Download Game Theory -- Lecture 5

GameTheory
-Lecture5
PatrickLoiseau
EURECOM
Fall2016
1
Lecture3-4recap
• DefinedmixedstrategyNashequilibrium
• ProvedexistenceofmixedstrategyNashequilibriumin
finitegames
• Discussedcomputationandinterpretationofmixed
strategiesNashequilibrium
• Definedanotherconceptofequilibriumfrom
evolutionarygametheory
àToday:introduceothersolutionconceptsfor
simultaneousmovesgames
àIntroducesolutionsforsequentialmovesgames
2
Outline
• Othersolutionconceptsforsimultaneous
moves
– Stabilityofequilibrium
• Trembling-handperfectequilibrium
– Correlatedequilibrium
– Minimax theoremandzero-sumgames
– ε-Nashequilibrium
• Thelenderandborrowergame:introduction
andconceptsfromsequentialmoves
3
Outline
• Othersolutionconceptsforsimultaneous
moves
– Stabilityofequilibrium
• Trembling-handperfectequilibrium
– Correlatedequilibrium
– Minimax theoremandzero-sumgames
– ε-Nashequilibrium
• Thelenderandborrowergame:introduction
andconceptsfromsequentialmoves
4
TheLocationModel
• Assumewehave2N playersinthisgame(e.g.,N=70)
– Playershavetwotypes:tallandshort
– ThereareN tallplayersandN shortplayers
• Playersarepeoplewhoneedtodecideinwhichtowntolive
• Therearetwotowns:EasttownandWesttown
– EachtowncanhostnomorethanN players
• Assume:
– Ifthenumberofpeoplechoosingaparticulartownislargerthanthe
towncapacity,thesurpluswillberedistributedrandomly
• Game:
– Players:2N people
– Strategies:EastorWesttown
– Payoffs
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TheLocationModel:payoffs
Utilityforplayeri
• Theideais:
– Ifyouareasmall
minority inyourtown
yougetapayoffofzero
– Ifyouareinlarge
majority inyourtown
yougetapayoffof½
– Ifyouarewell
integrated yougeta
payoffof1
1
1/2
0
• Peoplewouldliketolive
inmixedtowns,butif
theycannot,thenthey
prefertoliveinthe
majoritytown
35
70
#ofyourtype
inyourtown
6
Initialstate
• Assumetheinitial
pictureisthisone
• Whatwillplayersdo?
Tallplayer
Shortplayer
WestTown
EastTown
7
Firstiteration
• Fortallplayers
• There’saminorityof
easttown“giants”to
beginwith
à switchtoWesttown
Tallplayer
Shortplayer
WestTown
EastTown
• Forshortplayers
• There’saminorityof
westtown“dwarfs”to
beginwith
àswitchtoEasttown
8
Seconditeration
• Sametrend
• Stillafewplayerswho
didnotunderstand
– Whatistheirpayoff?
Tallplayer
Shortplayer
WestTown
EastTown
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Lastiteration
• Peoplegotsegregated
• Buttheywouldhave
preferredintegratedtowns!
– Why?Whathappened?
– Peoplethatstartedina
minority(eventhough
nota“bad”minority)
hadincentivestodeviate
Tallplayer
Shortplayer
WestTown
EastTown
10
TheLocationModel:Nashequilibria
• TwosegregatedNE:
– Short,E;Tall,W
– Short,W;Tall,E
• IsthereanyotherNE?
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Stabilityofequilibria
• Theintegratedequilibriumisnotstable
– Ifwemoveawayfromthe50%ratio,evenalittlebit,playershavean
incentivetodeviateevenmore
– Weendupinoneofthesegregatedequilibrium
• Thesegregatedequilibria arestable
– Introduceasmallperturbation:playerscomebacktosegregation
quickly
• NotionofstabilityinPhysics:ifyouintroduceasmallperturbation,
youcomebacktotheinitialstate
• Tippingpoint:
– IntroducedbyGrodzins (WhiteflightsinAmerica)
– ExtendedbyShelling(Nobelprizein2005)
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Trembling-handperfectequilibrium
Definition: Trembling-handperfectequilibrium
A (mixed)strategyprofilesisatrembling-hand
perfectequilibriumifthereexistsasequence
s(0),s(1),…offullymixedstrategyprofilesthat
convergestowardssandsuchthatforallkand
allplayeri,si isabestresponsetos(k)-i.
