大学生博弈论 课堂讲稿 UNDERGRADUATE GAME THEORY LECTURE NOTES

2018

UNDERGRADUATEGAMETHEORY

LECTURENOTES

BY

OMERTAMUZ

CaliforniaInstituteofTechnology

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Acknowledgments

TheselecturenotesarepartiallyadaptedfromOsborneandRubinstein[

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],Maschler,SolanandZamir[

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],lecturenotesbyFedericoEchenique,andslidesbyDaronAcemogluandAsuOzdaglar.IamindebtedtoSeoYoung(Silvia)KimandZhuofangLifortheirhelpinfindingandcorrectingmanyerrors.Anycommentsorsuggestionsarewelcome.

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Contents

1WhatisaGame?

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2Finiteextensiveformgameswithperfectinformation

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2.1Tic-Tac-Toe

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2.2TheSweetFifteenGame

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2.3Chess

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2.4Definitionoffiniteextensiveformgameswithperfectinformation

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2.5Theultimatumgame

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2.6Equilibria

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2.7Thecentipedegame

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2.8Subgamesandsubgameperfectequilibria

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2.9Thedollarauction

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2.10Backwardinduction,Kuhn’sTheoremandaproofofZermelo’sTheorem

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3Strategicformgames

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3.1Definition

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3.2Nashequilibria

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3.3Classicalexamples

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3.4Dominatedstrategies

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3.5Repeatedeliminationofdominatedstrategies

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3.6Dominantstrategies

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3.7MixedequilibriaandNash’sTheorem

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3.8ProofofNash’sTheorem

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3.9Bestresponses

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3.10Tremblinghandperfectequilibria

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3.10.1Motivatingexample

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3.10.2Definitionandresults

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3.11Correlatedequilibria

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3.11.1Motivatingexample

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3.11.2Definition

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3.12Zero-sumgames

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3.12.1Motivatingexample

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3.12.2Definitionandresults

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4Knowledgeandbelief

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4.1Knowledge

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4.1.1Thehatsriddle

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4.1.2Knowledgespaces

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4.1.3Knowledge

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4.1.4Knowledgeintermsofself-evidenteventalgebras

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4.1.5Commonknowledge

37

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4.1.6Backtothehatsriddle

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4.2Beliefs

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4.2.1Amotivatingexample

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4.2.2Beliefspaces

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4.3Rationalizability

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4.4Agreeingtodisagree

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4.5Notrade

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4.6Reachingcommonknowledge

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4.7Bayesiangames

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5Auctions

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5.1Classicalauctions

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5.1.1Firstprice,sealedbidauction

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5.1.2Secondprice,sealedbidauction

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5.1.3Englishauction

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5.1.4Socialwelfare

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5.2Bayesianauctions

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5.2.1Secondprice,sealedbidauction

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5.2.2Firstprice,sealedbidauction

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5.3Truthfulmechanismsandtherevelationprinciple

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6Extensiveformgameswithchancemovesandimperfectinformation

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6.1Motivatingexample:traininspections

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6.2Definition

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6.3Purestrategies,mixedstrategiesandbehavioralstrategies

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6.4Beliefsystemsandassessments

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6.5Sequentialrationalityandsequentialequilibria

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6.6Tremblinghandperfectequilibrium

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6.7PerfectBayesianequilibrium

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6.8Cheaptalk

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6.8.1Example1

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6.8.2Example2

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6.8.3Example3

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6.9TheWalmartgame

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6.9.1Perfectinformation,oneround

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6.9.2Perfectinformation,manyrounds

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6.9.3Imperfectinformation,oneround

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6.9.4Imperfectinformation,manyrounds

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7Repeatedgames

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7.1Definition

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7.2Payoffs

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7.2.1Finitelyrepeatedgames

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7.2.2Infinitelyrepeatedgames:discounting

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7.2.3Infinitelyrepeatedgames:limitofmeans

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7.3Folktheorems

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7.3.1Examples

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7.3.2Enforceableandfeasiblepayoffs

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7.4Nashfolktheorems

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7.5Perfectfolktheorems

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7.5.1Perfectfolktheoremforlimitingmeans

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7.5.2Perfectfolktheoremsfordiscounting

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7.6Finitelyrepeatedgames

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7.6.1Nashequilibriaandfolktheorems

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7.6.2PerfectNashequilibriaandfolktheorems

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8Sociallearning

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8.1Bayesianhypothesistesting

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8.2Herdbehavior

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9Betterresponsedynamicsandpotentialgames

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9.1Betterresponsedynamics

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9.2Acongestiongame

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9.3Potentialgames

91

9.4Theconformismgame

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10Socialchoice

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10.1Preferencesandconstitutions

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10.2TheCondorcetParadox

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10.3Arrow’sTheorem

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10.4TheGibbard-SatterthwaiteTheorem

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Disclaimer

Thisanotatextbook.Thesearelecturenotes.

