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GameTheory--
Lecture5
PatrickLoiseauEURECOMFall2016
1
Lecture3-4recap• DefinedmixedstrategyNashequilibrium• ProvedexistenceofmixedstrategyNashequilibriuminfinitegames
• DiscussedcomputationandinterpretationofmixedstrategiesNashequilibrium
• Definedanotherconceptofequilibriumfromevolutionarygametheory
àToday:introduceothersolutionconceptsforsimultaneousmovesgames
àIntroducesolutionsforsequentialmovesgames
2
Outline
• Othersolutionconceptsforsimultaneousmoves– Stabilityofequilibrium• Trembling-handperfectequilibrium
– Correlatedequilibrium–Minimax theoremandzero-sumgames– ε-Nashequilibrium
• Thelenderandborrowergame:introductionandconceptsfromsequentialmoves
3
Outline
• Othersolutionconceptsforsimultaneousmoves– Stabilityofequilibrium• Trembling-handperfectequilibrium
– Correlatedequilibrium–Minimax theoremandzero-sumgames– ε-Nashequilibrium
• Thelenderandborrowergame:introductionandconceptsfromsequentialmoves
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 5
TheLocationModel:payoffs
0
1
1/2
7035#ofyourtypeinyourtown
Utilityforplayeri • Theideais:– Ifyouareasmall
minority inyourtownyougetapayoffofzero
– Ifyouareinlargemajority inyourtownyougetapayoffof½
– Ifyouarewellintegrated yougetapayoffof1
• Peoplewouldliketoliveinmixedtowns,butiftheycannot,thentheyprefertoliveinthemajoritytown
6
Initialstate
• Assumetheinitialpictureisthisone
• Whatwillplayersdo?
7
TallplayerShortplayer
WestTown
EastTown
Firstiteration• Fortallplayers• There’saminorityof
easttown“giants”tobeginwith
à switchtoWesttown
• Forshortplayers• There’saminorityof
westtown“dwarfs”tobeginwith
àswitchtoEasttown
8
TallplayerShortplayer
WestTown
EastTown
Seconditeration
• Sametrend
• Stillafewplayerswhodidnotunderstand–Whatistheirpayoff?
9
TallplayerShortplayer
WestTown
EastTown
Lastiteration
• Peoplegotsegregated
• Buttheywouldhavepreferredintegratedtowns!–Why?Whathappened?– Peoplethatstartedinaminority(eventhoughnota“bad”minority)hadincentivestodeviate
10
TallplayerShortplayer
WestTown
EastTown
TheLocationModel:Nashequilibria
• TwosegregatedNE:– Short,E;Tall,W– Short,W;Tall,E
• IsthereanyotherNE?
11
Stabilityofequilibria• Theintegratedequilibriumisnotstable
– Ifwemoveawayfromthe50%ratio,evenalittlebit,playershaveanincentivetodeviateevenmore
– Weendupinoneofthesegregatedequilibrium• Thesegregatedequilibria arestable
– Introduceasmallperturbation:playerscomebacktosegregationquickly
• NotionofstabilityinPhysics:ifyouintroduceasmallperturbation,youcomebacktotheinitialstate
• Tippingpoint:– IntroducedbyGrodzins (WhiteflightsinAmerica)– ExtendedbyShelling(Nobelprizein2005)
12
Trembling-handperfectequilibrium
• Fully-mixedstrategy:positiveprobabilityoneachaction
• Informally:aplayer’sactionsi mustbeBRnotonlytoopponentsequilibriumstrategiess-i butalsotosmallperturbationsofthoses(k)-i. 13
Definition: Trembling-handperfectequilibriumA (mixed)strategyprofilesisatrembling-handperfectequilibriumifthereexistsasequences(0),s(1),…offullymixedstrategyprofilesthatconvergestowardssandsuchthatforallkandallplayeri,si isabestresponsetos(k)-i.
