Game Theory -- Lecture 3lig-membres.imag.fr/loiseapa/courses/GameTheory/slides/Lecture3.pdf ·...

GameTheory--

Lecture3

PatrickLoiseauEURECOMFall2016

1

Lecture2 recap

• DefinedParetooptimality– Coordinationgames

• Studiedgameswithcontinuousactionspace– AlwayshaveaNashequilibriumwithsomeconditions– Cournot duopolyexample

à CanwealwaysfindaNashequilibriumforallgames?à How?

2

Outline

1. Mixedstrategies– BestresponseandNashequilibrium

2. MixedstrategiesNashequilibriumcomputation3. Interpretationsofmixedstrategies

3

Outline



4

Example:installingcheckpoints

• Tworoad,Policechooseonwhichtocheck,Terroristschooseonwhichtopass

5

R1 R2

R1

R2

1,-1 -1,1

1,-1-1,1

Police

Terrorist• CanyoufindaNashequilibrium?

à Playersmustrandomize

Matchingpennies

• Similarexamples:– Checkpointplacement– Intrusiondetection– Penaltykick– Tennisgame

• Needtobeunpredictable

6

heads tails

heads

tails

1,-1 -1,1

1,-1-1,1

Player1

Player2

Purestrategies/Mixedstrategies

• Game• Ai:setofactionsofplayeri (whatwecalledSibefore)

• Action=purestrategy• Mixedstrategy:distributionoverpurestrategies

– Includepurestrategyasspecialcase– Support:

• Strategyprofile:7

N, Ai( )i∈N , ui( )i∈N( )

si ∈ Si = Δ(Ai )

s = (s1,, sn )∈ S = S1 ×× Sn

supp si = {ai ∈ Ai : si (ai )> 0}

Matchingpennies:payoffs• WhatisPlayer1’spayoffifPlayer2

playss2 =(1/4,3/4)andheplays:

– Heads?

– Tails?

– s1 =(½,½)?

8

heads tails

heads

tails

1,-1 -1,1

1,-1-1,1

Player1

Player2

Payoffsinmixedstrategies:generalformula

• Game,let• Ifplayersfollowamixed-strategyprofiles,theexpectedpayoffofplayeri is:

• a:purestrategy(oraction)profile• Pr(a|s):probabilityofseeingagiventhemixedstrategyprofiles

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ui (s) = uia∈A∑ (a)Pr(a | s) where Pr(a | s) = si (ai )

i∈N∏

N, Ai( )i∈N , ui( )i∈N( ) A = ×i∈N

Ai

Matchingpennies:payoffscheck• WhatarethepayoffsofPlayer1

andPlayer2ifs=((½,½),(¼,¾))?

• DoesthatlooklikeitcouldbeaNashequilibrium?

10

heads tails

heads

tails

1,-1 -1,1

1,-1-1,1

Player1

Player2

Bestresponse

• Thedefinitionformixedstrategiesisunchanged!

• BRi(s-i):setofbestresponsesofi tos-i11

Definition: Best ResponsePlayeri’s strategyŝi isaBRtostrategys-i ofotherplayersif:

ui(ŝi ,s-i)≥ui(s’i ,s-i)for alls’i inSi

Matchingpennies:bestresponse• Whatisthebestresponseof

Player1tos2 =(¼,¾)?

• Foralls1,u1(s1,s2)liebetweenu1(heads,s2)andu1(tails,s2)(theweightedaverageliesbetweenthepurestrategiesexp.Payoffs)

à Bestresponseistails!

12

heads tails

heads

tails

1,-1 -1,1

1,-1-1,1

Player1

Player2

Importantproperty

• Ifamixedstrategyisabestresponsetheneachofthepurestrategiesinthemixmustbebestresponses

è Theymustyieldthesameexpectedpayoff

13

Proposition:Forany (mixed)strategys-i,if,then

.

Inparticular,ui(ai,s-i) isthesameforallai suchthat

si ∈ BRi (s−i )ai ∈ BRi (s−i ) for all ai such that si (ai )> 0

si (ai )> 0

Wordyproof• Supposeitwerenottrue.Thentheremustbeatleastone

purestrategyai thatisassignedpositiveprobabilitybymybest-responsemixandthatyieldsalowerexpectedpayoffagainstsi

• Ifthereismorethanone,focusontheonethatyieldsthelowestexpectedpayoff.SupposeIdropthat(low-yield)purestrategyfrommymix,assigningtheweightIusedtogiveittooneoftheother(higher-yield)strategiesinthemix

• Thismustraisemyexpectedpayoff• Butthentheoriginalmixedstrategycannothavebeenabest

response:itdoesnotdoaswellasthenewmixedstrategy• Thisisacontradiction

14

Matchingpenniesagain• WhatisthebestresponseofPlayer1tos2 =(¼,¾)?