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1WhatisaGame?

Agameisamathematicalmodelofastrategicinteraction.Wewillbestudyingawidevarietyofgames,butallofthemwillhavethefollowingcommonelements.

•Players.Weoftenthinkofplayersaspeople,butsometimestheymodelbusinesses,teams,politicalparties,countries,etc.

•Choices.Playershavetomakeachoiceormultiplechoicesbetweendifferentactions.Aplayer’sstrategyisherruleforchoosingactions.

•Outcomes.Whentheplayersaredonechoosing,anoutcomeisrealizedandthegameends.Thisoutcomedependsonthechoices.Examplesofoutcomesinclude“player1wins,”“FloragetsadollarandMilesgetstwodollars,”or“anuclearwarstartsandeveryonedies.”

•Preferences.Playershavepreferencesoveroutcomes.Forexample,Floramayprefertheoutcome“FloragetstwodollarsandMilesgetsnothing”overtheoutcome“MilesgetsadollarandFloragetsnothing.”Milesmayhavetheoppositepreference.

Twoimportantfeaturesmakeagamestrategic:first,thefactthatoutcomesaredeter-minedbyeveryone’sactions,ratherthanbytheactionsofjustoneplayer.Second,thatplayershavedifferentpreferences.Thiscreatestensions,whichmakegamesinteresting.

Gamesdifferinmanyaspects.

•Timing.Doplayerschooseonce(e.g.,rock-paper-scissors),oragainandagainovertime(e.g.,chess)?Inthelattercase,doesthegameeventuallyend,ordoesitcontinueforever?Dotheychoosesimultaneously,orinturn?

•Observations.Canplayersobserveeachother’schoices?

•Uncertainty.Istheoutcomerandom,orisitadeterministicfunctionoftheplayers’actions?Dosomeplayershaveinformationthattheothersdonot?

Itisimportanttonotethatagamedoesnotspecifywhattheplayersactuallydo,butonlywhattheiroptionsareandwhattheconsequencesare.Unlikeanequationwhich(maybe)hasauniquesolution,theansweringamesismuchlessclearcut.Asolutionconceptisawaytothinkaboutwhattheyplayersmightdecidetodo.Itisnotpartofthedescriptionofthegame,anddifferentsolutionconceptscanyielddifferentpredictionsforthesamegame.

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2Finiteextensiveformgameswithperfectinformation

Wewillstartbystudyingasimplefamilyofgames,whichincludesmanythatareindeedgamesinthelaypersonmeaningoftheword.Inthesegamesplayerstaketurnsmakingmoves,allplayersobserveallpastmoves,nothingisrandom,andthegameendsaftersomefixednumberofmovesorless.Wewillpresentsomeexamplesandthendefinethisclassofgamesformally.

2.1Tic-Tac-Toe

Twopeopleplaythefollowinggame.Athree-by-threesquaregridisdrawnonapieceofpaper.Thefirstplayermarksasquarewithan“x”,thenthesecondplayermarksasquarefromthoseleftwithan“o”,etc.Thewinneristhefirstplayertohavemarksthatformeitherarow,acolumnoradiagonal.

Doesthefirstplayerhaveastrategythatassuresthatshewins?Whataboutthesecondplayer?

2.2TheSweetFifteenGame

Twopeopleplaythefollowinggame.Therearecardsonthetablenumberedonethroughnine,facingup,andarrangedinasquare.Thefirstplayermarksacardwithan“x”,thenthesecondplayermarksacardfromthoseleftwithan“o”,etc.Thewinneristhefirstplayertohavethreecards(outofthethreeormorethattheyhavepicked)thatsumtoexactlyfifteen.

Doesthefirstplayerhaveastrategythatassuresthatshewins?Whataboutthesecondplayer?

