GameTheory--

Lecture4

PatrickLoiseauEURECOMFall2016

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Lecture2-3recap• ProvedexistenceofpurestrategyNashequilibriumin

gameswithcompactconvexactionsetsandcontinuousconcaveutilities

• DefinedmixedstrategyNashequilibrium• ProvedexistenceofmixedstrategyNashequilibriumin

finitegames• Discussedcomputationandinterpretationofmixed

strategiesNashequilibrium

àNashequilibriumisnottheonlysolutionconceptàToday:Anothersolutionconcept:evolutionarystable

strategies

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Outline

• Evolutionarystablestrategies

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Evolutionarygametheory• Gametheoryßà evolutionarybiology• Idea:– Relatestrategiestophenotypes ofgenes– Relatepayoffstogeneticfitness– Strategiesthatdowell“grow”,thosethatobtainlowerpayoffs“dieout”

• Importantnote:– Strategiesarehardwired,theyarenotchosenbyplayers

• Assumptions:– Withinspeciescompetition:nomixtureofpopulation

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Examples• Usinggametheorytounderstandpopulationdynamics

– Evolutionofspecies– Groupsoflionsdecidingwhethertoattackingroupanantelope– Antsdecidingtorespondtoanattackofaspider

– TCPvariants,P2Papplications

• Usingevolutiontointerpreteconomicactions– Firmsinacompetitivemarket– Firmsarebounded,theycan’tcomputethebestresponse,but

haverulesofthumbsandadopthardwired(consistent)strategies

– Survivalofthefittest==riseoffirmswithlowcostsandhighprofits

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Asimplemodel• Assumesimplegame:two-playersymmetric• Assumerandomtournaments– Largepopulationofindividualswithhardwiredstrategies,picktwoindividualsatrandomandmakethemplaythesymmetricgame

– Theplayeradoptingthestrategyyieldinghigherpayoffwillsurvive(andeventuallygainnewelements)whereastheplayerwho“lost”thegamewill“dieout”

• Startwithentirepopulationplayingstrategys• Thenintroduceamutation:asmall groupofindividualsstartplayingstrategys’

• Question:willthemutantssurviveandgrowordieout? 6

Asimpleexample(1)

• Haveyoualreadyseenthisgame?• Examples:– Lionshuntinginacooperativegroup– Antsdefendingthenestinacooperativegroup

• Question:iscooperationevolutionarystable?

2,2 0,33,0 1,1

C

D

Cooperate Defect

ε 1- ε

Player1

Player2

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Asimpleexample(2)Playerstrategyhardwiredè C

“SpatialGame”

Allplayersarecooperativeandgetapayoffof2

Whathappenswithamutation?

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Asimpleexample(3)Playerstrategyhardwiredè C

Focusyourattentiononthisrandom“tournament”:

• Cooperatingplayerwillobtainapayoffof0• Defectingplayerwillobtainapayoffof3

Survivalofthefittest:DwinsoverC

Playerstrategyhardwiredè D

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Asimpleexample(4)Playerstrategyhardwiredè C

Playerstrategyhardwiredè D

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Asimpleexample(5)Playerstrategyhardwiredè C

Playerstrategyhardwiredè D

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Asimpleexample(6)Playerstrategyhardwiredè C

Playerstrategyhardwiredè D

Asmallinitialmutationisrapidlyexpandinginsteadofdyingout

Eventually,Cwilldieout

à Conclusion:CisnotES

Remark:wehaveassumedasexualreproductionandnogeneredistribution 12

ESSDefinition1[MaynardSmith1972]

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Definition1: EvolutionarystablestrategyInasymmetric2-playergame,thepurestrategyŝ isES(inpurestrategies)ifthereexistsε0 >0suchthat:

forallpossibledeviationss’andforallmutationsizesε <ε0.

[ ] [ ] [ ] [ ]),()ˆ,()1(),ˆ()ˆ,ˆ()1( ssussussussu ¢¢+¢->¢+- eeeePayofftoESŝ Payofftomutants’

ESstrategiesinthesimpleexample

• IscooperationES?Cvs.[(1-ε)C+εD]à (1-ε)2+ε0=2(1-ε)Dvs.[(1-ε)C+εD]à (1-ε)3+ε1=3(1-ε)+ ε3(1-ε)+ ε >2(1-ε)

èCisnotES becausetheaveragepayofftoCislowerthantheaveragepayofftoD

èAstrictlydominatedisneverEvolutionarilyStable– Thestrictlydominantstrategywillbeasuccessfulmutation

2,2 0,33,0 1,1

C

D

Cooperate Defect

ε 1- ε

Player1

Player2

1- ε ε ForCbeingamajorityForDbeingamajority

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ESstrategiesinthesimpleexample

• IsdefectionES?Dvs.[εC+(1-ε)D]à (1-ε)1+ε3=(1-ε)+3εCvs.[εC+(1-ε)D]à (1-ε)0+ε2=2ε(1-ε)+3>2 ε

è DisES:anymutationfromDgetswipedout!