TheLocationModel
• Thesegregatedequilibria aretrembling-handperfect
• Theintegratedequilibriumisnottrembling-handperfect
14
Outline
• Othersolutionconceptsforsimultaneousmoves– Stabilityofequilibrium• Trembling-handperfectequilibrium
– Correlatedequilibrium–Minimax theoremandzero-sumgames– ε-Nashequilibrium
• Thelenderandborrowergame:introductionandconceptsfromsequentialmoves
15
Example:battleofthesexes
• NE:(O,O),(S,S)and((1/3,2/3),(2/3,1/3))– Themixedequilibriumhaspayoff2/3each
• Supposetheplayerscanobservetheoutcomeofafairtosscoinandconditiontheirstrategiesonthisoutcome– Newstrategiespossible:Oifhead,Siftails– Payoff1.5each
• Thefaircoinactsasacorrelatingdevice
2,1 0,00,0 1,2
Opera
Soccer
Opera
Player1
Player2Soccer
16
Correlatedequilibrium:generalcase
• Inthepreviousexample:bothplayersobservetheexactsamesignal(outcomeofthecointossrandomvariable)
• Generalcase:eachplayerreceivesasignalwhichcanbecorrelatedtotherandomvariable(cointoss)andtotheotherplayerssignal
• Model:– nrandomvariables(oneperplayer)– AjointdistributionoverthenRVs– NaturechoosesaccordingtothejointdistributionandrevealstoeachplayeronlyhisRV
à AgentcanconditionhisactiontohisRV(hissignal)
17
Correlatedequilibrium:definition
18
Definition: CorrelatedequilibriumAcorrelatedequilibriumofthegame(N,(Ai),(ui))isatuple(v,π,σ)where• v=(v1,…,vn)isatupleofrandomvariableswith
domains(D1,…,Dn)• πisajointdistributionoverv• σ=(σ1,…,σn)isavectorofmappingsσi:DiàAisuchthatforalli andanymappingσi’:DiàAi,
π (d)u(σ1(d1),,σ i (di ),,σ n (dn )) ≥d∈D1××Dn
∑ π (d)u(σ1(d1),, %σ i (di ),,σ n (dn ))d∈D1××Dn
∑
• Thesetofcorrelatedequilibria containsthesetofNashequilibria
• Proof:constructitwithDi=Ai,independentsignals(π(d)=σ*1(d1)x…xσ*n(dn))andidentitymappingsσi
Correlatedvs Nashequilibrium
19
Theorem:ForeveryNashequilibriumσ*,thereexistsacorrelatedequilibrium(v,π,σ)suchthatforeachplayeri,thedistributioninducedonAiisσi*.
Correlatedvs Nashequilibrium(2)
20
• Notallcorrelatedequilibria correspondtoaNashequilibrium
• Example,thecorrelatedequilibriuminthebattle-of-sexgame
à CorrelatedequilibriumisastrictlyweakernotionthanNE
Outline
• Othersolutionconceptsforsimultaneousmoves– Stabilityofequilibrium• Trembling-handperfectequilibrium
– Correlatedequilibrium–Minimax theoremandzero-sumgames– ε-Nashequilibrium
• Thelenderandborrowergame:introductionandconceptsfromsequentialmoves
21
Maxmin strategy
• Maximize“worst-casepayoff”
• Example– Attacker:Notattack– Defender:Defend
• ThisisnotaNashequilibrium!22
-2,1 2,-20,-1 0,0
Attack
Notatt
Defend
attacker
defenderNotdef
Definition:Maxmin strategyThemaxmin strategyforplayeri is argmax
simins− iui (si, s−i )
Maxmin strategy:intuition
• Playeri commitstostrategysi (possiblymixed)• Player–i observesi andchooses-i tominimizei’spayoff
• Playeri guaranteespayoffatleastequaltothemaxmin value
23
maxsimins− iui (si, s−i )
Twoplayerszero-sumgames
• Definition:a2-playerszero-sumgameisagamewhereu1(s)=-u2(s)forallstrategyprofiles– Sumofpayoffsconstantequalto0
• Example:Matchingpennies• Defineu(s)=u1(s)– Player1:maximizer– Player2:minimizer
24
heads tails
heads
tails
1,-1 -1,1
1,-1-1,1
Player1
Player2
Minimax theorem
• Thisquantityiscalledthevalue ofthegame– correspondstothepayoffofplayer1atNE
• Maxmin strategiesó NEstrategies• Canbecomputedinpolynomialtime(throughlinearprogramming) 25
Theorem:Minimax theorem(VonNeumann1928)Foranytwo-playerzero-sumgamewithfiniteactionspace: max
s1mins2u(s1, s2 ) =mins2
maxs1u(s1, s2 )
Outline
• Othersolutionconceptsforsimultaneousmoves– Stabilityofequilibrium• Trembling-handperfectequilibrium
– Correlatedequilibrium–Minimax theoremandzero-sumgames– ε-Nashequilibrium
• Thelenderandborrowergame:introductionandconceptsfromsequentialmoves
26
ε-Nashequilibrium
• ItisanapproximateNashequilibrium– Agentsindifferenttosmallgains(couldnotgainmorethanε byunilateraldeviation)
• ANashequilibriumisanε-Nashequilibriumforallε!
