GameTheory
-Lecture3
PatrickLoiseau
EURECOM
Fall2016
1
Lecture2 recap
• DefinedParetooptimality
– Coordinationgames
• Studiedgameswithcontinuousactionspace
– AlwayshaveaNashequilibriumwithsomeconditions
– Cournot duopolyexample
à CanwealwaysfindaNashequilibriumforall
games?
à How?
2
Outline
1. Mixedstrategies
– BestresponseandNashequilibrium
2. MixedstrategiesNashequilibriumcomputation
3. Interpretationsofmixedstrategies
3
Outline
1. Mixedstrategies
– BestresponseandNashequilibrium
2. MixedstrategiesNashequilibriumcomputation
3. Interpretationsofmixedstrategies
4
Example:installingcheckpoints
• Tworoad,Policechooseonwhichtocheck,
Terroristschooseonwhichtopass
Terrorist
R1
R1
R2
1,-1
-1,1
Police
R2
-1,1
1,-1
• CanyoufindaNash
equilibrium?
à Playersmust
randomize
5
Matchingpennies
• Similarexamples:
Player2
heads
tails
heads
1,-1
-1,1
– Checkpointplacement
– Intrusiondetection
– Penaltykick
– Tennisgame
Player1
tails
-1,1
1,-1
• Needtobeunpredictable
6
Purestrategies/Mixedstrategies
• Game ( N, ( Ai )i∈N , (ui )i∈N )
• Ai:setofactionsofplayeri (whatwecalledSi
before)
• Action=purestrategy
• Mixedstrategy:distributionoverpurestrategies
si ∈ Si = Δ(Ai )
– Includepurestrategyasspecialcase
– Support: supp si = {ai ∈ Ai : si (ai ) > 0}
• Strategyprofile: s = (s1,, sn ) ∈ S = S1 ×× Sn
7
Matchingpennies:payoffs
• WhatisPlayer1’spayoffifPlayer2
playss2 =(1/4,3/4)andheplays:
Player2
heads
tails
heads
1,-1
-1,1
– Heads?
– Tails?
Player1
tails
-1,1
1,-1
– s1 =(½,½)?
8
Payoffsinmixedstrategies:general
formula
• Game,let
A = × Ai
( N, ( Ai )i∈N , (ui )i∈N )
i∈N
• Ifplayersfollowamixed-strategyprofiles,the
expectedpayoffofplayeri is:
ui (s) = ∑ ui (a)Pr(a | s) where Pr(a | s) = ∏ si (ai )
a∈A
i∈N
• a:purestrategy(oraction)profile
• Pr(a|s):probabilityofseeingagiventhe
mixedstrategyprofiles
9
Matchingpennies:payoffscheck
• WhatarethepayoffsofPlayer1
andPlayer2ifs=((½,½),(¼,¾))?
Player2
heads
tails
heads
1,-1
-1,1
Player1
tails
-1,1
1,-1
• Doesthatlooklikeitcouldbea
Nashequilibrium?
10
Bestresponse
• Thedefinitionformixedstrategiesis
unchanged!
Definition: Best Response
Playeri’s strategyŝi isaBRtostrategys-i ofother
playersif:
ui(ŝi ,s-i)≥ui(s’i ,s-i)for alls’i inSi
• BRi(s-i):setofbestresponsesofi tos-i
11
Matchingpennies:bestresponse
• Whatisthebestresponseof
Player1tos2 =(¼,¾)?
Player2
heads
tails
heads
1,-1
-1,1
Player1
tails
-1,1
• Foralls1,u1(s1,s2)liebetween
u1(heads,s2)andu1(tails,s2)
(theweightedaveragelies
betweenthepurestrategies
exp.Payoffs)
1,-1
à Bestresponseistails!
12
Importantproperty
• Ifamixedstrategyisabestresponsethen
eachofthepurestrategiesinthemixmustbe
bestresponses
è Theymustyieldthesameexpectedpayoff
Proposition:
Forany (mixed)strategys-i,if,then
si ∈ BRi (s−i )
ai ∈ BRi (s−i ) for all ai such that si (ai ) > 0 .
Inparticular,ui(ai,s-i) isthesameforallai suchthat
si (ai ) > 0
13
Wordyproof
• Supposeitwerenottrue.Thentheremustbeatleastone
purestrategyai thatisassignedpositiveprobabilitybymy
best-responsemixandthatyieldsalowerexpectedpayoff
againstsi
• Ifthereismorethanone,focusontheonethatyieldsthe
lowestexpectedpayoff.SupposeIdropthat(low-yield)pure
strategyfrommymix,assigningtheweightIusedtogiveitto
oneoftheother(higher-yield)strategiesinthemix
• Thismustraisemyexpectedpayoff
• Butthentheoriginalmixedstrategycannothavebeenabest
response:itdoesnotdoaswellasthenewmixedstrategy
• Thisisacontradiction
14
Matchingpenniesagain
Player2
heads
tails
heads
1,-1
-1,1
• Whatisthebestresponse
ofPlayer1tos2 =(¼,¾)?