• Fully-mixedstrategy:positiveprobabilityoneach
action
• Informally:aplayer’sactionsi mustbeBRnot
onlytoopponentsequilibriumstrategiess-i but
alsotosmallperturbationsofthoses(k)-i.
13
TheLocationModel
• Thesegregatedequilibria aretrembling-hand
perfect
• Theintegratedequilibriumisnottremblinghandperfect
14
Outline
• Othersolutionconceptsforsimultaneous
moves
– Stabilityofequilibrium
• Trembling-handperfectequilibrium
– Correlatedequilibrium
– Minimax theoremandzero-sumgames
– ε-Nashequilibrium
• Thelenderandborrowergame:introduction
andconceptsfromsequentialmoves
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Example:battleofthesexes
Player2
Player1
Opera
Soccer
Opera
Soccer
2,1
0,0
0,0
1,2
• NE:(O,O),(S,S)and((1/3,2/3),(2/3,1/3))
– Themixedequilibriumhaspayoff2/3each
• Supposetheplayerscanobservetheoutcomeofafair
tosscoinandconditiontheirstrategiesonthis
outcome
– Newstrategiespossible:Oifhead,Siftails
– Payoff1.5each
• Thefaircoinactsasacorrelatingdevice
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Correlatedequilibrium:generalcase
• Inthepreviousexample:bothplayersobservethe
exactsamesignal(outcomeofthecointossrandom
variable)
• Generalcase:eachplayerreceivesasignalwhichcan
becorrelatedtotherandomvariable(cointoss)andto
theotherplayerssignal
• Model:
– nrandomvariables(oneperplayer)
– AjointdistributionoverthenRVs
– Naturechoosesaccordingtothejointdistributionand
revealstoeachplayeronlyhisRV
à AgentcanconditionhisactiontohisRV(hissignal)
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Correlatedequilibrium:definition
Definition: Correlatedequilibrium
Acorrelatedequilibriumofthegame(N,(Ai),(ui))is
atuple(v,π,σ)where
• v=(v1,…,vn)isatupleofrandomvariableswith
domains(D1,…,Dn)
• πisajointdistributionoverv
• σ=(σ1,…,σn)isavectorofmappingsσi:DiàAi
suchthatforalli andanymappingσi’:DiàAi,
∑
d∈D1××Dn
π (d)u(σ 1 (d1 ),, σ i (di ),, σ n (dn )) ≥
∑
d∈D1××Dn
π (d)u(σ 1 (d1 ),, σ i%(di ),, σ n (dn ))
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Correlatedvs Nashequilibrium
• Thesetofcorrelatedequilibria containsthe
setofNashequilibria
Theorem:
ForeveryNashequilibriumσ*,thereexistsa
correlatedequilibrium(v,π,σ)suchthatforeach
playeri,thedistributioninducedonAiisσi*.
• Proof:constructitwithDi=Ai,independent
signals(π(d)=σ*1(d1)x…xσ*n(dn))andidentity
mappingsσi
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Correlatedvs Nashequilibrium(2)
• Notallcorrelatedequilibria correspondtoa
Nashequilibrium
• Example,thecorrelatedequilibriuminthe
battle-of-sexgame
à Correlatedequilibriumisastrictlyweaker
notionthanNE
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Outline
• Othersolutionconceptsforsimultaneous
moves
– Stabilityofequilibrium
• Trembling-handperfectequilibrium
– Correlatedequilibrium
– Minimax theoremandzero-sumgames
– ε-Nashequilibrium
• Thelenderandborrowergame:introduction
andconceptsfromsequentialmoves
21
Maxmin strategy
• Maximize“worst-casepayoff”
Definition: Maxmin strategy
Themaxmin strategyforplayeri is arg max min ui (si , s−i )
s−i
si
defender
• Example
– Attacker:Notattack
– Defender:Defend
Defend Notdef
attacker
Attack
Notatt
-2,1
0,-1
2,-2
0,0
• ThisisnotaNashequilibrium!