2.3Chess

Weassumethestudentsarefamiliarwithchess,butthedetailsofthegamewill,infact,notbeimportant.Wewillchoosethefollowing(non-standard)rulesfortheendingofchess:thegameendseitherbythecapturingofaking,inwhichcasethecapturingsidewinsandtheotherloses,orelseinadraw,whichhappenswhenthereaplayerhasnolegalmoves,ormorethan100turnshaveelapsed.

Assuch,thisgameshasthefollowingfeatures:

•Therearetwoplayers,whiteandblack.

•Thereare(atmost)100timesperiods.

•Ineachtimeperiodoneoftheplayerschoosesanaction.Thisactionisobservedbytheotherplayer.

•Thesequenceofactionstakenbytheplayerssofardetermineswhatactionstheactiveplayerisallowedtotake.

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•Everysequenceofalternatingactionseventuallyendswitheitheradraw,oroneoftheplayerswinning.

Wesaythatwhitecanforceavictoryif,foranymovesthatblackchooses,whitecanchoosemovesthatwillendinitsvictory.Zermeloshowedin1913[

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]thatinthegameofchess,asdescribedabove,oneofthefollowingthreeholds:

•Whitecanforceavictory.

•Blackcanforceavictory.

•Bothwhiteandblackcanforceadraw.

Wewillprovethislater.

Exercise2.1.Thesametheoremappliestotic-tac-toe.Whichofthethreeholdsthere?

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2.4Definitionoffiniteextensiveformgameswithperfectinforma-

tion

Ingeneral,anextensiveformgame(withperfectinformation)GisatupleG=(N,A,H,P,{ui}i∈N)where

1.Nisafinitesetofplayers.

2.Aisafinitesetofactions.

3.Hisafinitesetofallowedhistories.ThisisasetofsequencesofelementsofAsuch

thatifh∈HtheneveryprefixofhisalsoinH.eachh=(a1,a2,...,an)isaseriesofallowedlegalmovesinthegame.

4.ThesetofterminalhistoriesZ⊆HisthesetofsequencesinHthatarenotsubse-quencesofothersinH.ThusZisthesetofhistoriesatwhichthegameends.NotethatwecanspecifyHbyspecifyingZ;HisthesetofsubsequencesofsequencesinZ.

5.PisafunctionfromH\ZtoN.WhenP(h)=ithenitisplayeri’sturntoplayafterhistoryh.

6.Foreachplayeri∈N,uiisafunctionfromtheterminalhistoriestoR.Thenumberui(h)istheutilitythatplayeriassignstotheterminalhistoryh.Playersareassumedtopreferhigherutilities.Notethatthenumbersthemselvesdonotmatter(fornow);onlytheirorderingmatters.

WedenotebyA(h)theactionsavailabletoplayerP(h)afterhistoryh:

A(h)={a∈A:ha∈H}.

Astrategyforplayeriisamapσifromtheset{h∈H:P(h)=i}ofhistorieshatwhichP(h)=itothesetofactionsA.Astrategyprofiles={si}i∈Nconstitutesastrategyforeachplayer.Givenastratgyprofileweknowhowplayersaregoingtoplay,andwedenotebyh(s)thepathofplay,i.e.,thehistorythatisrealized.Wealsodenotebyui(s)theutilitythatplayerirecievesunderthishistory.

2.5Theultimatumgame

Intheultimatumgameplayer1makesanoffera∈{0,1,2,3,4}toplayer2.Player2eitheracceptsorrejects.Ifplayer2acceptsthenshereceivesadollarsandplayer1receives4−adollars.If2rejectsthenbothgetnothing.Thisishowthisgamecanbewritteninextensiveform:

1.N={1,2}.

2.A={0,1,2,3,4,a,r}.

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3.Z={0a,1a,2a,3a,4a,0r,1r,2r,3r,4r}.

4.O={(0,0),(0,4),(1,3),(2,2),(3,1),(4,0)}.Eachpaircorrespondstowhatplayers1re-ceivesandwhatplayer2receives.

5.Forb∈{0,1,2,3,4},u1(ba)=4−b,u2(ba)=bandu1(br)=u2(br)=0.

6.P(Φ)=1,P(0)=P(1)=P(2)=P(3)=P(4)=2.

Astrategyforplayer1isjustachoiceamong{0,1,2,3,4}.Astrategyforplayer2isamapfrom{0,1,2,3,4}to{a,r}:player2’sstrategydescribeswhetherornotsheacceptsorrejectsanygivenoffer.

Remark2.2.Acommonmistakeistothinkthatastrategyofplayer2isjusttochooseamong{a,r}.Butactuallyastrategyisacompletecontingencyplan,whereanactionischosenforeverypossiblehistoryinwhichtheplayerhastomove.

