConfessstrictlydominatesSilent.

TheonlyNEis(Confess,Confess).-10,0-2,-2-5,-50,-10But(Silence,Silence)

isbetterforbothTim

andTom.CoordinationGames;ThePrisoner’sDilemmaConfessSilentTomTimSilentConfessPossiblemeansincludefuturepunis