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Maxmin strategy:intuition
• Playeri commitstostrategysi (possiblymixed)
• Player–i observesi andchooses-i tominimize
i’spayoff
• Playeri guaranteespayoffatleastequaltothe
maxmin value max min ui (si , s−i )
si
s−i
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Twoplayerszero-sumgames
• Definition:a2-playerszero-sumgameisagame
whereu1(s)=-u2(s)forallstrategyprofiles
– Sumofpayoffsconstantequalto0
• Example:Matchingpennies
• Defineu(s)=u1(s)
– Player1:maximizer
– Player2:minimizer
heads
Player2
heads
tails
1,-1
-1,1
Player1
tails
-1,1
1,-1
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Minimax theorem
Theorem: Minimax theorem(VonNeumann1928)
Foranytwo-playerzero-sumgamewithfiniteaction
space:
max min u(s1, s2 ) = min max u(s1, s2 )
s1
s2
s2
s1
• Thisquantityiscalledthevalue ofthegame
– correspondstothepayoffofplayer1atNE
• Maxmin strategiesó NEstrategies
• Canbecomputedinpolynomialtime(through
linearprogramming)
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Outline
• Othersolutionconceptsforsimultaneous
moves
– Stabilityofequilibrium
• Trembling-handperfectequilibrium
– Correlatedequilibrium
– Minimax theoremandzero-sumgames
– ε-Nashequilibrium
• Thelenderandborrowergame:introduction
andconceptsfromsequentialmoves
26
ε-Nashequilibrium
Definition: ε-Nashequilibrium
Forε>0,astrategyprofile(s1*,s2*,…,sN*)isanεNashequilibriumif,foreachplayeri,
ui(si*,s-i*)≥ui(si,s-i*)- ε forallsi ≠si*
• ItisanapproximateNashequilibrium
– Agentsindifferenttosmallgains(couldnotgain
morethanε byunilateraldeviation)
• ANashequilibriumisanε-Nashequilibrium
forallε!
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Outline
• Othersolutionconceptsforsimultaneous
moves
– Stabilityofequilibrium
• Trembling-handperfectequilibrium
– Correlatedequilibrium
– Minimax theoremandzero-sumgames
– ε-Nashequilibrium
• Thelenderandborrowergame:introduction
andconceptsfromsequentialmoves
28
“CashinaHat”game(1)
• Twoplayers,1and2
• Player1strategies:put$0,$1or$3inahat
• Then,thehatispassedtoplayer2
• Player2strategies:either“match”(i.e.,add
thesameamountofmoneyinthehat)ortake
thecash
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“CashinaHat”game(2)
Payoffs:
• Player1:
$0à $0
$1à ifmatchnetprofit$1,-$1ifnot
$3à ifmatchnetprofit$3,-$3ifnot
• Player2:
Match$1 à Netprofit$1.5
Match$3à Netprofit$2
Takethecashà $inthehat
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Lender&Borrowergame
• The“cashinahat”gameisatoyversionofthe
moregeneral“lenderandborrower”game:
– Lenders:Banks,VCFirms,…
– Borrowers:entrepreneurswithprojectideas
• Thelenderhastodecidehowmuchmoneyto
invest intheproject
• Afterthemoneyhasbeeninvested,theborrower
could
– Goforwardwiththeprojectandworkhard
– Shirk,andruntoMexicowiththemoney
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Simultaneousvs.SequentialMoves
• Whatisdifferentaboutthisgamewrt games
studieduntilnow?