2.6Equilibria

Givenastrategyprofiles={si}i∈N,wedenoteby(s−i,s)thestrategyprofileinwhichi’sstrategyischangedfromsitosandtherestremainthesame.

Astrategyprofiles*isaNashequilibriumifforalli∈Nandstrategysiofplayeriitholdsthat

i,si)≤ui(s*).

Whensistheequilibriumh(s)isalsoknownastheequilibriumpathassociatedwiths.

Example:intheultimatumgame,considerthestrategyprofileinwhichplayer1offers3,andplayer2accepts3or4andrejects0,1or2.Itiseasytocheckthatthisisanequilibriumprofile.

2.7Thecentipedegame

Inthecentipedegametherearentimeperiodsand2players.Theplayersalternateinturns,andateachturneachplayercaneitherstop(S)orcontinue(C),exceptatthelastturn,wheretheymuststop.Now,thereisapiggybankwhichinitiallyhasinit2dollars.Inthebeginningofeachturn,thisamountdoubles.Ifaplayerdecidestostop(whichshemustdoinperiodn),sheisawardedthreefourthofwhat’sinthebank,andtheotherplayerisawardedtheremainder.Ifaplayerdecidestocontinue,theamountinthebankdoubles.

Hence,inperiodm,aplayerisawarded3

4·2·2mifshedecidedtostop,andtheotherplayer

4·2·2m.

isgiven1

m=1

m=2

m=3

m=4

m=5

m=6

m=7

m=8

m=9

3

2

12

8

48

32

192

128

768

512

player1

1

6

4

24

16

96

64

384

256

1536

player2

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Exercise2.3.Definethecentipedegameformally,forn=4.Howmanystrategiesdoeseachplayerhave?MakesureyouunderstandRemark

2.2

beforeansweringthis.

Exercise2.4.ShowthatthestrategyprofileinwhichbothplayersplaySineverytimeperiodisanequilibrium.

Theorem2.5.IneveryNashequilibrium,player1playsSinthefirstperiod.

Proof.Assumebycontradictionthatplayer1playsCinthefirstperiodundersomeequi-libriums.Thenthereissomeperiodm>1inwhichSisplayedforthefirsttimeontheequilibriumpath.ItfollowsthattheplayerwhoplayedCinthepreviousperiodisawarded

period,

·2·2m.Butshecouldhavebeenawarded=·2·2mbyplayingSintheprevious

andthereforesisnotanequilibrium.

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2.8Subgamesandsubgameperfectequilibria

AsubgameofagameG=(N,A,H,P,{ui}i∈N)isagamethatstartsafteragivenfinitehistoryh∈H.Formally,thesubgameG(h)associatedwithh=(h1,...,hn)∈HisG(h)=(N,A,Hh,P,{ui}i∈N),where

Hh={(a1,a2,...):(h1,...,hn,a1,a2,...)∈H}.

ThefunctionsPanduiareasbefore,justrestrictedtotheappropriatesubdomains.

AstrategysofGcanlikewiseusedtodefineastrategyshofG(h).Wewilldropthehsubscriptswheneverthisdoesnotcreate(toomuch)confusion.

AsubgameperfectequilibriumofGisastrategyprofiles*suchthatforeverysubgameG(h)itholdsthats*(moreprecisely,itsrestrictiontoHh)isaNashequilibriumofG(h).WewillproveKuhn’sTheorem,whichstatesthateveryfiniteextensiveformgamewithperfectinformationhasasubgameperfectequilibrium.WewillthenshowthatZermelo’sTheoremfollowsfromKuhn’s.

Asanexample,considerthefollowingColdWargameplayedbetweentheUSAandtheUSSR.First,theUSSRdecideswhetherornottostationmissilesinCuba.Ifitdoesnot,thegameendswithutility0forall.Ifitdoes,theUSAhastodecideiftodonothing,inwhichcasetheutilityis1fortheUSSRand-1fortheUSA,ortostartanuclearwar,inwhichcasetheutilityis-1,000,000forall.

Exercise2.6.Findtwoequilibriaforthisgame,oneofwhichissubgameperfect,andonewhichisnot.

Exercise2.7.Findtwoequilibriaoftheultimatumgame,oneofwhichissubgameperfect,andonewhichisnot.