27
Definition: ε-NashequilibriumForε>0,astrategyprofile(s1*,s2*,…,sN*)isanε-Nashequilibriumif,foreachplayeri,
ui(si*,s-i*)≥ui(si,s-i*)- ε forallsi ≠si*
Outline
• Othersolutionconceptsforsimultaneousmoves– Stabilityofequilibrium• Trembling-handperfectequilibrium
– Correlatedequilibrium–Minimax theoremandzero-sumgames– ε-Nashequilibrium
• Thelenderandborrowergame:introductionandconceptsfromsequentialmoves
28
“CashinaHat”game(1)
• Twoplayers,1and2• Player1strategies:put$0,$1or$3inahat
• Then,thehatispassedtoplayer2
• Player2strategies:either“match”(i.e.,addthesameamountofmoneyinthehat)ortakethecash
29
“CashinaHat”game(2)
Payoffs:
• Player1:
• Player2:
$0à $0$1à ifmatchnetprofit$1,-$1ifnot$3à ifmatchnetprofit$3,-$3ifnot
Match$1à Netprofit$1.5Match$3à Netprofit$2Takethecashà $inthehat
30
Lender&Borrowergame
• The“cashinahat”gameisatoyversionofthemoregeneral“lenderandborrower”game:– Lenders:Banks,VCFirms,…– Borrowers:entrepreneurswithprojectideas
• Thelenderhastodecidehowmuchmoneytoinvest intheproject
• Afterthemoneyhasbeeninvested,theborrowercould– Goforwardwiththeprojectandworkhard– Shirk,andruntoMexicowiththemoney
31
Simultaneousvs.SequentialMoves
• Whatisdifferentaboutthisgamewrt gamesstudieduntilnow?
• Itisasequentialmovegame– Playerchoosesfirst,thenplayer2
• Timingisnotthekey– ThekeyisthatP2observesP1’schoicebeforechoosing
– AndP1knowsthatthisisgoingtobethecase
32
Extensiveformgames
• Ausefulrepresentationofsuchgamesisgametrees alsoknownastheextensiveform– Eachinternalnodeofthetreewillrepresenttheabilityofaplayertomakechoicesatacertainstage,andtheyarecalleddecisionnodes
– Leafsofthetreearecalledendnodes andrepresentpayoffstobothplayers
• Normalformgamesàmatrices• Extensiveformgamesà trees
33
“Cashinahat”representation
1
2
2
2
(0,0)
(1,1.5)
(-1,1)
(3,2)
(-3,3)
$0
$1
$3
$1
- $1
$3
- $3
Howtoanalyzesuchgame?34
BackwardInduction• Fundamentalconceptingametheory
• Idea:playersthatmoveearlyoninthegameshouldputthemselvesintheshoesofotherplayersplayinglater
à anticipation
• Lookattheendofthetreeandworkbacktowardstheroot– Startwiththelastplayerandchosethestrategiesyielding
higherpayoff– Thissimplifiesthetree– Continuewiththebefore-lastplayeranddothesamething– Repeatuntilyougettotheroot
35
BackwardInductioninpractice(1)
1
2
2
2
(0,0)
(1,1.5)
(-1,1)
(3,2)
(-3,3)
$0
$1
$3
$1
- $1
$3
- $3
36
BackwardInductioninpractice(2)
1
2
2
2
(0,0)
(1,1.5)
(-3,3)
$0
$1
$3
37
BackwardInductioninpractice(3)
1
2
2
2
(0,0)
(1,1.5)
(-1,1)
(3,2)
(-3,3)
$0
$1
$3
$1
- $1
$3
- $3
Outcome:Player1choosestoinvest$1,Player2matches38
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”)
• Onepartyhasincentivetotakeariskbecausethecostisfeltbyanotherparty
• HowcanwesolvetheMoralHazardproblem?40
Solution(1):Introducelaws
• Todaywehavesuchlaws:bankruptcylaws
• But,therearelimitstothedegreetowhichborrowerscanbepunished– Theborrowercansay:Ican’trepay,I’mbankrupt– Andhe/she’smoreorlessallowedtohaveafreshstart
41
Solution(2):Limits/restrictionsonmoney
• Asktheborrowersaconcreteplan(businessplan)onhowhe/shewillspendthemoney
• Thisboilsdowntochangingtheorderofplay!
• Alsofacessomeissues:– Lackofflexibility,whichisthemotivationtobeanentrepreneurinthefirstplace!