2,2 0,33,0 1,1

C

D

Cooperate Defect

ε 1- ε

Player1

Player2

1- ε ε ForCbeingamajorityForDbeingamajority

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Anotherexample(1)

• 2-playerssymmetricgamewith3strategies• Is“c”ES? cvs.[(1-ε)c+εb]à (1-ε)0+ε 1=ε

bvs.[(1-ε)c+εb]à (1-ε)1+ε 0=1- ε >ε

è “c”isnotevolutionarystable,as“b”caninvadeit

• Note:“b”,theinvader,isitselfnotES!– ItisnotnecessarilytruethataninvadingstrategymustitselfbeES– Butitstillavoidsdyingoutcompletely(growsto50%here)

2,2 0,0 0,00,0 0,0 1,10,0 1,1 0,0

a

b

c

a b c

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Anotherexample(3)

• Is(c,c)aNE?

2,2 0,0 0,00,0 0,0 1,10,0 1,1 0,0

a

b

c

a b c

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Observation

• Ifs isnotNash(thatis(s,s) isnotaNE),thens isnotevolutionarystable (ES)

Equivalently:• Ifs isES,then(s,s) isaNE

• Question:istheoppositetrue?Thatis:– If(s,s) isaNE,thens isES

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Yetanotherexample(1)

• NEofthisgame:(a,a) and(b,b)• Isb ES? bà 0

aà (1-ε)0+ε 1=ε >0

è (b,b) isaNE,butitisnotES!• ThisrelatestotheideaofaweakNE

è If(s,s) isastrictNE thens isES

1,1 0,00,0 0,0

a

b

a b

ε 1- ε

Player1

Player2

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StrictNashequilibrium

• WeakNE:theinequalityisanequalityforatleastonealternativestrategy

• StrictNEissufficientbutnotnecessaryforES20

Definition: StrictNashequilibriumAstrategyprofile(s1*,s2*,…,sN*)isastrictNashEquilibriumif,foreachplayeri,

ui(si*,s-i*)>ui(si,s-i*)forallsi ≠si*

ESSDefinition2

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Definition2: EvolutionarystablestrategyInasymmetric2-playergame,thepurestrategyŝ isES(inpurestrategies)if:

A)

AND

B)

sssussuss

¢"¢³ )ˆ,()ˆ,ˆ(mEquilibriuNash symmetric a is )ˆ,ˆ(

),(),ˆ( then )ˆ,()ˆ,ˆ( if

ssussussussu¢¢>¢¢=

Linkbetweendefinitions1and2

• Proofsketch:

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TheoremDefinition1Definition2

€

⇔

Recap:checkingforESstrategies

• WehaveseenadefinitionthatconnectsEvolutionaryStabilitytoNashEquilibrium

• Bydef 2,tocheckthatŝ isES,weneedtodo:

– Firstcheckif(ŝ,ŝ)isasymmetric NashEquilibrium– Ifitisastrict NE,we’redone– Otherwise,weneedtocomparehowŝperformsagainstamutation,andhowamutationperformsagainstamutation

– Ifŝperformsbetter,thenwe’redone

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Example:Is“a”evolutionarystable?

• Is(a,a)aNE?Isitstrict?• Is“a”evolutionarystable?

1,1 1,11,1 0,0

a

b

a b

ε 1- ε

Player1

Player2

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Evolutionofsocialconvention• Evolutionisoftenappliedtosocialsciences• Let’shavealookathowdrivingtotheleftorrighthandsideoftheroadmightevolve

• WhataretheNE?aretheystrict?WhataretheESS?

• Conclusion:wecanhaveseveralESS– Theyneednotbeequallygood

2,2 0,00,0 1,1

L

R

L R

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ThegameofChicken

• Thisisasymmetric coordinationgame• Biologyinterpretation:– “a”:individualsthatareaggressive– “b”:individualsthatarenon-aggressive

• WhatarethepurestrategyNE?– Theyarenotsymmetricà nocandidateforESS

0,0 2,11,2 0,0

a

b

a b

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ThegameofChicken:mixedstrategyNE

• What’sthemixedstrategyNEofthisgame?– MixedstrategyNE=[(2/3,1/3),(2/3,1/3)]è Thisisasymmetric NashEquilibrium

èInterpretation:thereisanequilibriuminwhich2/3ofthegenesareaggressiveand1/3arenon-aggressive

• IsitastrictNashequilibrium?• IsitanESS?