• WhatisthebestresponseofPlayer1tos2 =(½,½)?

15

heads tails

heads

tails

1,-1 -1,1

1,-1-1,1

Player1

Player2

Nashequilibriumdefinition

• Samedefinitionasforpurestrategies!– Butherethestrategiessi* aremixedstrategies

16

Definition: NashEquilibriumAstrategyprofile(s1*,s2*,…,sN*)isaNashEquilibrium(NE)if,foreachi,herchoicesi*isabestresponsetotheotherplayers’choicess-i*

Matchingpenniesagain

• Nashequilibrium:((½,½),(½,½))

17

heads tails

heads

tails

1,-1 -1,1

1,-1-1,1

Player1

Player2

Nashequilibriumexistencetheorem

• Inmixedstrategy!– Nottrueinpurestrategy

• Finitegame:finitesetofplayerandfiniteactionsetforallplayers– Botharenecessary!

• Proof:reductiontoKakutani’s fixed-pointthm18

Theorem: Nash(1951)EveryfinitegamehasaNashequilibrium.

Outline



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ComputationofmixedstrategyNE

• Hardifthesupportisnotknown• Ifyoucanguessthesupport,itbecomesveryeasy,usingthepropertyshownearlier:

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Proposition:Forany (mixed)strategys-i,if,then

.

Inparticular,ui(ai,s-i) isthesameforallai suchthat(i.e.,ai inthesupportofsi)

si ∈ BRi (s−i )ai ∈ BRi (s−i ) for all ai such that si (ai )> 0

si (ai )> 0

Example:battleofthesexes

• Wehaveseenthat(O,O)and(S,S)areNE

• IsthereanyotherNE(inmixedstrategies)?– Let’strytofindaNEwithsupport{O,S}foreachplayer

2,1 0,00,0 1,2

Opera

Soccer

Opera

Player1

Player2Soccer

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Example:battleofthesexes(2)

• Lets2 =(p,1-p)• Ifs1 isaBRwithsupport{O,S},thenPlayer1mustbeindifferentbetweenOandS

à p=1/3

2,1 0,00,0 1,2

Opera

Soccer

Opera

Player1

Player2Soccer

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• Similarly,lets1 =(q,1-q)• Ifs2 isaBRwithsupport{O,S},thenPlayer2mustbeindifferentbetweenOandS

à q=2/3

2,1 0,00,0 1,2

Opera

Soccer

Opera

Player1

Player2Soccer

23


• Conclusion:((2/3,1/3),(1/3,2/3))isaNE

2,1 0,00,0 1,2

Opera

Soccer

Opera

Player1

Player2Soccer

24

Example:prisoner’sdilemma

• Weknowthat(D,D)isNE• CanwefindaNEwithsupport{C,D}witheach?

• ANEinstrictlydominantstrategiesisunique! 25

D C

D

C

-5,-5 0,-6

-2,-2-6,0

Prisoner1

Prisoner2

GeneralmethodstocomputeNashequilibrium

• Ifyouknowthesupport,writetheequationstranslatingindifferencebetweenstrategiesinthesupport(worksforanynumberofactions!)

• Otherwise:– TheLemke-Howson Algorithm(1964)– Supportenumerationmethod(Porteretal.2004)• Smartheuristicsearchthroughallsetsofsupport

• Exponentialtimeworstcasecomplexity

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ComplexityoffindingNashequilibrium

• IsitNP-complete?– No,weknowthereisasolution– Butmanyderivedproblemsare(e.g.,doesthereexistsastrictlyParetooptimalNashequilibrium?)