• Whatisthebestresponse
ofPlayer1tos2 =(½,½)?
Player1
tails
-1,1
1,-1
15
Nashequilibriumdefinition
Definition: NashEquilibrium
Astrategyprofile(s1*,s2*,…,sN*)isaNash
Equilibrium(NE)if,foreachi,herchoicesi*isa
bestresponsetotheotherplayers’choicess-i*
• Samedefinitionasforpurestrategies!
– Butherethestrategiessi* aremixedstrategies
16
Matchingpenniesagain
Player2
heads
tails
heads
1,-1
• Nashequilibrium:
((½,½),(½,½))
-1,1
Player1
tails
-1,1
1,-1
17
Nashequilibriumexistencetheorem
Theorem: Nash(1951)
EveryfinitegamehasaNashequilibrium.
• Inmixedstrategy!
– Nottrueinpurestrategy
• Finitegame:finitesetofplayerandfinite
actionsetforallplayers
– Botharenecessary!
• Proof:reductiontoKakutani’s fixed-pointthm
18
Outline
1. Mixedstrategies
– BestresponseandNashequilibrium
2. MixedstrategiesNashequilibriumcomputation
3. Interpretationsofmixedstrategies
19
ComputationofmixedstrategyNE
• Hardifthesupportisnotknown
• Ifyoucanguessthesupport,itbecomesvery
easy,usingthepropertyshownearlier:
Proposition:
Forany (mixed)strategys-i,if,then
si ∈ BRi (s−i )
ai ∈ BRi (s−i ) for all ai such that si (ai ) > 0 .
Inparticular,ui(ai,s-i) isthesameforallai suchthat
si (ai ) > 0 (i.e.,ai inthesupportofsi)
20
Example:battleofthesexes
Player2
Player1
Opera
Soccer
Opera
Soccer
2,1
0,0
0,0
1,2
• Wehaveseenthat(O,O)and(S,S)areNE
• IsthereanyotherNE(inmixedstrategies)?
– Let’strytofindaNEwithsupport{O,S}foreach
player
21
Example:battleofthesexes(2)
Player2
Player1
Opera
Soccer
Opera
Soccer
2,1
0,0
0,0
1,2
• Lets2 =(p,1-p)
• Ifs1 isaBRwithsupport{O,S},thenPlayer1
mustbeindifferentbetweenOandS
à p=1/3
22
Example:battleofthesexes(3)
Player2
Player1
Opera
Soccer
Opera
Soccer
2,1
0,0
0,0
1,2
• Similarly,lets1 =(q,1-q)
• Ifs2 isaBRwithsupport{O,S},thenPlayer2
mustbeindifferentbetweenOandS
à q=2/3
23
Example:battleofthesexes(4)
Player2
Player1
Opera
Soccer
Opera
Soccer
2,1
0,0
0,0
1,2
• Conclusion:((2/3,1/3),(1/3,2/3))isaNE
24
Example:prisoner’sdilemma
• Weknowthat(D,D)isNE
• CanwefindaNEwith
support{C,D}witheach?
Prisoner2
D
D
-5,-5
C
0,-6
Prisoner1
C
• ANEinstrictlydominant
strategiesisunique!
-6,0
-2,-2
25
GeneralmethodstocomputeNash
equilibrium
• Ifyouknowthesupport,writetheequations
translatingindifferencebetweenstrategiesin
thesupport(worksforanynumberof
actions!)
• Otherwise:
– TheLemke-Howson Algorithm(1964)
– Supportenumerationmethod(Porteretal.2004)
• Smartheuristicsearchthroughallsetsofsupport
• Exponentialtimeworstcasecomplexity
26
ComplexityoffindingNashequilibrium
• IsitNP-complete?
– No,weknowthereisasolution
– Butmanyderivedproblemsare(e.g.,doesthere
existsastrictlyParetooptimalNashequilibrium?)