0,0 2,11,2 0,0

a

b

a b

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Remark

• Amixed-strategyNashequilibrium(withasupportofatleast2actionsforoneoftheplayers)canneverbeastrictNashequilibrium

• ThedefinitionofESSisthesame!

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ESSDefinition2bis

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Definition2: EvolutionarystablestrategyInasymmetric2-playergame,themixedstrategyŝisES(inmixedstrategies)if:

A)

AND

B)

sssussuss

¢"¢³ )ˆ,()ˆ,ˆ(mEquilibriuNash symmetric a is )ˆ,ˆ(

),(),ˆ( then )ˆ,()ˆ,ˆ( if

ssussussussu¢¢>¢¢=

ThegameofChicken:ESS

• MixedstrategyNE=[(2/3,1/3),(2/3,1/3)].• IsitanESS?weneedtocheckforallpossiblemixed

mutationss’:• Yes,itis(doitathome!)

• Inmanycasesthatariseinnature,theonlyequilibriumisamixedequilibrium– Itcouldmeanthatthegeneitselfisrandomizing,whichis

plausible– Itcouldbethatthereareactuallytwotypessurvivinginthe

population(cf.ourinterpretationofmixedstrategies)

0,0 2,11,2 0,0

a

b

a b

u(s, !s )> u( !s , !s ) ∀ !s ≠ s

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Hawksanddoves

DoveHawk

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TheHawksandDovegame(1)

• Moregeneralgameofaggressionvs.non-aggression– Theprizeisfood,anditsvalueisv>0– There’sacostforfighting,whichisc>0

• Note:we’restillinthecontextofwithinspicescompetition– Soit’snotabattleagainsttwodifferentanimals,hawksand

doves,wetalkaboutstrategies• “Actdovishvs.acthawkish”

• WhataretheESS?Howdotheychangewithc,v?

(v-c)/2,(v-c)/2 v,00,v v/2, v/2

H

D

H D

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TheHawksandDovegame(2)

• CanwehaveaESpopulationofdoves?• Is(D,D) aNE?– No,hence“D”isnotESS– Indeed,amutationofhawksagainstdoveswouldbeprofitableinthatitwouldobtainapayoffofv

(v-c)/2,(v-c)/2 v,00,v v/2, v/2

H

D

H D

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TheHawksandDovegame(3)

• CanwehaveaESpopulationofHawks?• Is(H,H) aNE?Itdepends:itisasymmetricNEif(v-c)/2≥0

• Case1:v>cè (H,H) isastrict NEè “H” isESS• Case2:v=cè (v-c)/2=0è u(H,H) =u(D,H) -- (H,H)isaweakNE

– Isu(H,D)=v largerthanu(D,D)=v/2?Yesè “H” isESS

èHisESS ifv≥c• Iftheprizeishighandthecostforfightingislow,thenyou’llsee

fightsarisinginnature

(v-c)/2,(v-c)/2 v,00,v v/2, v/2

H

D

H D

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TheHawksandDovegame(4)

• Whatifc>v?– “H”isnotESSand“D”isnotESS(theyarenotNE)

• Step1:findamixedNE

• Step2:verifytheESScondition

(v-c)/2,(v-c)/2 v,00,v v/2, v/2

H

D

H D

s 1− s

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TheHawksandDovegame:results

• Incasev<cwehaveanevolutionarilystablestateinwhichwehavev/chawks1. Asv↗wewillhavemorehawksinESS2. Asc↗wewillhavemoredovesinESS

• BymeasuringtheproportionofHandD,wecangetthevalueofv/c

• Payoff: E u(D, s)[ ] = E u(H, s)[ ] = 0 vc+ 1− v

c"

#$

%

&'v236

Onelastexample(1)

• Assume1<v<2– ~Rock,paper,scissors

• OnlyNE:ŝ =(1/3,1/3,1/3)– mixed,notstrict• IsitanESS?

– Supposes’=R– u(ŝ,R)=(1+v)/3<1– u(R,R)=1

• Conclusion:NotallgameshaveanESS!

1,1 v,0 0,v0,v 1,1 v,0v,0 0,v 1,1

R

P

S

R P S

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