– Problemoftiming:itissometimeshardtopredictup-frontalltheexpensesofaproject
42
Solution(3):Breaktheloanup
• Lettheloancomeinsmallinstallments• Ifaborrowerdoeswellonthefirstinstallment,thelenderwillgiveabiggerinstallmentnexttime
• Itissimilartotakingthisone-shotgameandturnitintoarepeatedgame
43
Solution(4):Changecontracttoavoidshirk-- Incentives
• Theborrowercouldre-designthepayoffsofthegameincasetheprojectissuccessful
• Profitdoesn’tmatchinvestmentbuttheoutcomeisbetter– Sometimesasmallershareofalargerpiecanbe
biggerthanalargershareofasmallerpie
1
2
2
2
(0,0)
(1,1.5)
(-1,1)
(1.9,3.1)
(-3,3)
$0
$1
$3
$1
- $1
$3
- $3
1
2
2
2
(0,0)
(1,1.5)
(-1,1)
(3,2)
(-3,3)
$0
$1
$3
$1
- $1
$3
- $3
44
Absolutepayoffvs ROI
• Previousexample:largerabsolutepayoffinthenewgameontheright,butsmallerreturnoninvestment(ROI)
• Whichmetric(absolutepayofforROI)shouldaninvestmentbanklookat?
45
Solution(5):Beyondincentives,collaterals
• Theborrowercouldre-designthepayoffsofthegameincasetheprojectissuccessful– Example:subtracthousefromrunawaypayoffs
– Lowersthepayoffstoborroweratsometreepoints,yetmakestheborrowerbetteroff!
1
2
2
2
(0,0)
(1,1.5)
(-1,1- HOUSE)
(3,2)
(-3,3- HOUSE)
$0
$1
$3
$1
- $1
$3
- $3
46
Collaterals
• Theydohurtaplayerenoughtochangehis/herbehavior
èLoweringthepayoffsatcertainpointsofthegame,doesnotmeanthataplayerwillbeworseoff!!
• Collateralsarepartofalargerbranchcalledcommitmentstrategies– Next,anexampleofcommitmentstrategies
47
NormanArmyvs.SaxonArmyGame
• Collateralsarepartofalargerbranchcalledcommitmentstrategies
• Backin1066,WilliamtheConquerorleadaninvasionfromNormandyontheSussexbeaches
• We’retalkingaboutmilitarystrategy• Sobasicallywehavetwoplayers(thearmies)andthestrategiesavailabletotheplayersarewhetherto“fight”or“run”
48
NormanArmyvs.SaxonArmyGame
N S
N
N
(0,0)
(1,2)
(2,1)
(1,2)
invade
fight
run
fight
fight
run
run
Let’sanalyzethegamewithBackwardInduction
49
NormanArmyvs.SaxonArmyGame
N S
N
N
(0,0)
(1,2)
(2,1)
(1,2)
invade
fight
run
fight
fight
run
run
50
NormanArmyvs.SaxonArmyGame
N S
N
N
(1,2)
(2,1)
invade
fight
run
51
NormanArmyvs.SaxonArmyGame
N S
N
N
(0,0)
(1,2)
(2,1)
(1,2)
invade
fight
run
fight
fight
run
run
BackwardInductiontellsus:• Saxonswillfight• Normanswillrunaway
WhatdidWilliamtheConquerordo?
52
NormanArmyvs.SaxonArmyGame
N
S
N
N
(0,0)
(1,2)
(2,1)
(1,2)
fight
run
fight
fight
run
run
S
Notburnboats
Burnboats
fight
run
N
N
fight
fight
(0,0)
(2,1)
53
NormanArmyvs.SaxonArmyGame
N
S
N
N
(1,2)
(2,1)
fight
run fight
run
S
Notburnboats
Burnboats
fight
run
N
N
fight
fight
(0,0)
(2,1)
54
NormanArmyvs.SaxonArmyGame
N
S(1,2)
S
Notburnboats
Burnboats
(2,1)
55
NormanArmyvs.SaxonArmyGame
N
S
N
N
(0,0)
(1,2)
(2,1)
(1,2)
fight
run
fight
fight
run
run
S
Notburnboats
Burnboats
fight
run
N
N
fight
fight
(0,0)
(2,1)
56
Commitment
• Sometimes,gettingridofchoicescanmakemebetteroff!
• Commitment:– Feweroptionschangethebehaviorofothers
• Theotherplayersmustknow aboutyourcommitments– Example:Dr.Strangelovemovie
57