• PPAD(“PolynomialParityArgumentsonDirectedgraphs”)[Papadimitriou1994]

• Theorem:ComputingaNashequilibriumisPPAD-complete[Chen,Deng2006]

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ComplexityoffindingNashequilibrium(2)

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P

NPPPAD

NP-complete

NP-hard

Outline



29

Mixedstrategiesinterpretations

• Playersrandomize• Beliefofothers’actions(thatmakeyouindifferent)

• Empiricalfrequencyofplayinrepeatedinteractions

• Fractionofapopulation– Let’sseeanexampleofthisone

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TheIncomeTaxGame(1)

• Assumesimultaneousmovegame• IsthereapurestrategyNE?• FindmixedstrategyNE

2,0 4,-104,0 0,4

A

N

Honest Cheat

q 1-q

p

(1-p)

Auditor

Taxpayer

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TheIncomeTaxGame:NEcomputation

• MixedstrategiesNE:

( )( )[ ]( )( )[ ]( )( )[ ]( )( )[ ] 7

2144)1(4101,,

01,,

32)1(42

)1(041,,)1(421,,

2

2

1

1

=Þ=þýü

-+-=-=-

=Þ-=þýü

-+=--+=-

ppppppCUE

ppHUE

qqqqqqqNUEqqqqAUE

Lookattaxpayerspayoffs

Tofindauditorsmixing

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TheIncomeTaxGame:mixedstrategyinterpretation

• Fromtheauditor’spointofview,he/sheisgoingtoauditasingletaxpayer2/7ofthetime

èThisisactuallyarandomization(which isappliedbylaw)

• Fromthetaxpayerperspective,he/sheisgoingtobehonest2/3ofthetime

è Thisinrealityimpliesthat2/3rdofpopulationisgoingtopaytaxeshonestly,i.e.,thisisafractionofalargepopulation payingtaxes

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TheIncomeTaxGame(6)

• Whatcouldeverbedoneifonepolicymaker(e.g.thegovernment)wouldliketoincreasetheproportionofhonesttaxpayers?

• Oneideacouldbeforexampleto“prevent”fraudbyincreasingthenumberofyearsataxpayerwouldspendinjailiffoundguilty

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TheIncomeTaxGame:Tryingtomakepeoplepay

• Howtomakepeoplepaytheirtaxes?

• Oneidea:increasepenaltyforcheating

• Whatisthenewequilibrium?

2,0 4,-204,0 0,4

A

N

Honest Cheat

q 1-q

p

(1-p)

Auditor

Taxpayer

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TheIncomeTaxGame:newNE

( )( )[ ]( )( )[ ]( )( )[ ]( )( )[ ] 7

261424

)1(4201,,01,,

32)1(42

)1(041,,)1(421,,

2

2

1

1

<=Þ=þýü

-+-=-=-

=Þ-=þýü

-+=--+=-

ppppppCUE

ppHUE

qqqqqqqNUEqqqqAUE

• Theproportionofhonesttaxpayersdidn’tchange!– Whatdeterminestheequilibriummixforthecolumnplayeristherowplayer’spayoffs

• Theprobabilityofauditdecreased– Stillgood,auditsareexpensive

• Tomakepeoplepaytax:changeauditor’spayoff– Makeauditscheaper,moreprofitable 36

Importantremark

• Rowplayer’sNEmixdeterminedbycolumnplayer’spayoffandviceversa

• Neutralizetheopponent(makehimindifferent)

• Insomesensetheoppositeofoptimization(mychoiceisindependentofmyownpayoff)

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Thepenaltykickgame

• 2players:kickerandgoalkeeper• 2actionseach– Kicker:kickleft,kickright– Goalkeeper:jumpleft,jumpright

• Payoff:probabilitytoscoreforthekicker,probabilitytostopitforthegoalkeeper

• Scoringprobabilities:

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58.30 94.9792.91 69.92

L

R

L R

Kicker

Goalkeeper

Thepenaltykickgame:results

• IgnacioPalacios-Huerta.ProfessionalsPlayMinimax.ReviewofEconomicsStudies(2003).

• Result:

• Foragivenkicker,hisstrategyisalsoseriallyindependent

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41.99 58.01 38.54 61.46

42.31 57.69 39.98 60.02

NEprediction

Observedfreq.

GoalL GoalR KickerL KickerR

Summary

• Mixedstrategies:distributionoveractions– ANashequilibriuminmixedstrategiesalwaysexistsforfinitegames

– Computationiseasyifthesupportisknown• Allpurestrategiesinthesupportofabestresponsearealsobestresponses• Makesotherplayerindifferentinhissupport

– Computationishardifthesupportisnotknown– Severalinterpretationsdependingonthegameatstake

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Game Theory -- Lecture 3lig-membres.imag.fr/loiseapa/courses/GameTheory/slides/Lecture3.pdf ·...

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