• Itisasequentialmovegame
– Playerchoosesfirst,thenplayer2
• Timingisnotthekey
– ThekeyisthatP2observesP1’schoicebefore
choosing
– AndP1knowsthatthisisgoingtobethecase
32
Extensiveformgames
• Ausefulrepresentationofsuchgamesisgame
trees alsoknownastheextensiveform
– Eachinternalnodeofthetreewillrepresentthe
abilityofaplayertomakechoicesatacertain
stage,andtheyarecalleddecisionnodes
– Leafsofthetreearecalledendnodes and
representpayoffstobothplayers
• Normalformgamesà matrices
• Extensiveformgamesà trees
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“Cashinahat”representation
2
(0,0)
$0
1
$1 2
$3
2
$1
(1,1.5)
- $1
(-1,1)
$3
(3,2)
- $3
(-3,3)
Howtoanalyzesuchgame?
34
BackwardInduction
• Fundamentalconceptingametheory
• Idea:playersthatmoveearlyoninthegameshouldput
themselvesintheshoesofotherplayersplayinglater
à anticipation
• Lookattheendofthetreeandworkbacktowardstheroot
– Startwiththelastplayerandchosethestrategiesyielding
higherpayoff
– Thissimplifiesthetree
– Continuewiththebefore-lastplayeranddothesamething
– Repeatuntilyougettotheroot
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BackwardInductioninpractice(1)
2
(0,0)
$0
1
$1 2
$3
2
$1
(1,1.5)
- $1
(-1,1)
$3
(3,2)
- $3
(-3,3)
36
BackwardInductioninpractice(2)
2
(0,0)
$0
1
$1 2
(1,1.5)
$3
2
(-3,3)
37
BackwardInductioninpractice(3)
2
(0,0)
$0
1
$1 2
$3
2
$1
(1,1.5)
- $1
(-1,1)
$3
(3,2)
- $3
(-3,3)
Outcome:
Player1choosestoinvest$1,Player2matches
38
Theproblemwiththe
“lendersandborrowers”game
• Itisnotadisaster:
– Thelenderdoubledhermoney
– Theborrowerwasabletogoaheadwithasmallscaleproject
andmakesomemoney
• But,wewouldhavelikedtoendupinanotherbranch:
– Largerprojectfundedwith$3andanoutcomebetterforboth
thelenderandtheborrower
• Verysimilartoprisoner’sdilemna
• Whatpreventsusfromgettingtothislattergoodoutcome?
39
MoralHazard
• Oneplayer(theborrower)hasincentivestodothingsthatarenot
intheinterestsoftheotherplayer(thelender)
– Bygivingatoobigloan,theincentivesfortheborrowerwillbesuch
thattheywillnotbealignedwiththeincentivesonthelender
– Noticethatmoralhazardhasalsodisadvantagesfortheborrower
• Example:Insurancecompaniesoffers“full-risk”policies
– Peoplesubscribingforthispoliciesmayhavenoincentivestotake
care!
– Inpractice,insurancecompaniesforcemetobearsomedeductible
costs(“franchise”)
• Onepartyhasincentivetotakeariskbecausethecostisfeltby
anotherparty
• HowcanwesolvetheMoralHazardproblem?
40
Solution(1):Introducelaws
• Todaywehavesuchlaws:bankruptcylaws
• But,therearelimitstothedegreetowhich
borrowerscanbepunished
– Theborrowercansay:Ican’trepay,I’mbankrupt
– Andhe/she’smoreorlessallowedtohaveafresh
start
41
Solution(2):Limits/restrictionson
money
• Asktheborrowersaconcreteplan(business
plan)onhowhe/shewillspendthemoney
• Thisboilsdowntochangingtheorderofplay!
• Alsofacessomeissues:
– Lackofflexibility,whichisthemotivationtobean
entrepreneurinthefirstplace!