• PPAD(“PolynomialParityArgumentson
Directedgraphs”)[Papadimitriou1994]
• Theorem:ComputingaNashequilibriumis
PPAD-complete[Chen,Deng2006]
27
ComplexityoffindingNashequilibrium
(2)
NP-hard
NP-complete
NP
PPAD
P
28
Outline
1. Mixedstrategies
– BestresponseandNashequilibrium
2. MixedstrategiesNashequilibriumcomputation
3. Interpretationsofmixedstrategies
29
Mixedstrategiesinterpretations
• Playersrandomize
• Beliefofothers’actions(thatmakeyou
indifferent)
• Empiricalfrequencyofplayinrepeated
interactions
• Fractionofapopulation
– Let’sseeanexampleofthisone
30
TheIncomeTaxGame(1)
Taxpayer
Cheat
Honest
Auditor
A
N
2,0 4,-10
4,0 0,4
q
p
(1-p)
1-q
• Assumesimultaneousmovegame
• IsthereapurestrategyNE?
• FindmixedstrategyNE
31
TheIncomeTaxGame:NE
computation
• MixedstrategiesNE:
E[U1 ( A, (q,1 - q ))] = 2q + 4(1 - q) ü
2
ý2q = 4(1 - q) Þ q =
E[U1 (N , (q,1 - q ))] = 4q + 0(1 - q)þ
3
E[U 2 (H , ( p,1 - p ))] = 0
ü
2
ý4 = 14 p Þ p =
E[U 2 (C , ( p,1 - p ))] = -10 p + 4(1 - p)þ
7
Lookattax
payers
payoffs
Tofind
auditors
mixing
32
TheIncomeTaxGame:mixedstrategy
interpretation
• Fromtheauditor’spointofview,he/sheisgoing
toauditasingletaxpayer2/7ofthetime
èThisisactuallyarandomization(which isapplied
bylaw)
• Fromthetaxpayerperspective,he/sheisgoingto
behonest2/3ofthetime
è Thisinrealityimpliesthat2/3rdofpopulationis
goingtopaytaxeshonestly,i.e.,thisisafraction
ofalargepopulation payingtaxes
33
TheIncomeTaxGame(6)
• Whatcouldeverbedoneifonepolicymaker
(e.g.thegovernment)wouldliketoincrease
theproportionofhonesttaxpayers?
• Oneideacouldbeforexampleto“prevent”
fraudbyincreasingthenumberofyearsatax
payerwouldspendinjailiffoundguilty
34
TheIncomeTaxGame:Tryingtomake
peoplepay
Taxpayer
Cheat
Honest
Auditor
A
N
2,0 4,-20
4,0 0,4
q
p
(1-p)
1-q
• Howtomakepeoplepaytheirtaxes?
• Oneidea:increasepenaltyforcheating
• Whatisthenewequilibrium?
35
TheIncomeTaxGame:newNE
E[U1 ( A, (q,1 - q ))] = 2q + 4(1 - q) ü
2
ý2q = 4(1 - q) Þ q =
E[U1 (N , (q,1 - q ))] = 4q + 0(1 - q)þ
3
E[U 2 (H , ( p,1 - p ))] = 0
ü
1 2
ý24 p = 4 Þ p = <
E[U 2 (C , ( p,1 - p ))] = -20 p + 4(1 - p)þ
6 7
• Theproportionofhonesttaxpayersdidn’tchange!
– Whatdeterminestheequilibriummixforthecolumn
playeristherowplayer’spayoffs
• Theprobabilityofauditdecreased
– Stillgood,auditsareexpensive
• Tomakepeoplepaytax:changeauditor’spayoff
– Makeauditscheaper,moreprofitable
36
Importantremark
• Rowplayer’sNEmixdeterminedbycolumn
player’spayoffandviceversa
• Neutralizetheopponent(makehim
indifferent)
• Insomesensetheoppositeofoptimization
(mychoiceisindependentofmyownpayoff)
37
Thepenaltykickgame
• 2players:kickerandgoalkeeper
• 2actionseach
– Kicker:kickleft,kickright
– Goalkeeper:jumpleft,jumpright
• Payoff:probabilitytoscoreforthekicker,
probabilitytostopitforthegoalkeeper
Goalkeeper
• Scoringprobabilities:
R
L
Kicker
L
58.30 94.97
R
92.91 69.92
38
Thepenaltykickgame:results
• IgnacioPalacios-Huerta.ProfessionalsPlay
Minimax.ReviewofEconomicsStudies(2003).
• Result:
GoalL
NEprediction
GoalR
KickerL
KickerR
41.99
58.01
38.54
61.46
Observedfreq. 42.31
57.69
39.98
60.02
• Foragivenkicker,hisstrategyisalsoserially
independent
39
Summary
• Mixedstrategies:distributionoveractions
– ANashequilibriuminmixedstrategiesalways
existsforfinitegames
– Computationiseasyifthesupportisknown
• Allpurestrategiesinthesupportofabestresponseare
alsobestresponses
• Makesotherplayerindifferentinhissupport
– Computationishardifthesupportisnotknown
– Severalinterpretationsdependingonthegameat
stake
40