– Problemoftiming:itissometimeshardtopredict
up-frontalltheexpensesofaproject
42
Solution(3):Breaktheloanup
• Lettheloancomeinsmallinstallments
• Ifaborrowerdoeswellonthefirst
installment,thelenderwillgiveabigger
installmentnexttime
• Itissimilartotakingthisone-shotgameand
turnitintoarepeatedgame
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Solution(4):Changecontracttoavoid
shirk-- Incentives
• Theborrowercouldre-designthepayoffsofthegamein
casetheprojectissuccessful
2
2
(0,0)
$0
1
$1 2
$3
2
$1
$0
(1,1.5)
1
- $1
(-1,1)
$3
(3,2)
- $3
(-3,3)
(0,0)
$1 2
$3
2
$1
(1,1.5)
- $1
(-1,1)
$3
(1.9,3.1)
- $3
(-3,3)
• Profitdoesn’tmatchinvestmentbuttheoutcomeisbetter
– Sometimesasmallershareofalargerpiecanbe
biggerthanalargershareofasmallerpie
44
Absolutepayoffvs ROI
• Previousexample:largerabsolutepayoffin
thenewgameontheright,butsmallerreturn
oninvestment(ROI)
• Whichmetric(absolutepayofforROI)should
aninvestmentbanklookat?
45
Solution(5):Beyondincentives,
collaterals
• Theborrowercouldre-designthepayoffsofthe
gameincasetheprojectissuccessful
– Example:subtracthousefromrunawaypayoffs
2
(0,0)
$0
1
$1 2
$1
- $1
$3
2
(1,1.5)
(-1,1- HOUSE)
$3
(3,2)
- $3
(-3,3- HOUSE)
– Lowersthepayoffstoborroweratsometreepoints,
yetmakestheborrowerbetteroff!
46
Collaterals
• Theydohurtaplayerenoughtochange
his/herbehavior
èLoweringthepayoffsatcertainpointsofthe
game,doesnotmeanthataplayerwillbe
worseoff!!
• Collateralsarepartofalargerbranchcalled
commitmentstrategies
– Next,anexampleofcommitmentstrategies
47
NormanArmyvs.SaxonArmyGame
• Collateralsarepartofalargerbranchcalled
commitmentstrategies
• Backin1066,WilliamtheConquerorleadan
invasionfromNormandyontheSussexbeaches
• We’retalkingaboutmilitarystrategy
• Sobasicallywehavetwoplayers(thearmies)and
thestrategiesavailabletotheplayersare
whetherto“fight”or“run”
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NormanArmyvs.SaxonArmyGame
N
S
N
invade
fight
run
N
fight
run
fight
run
(0,0)
(1,2)
(2,1)
(1,2)
Let’sanalyzethegamewith
BackwardInduction
49
NormanArmyvs.SaxonArmyGame
N
S
N
invade
fight
run
N
fight
run
fight
run
(0,0)
(1,2)
(2,1)
(1,2)
50
NormanArmyvs.SaxonArmyGame
N
S
N
invade
fight
run
(1,2)
N
(2,1)
51
NormanArmyvs.SaxonArmyGame
N
S
N
invade
fight
run
N
fight
run
fight
run
BackwardInductiontellsus:
• Saxonswillfight
• Normanswillrunaway
(0,0)
(1,2)
(2,1)
(1,2)
WhatdidWilliamthe
Conquerordo?
52
NormanArmyvs.SaxonArmyGame
N
S
fight
run
N
N
Notburn
boats
run
fight
run
Burnboats
S
fight
fight
run
N
fight
N
fight
(0,0)
(1,2)
(2,1)
(1,2)
(0,0)
(2,1)
53
NormanArmyvs.SaxonArmyGame
N
S
fight
run
N
run
N
fight
N
fight
N
fight
(1,2)
(2,1)
Notburn
boats
Burnboats
S
fight
run
(0,0)
(2,1)
54
NormanArmyvs.SaxonArmyGame
S
N
(1,2)
Notburn
boats
Burnboats
S
(2,1)
55
NormanArmyvs.SaxonArmyGame
N
S
fight
run
N
N
Notburn
boats
run
fight
run
Burnboats
S
fight
fight
run
N
fight
N
fight
(0,0)
(1,2)
(2,1)
(1,2)
(0,0)
(2,1)
56
Commitment
• Sometimes,gettingridofchoicescanmakeme
betteroff!
• Commitment:
– Feweroptionschangethebehaviorofothers
• Theotherplayersmustknow aboutyour
commitments
– Example:Dr.Strangelovemovie
57