Animportantpropertyoffinitehorizongamesistheonedeviationproperty.Beforeintroducingitwemakethefollowingdefinition.

Letsbeastrategyprofile.Wesaythatsisaprofitabledeviationfromsforplayeriat

historyhifsisastrategyforthesubgameGsuchthatui(s−i,s)>ui(s).

Notethatastrategyprofilehasnoprofitabledeviationsifandonlyifitisasubgameperfectequilibrium.

Theorem2.8(Theonedeviationprinciple).LetG=(N,A,H,P,{ui}i∈N)beafiniteextensiveformgamewithperfectinformation.Letsbeastrategyprofilethatisnotasubgameperfectequilibrium.Therethereexistssomehistoryhandaprofitabledeviations-iforplayeri=P(h)inthesubgameG(h)suchthats-i(k)=si(k)forallk=/h.

Proof.Letsbeastrategyprofilethatisnotasubgameperfectequilibrium.Thenthereisa

subgameG(h)andastrategysforplayeri=P(h)suchthatsisaprofitabledeviationfori

inG(h).Denotes′=(s−i,s),andnotethatui(s′)>ui(s).Lethbeahistorythatismaximal

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inlengthamongallhistorieswiththisproperty.Letibegivenbys-i(k)=si(k)forallk=/h,

ands-i(h)=s(h).Bythemaximaldepthpropertyofhwehavethatiisstillaprofitable

deviation,sinceotherwiseiwouldhaveaprofitabledeviationinsomesubgameofG(h).Wethushavethats-iisaprofitabledeviationforG(h)thatdiffersfromsiinjustonehistory.

2.9Thedollarauction

Twoplayersparticipateinanauctionforadollarbill.Player1actsintheoddperiods,andplayer2intheevenperiods.Bothplayersstartwithazerobid.Ineachperiodtheplayingplayercaneitherstayorquit.Ifshequitstheotherplayergetsthebill,bothpaythehighesttheyhavebidsofar,andthegameends.Ifshestays,shemustbid10centshigherthantheotherplayer’slastbid(exceptinthefirstperiod,whenshemustbid5cents)andthegamecontinues.Ifoneofthebidsexceeds100dollarsthegameends,thepersonwhomadethehighestbidgetsthedollar,andbothpaythehighesttheyhavebidsofar.Soassumingbothplayersstay,inthefirstperiodthefirstplayerbids5cents.Inthesecondperiodthesecondplayerbids15cents.Inthethirdperiodthefirstplayerbids25cents,etc.

Exercise2.9.Doesthisgamehaveequilibria?Subgameperfectequilibria?

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2.10Backwardinduction,Kuhn’sTheoremandaproofofZermelo’sTheorem

LetG=(N,A,H,P,{ui}i∈N)beanextensiveformgamewithperfectinformation.RecallthatA(Φ)isthesetofallowedinitialactionsforplayeri=P(Φ).Foreachb∈A(Φ),letsG(b)besomestrategyprofileforthesubgameG(b).Givensomea∈A(Φ),wedenotebysathestrategyprofileforGinwhichplayeri=P(Φ)choosestheinitialactiona,andforeach

actionb∈A(Φ)thesubgameG(b)isplayedaccordingtosG(b).Thatis,s(Φ)=aandfor

everyplayerj,b∈A(Φ)andbh∈H\Z,s(bh)=s(b)(h).

Lemma2.10(BackwardInduction).LetG=(N,A,H,P,{ui}i∈N)beafiniteextensiveformgamewithperfectinformation.Assumethatforeachb∈A(Φ)thesubgameG(b)hasasub-gameperfectequilibriumsG(b).Leti=P(Φ)andletabeamaximizeroverA(Φ)ofui(sG(a)).ThensaisasubgameperfectequilibriumofG.

Proof.Bytheonedeviationprinciple,weonlyneedtocheckthatsadoesnothavedeviationsthatdifferatasinglehistory.Soletsdifferfromsaatasinglehistoryh.

Ifhistheemptyhistorythens=sG(b)forb=si(Φ).Inthiscaseui(sa)>ui(s)=ui(sG(b)),bythedefinitionofaasthemaximizerofui(sG(a)).

Otherwise,hisequaltobh′forsomeb∈A(Φ)andh′∈Hb,andui(s)=ui(s).ButsincesaisasubgameperfectequilibriumwhenrestrictedtoG(b)therearenoprofitabledeviations,

andtheproofiscomplete.

Kuhn[

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]provedthefollowingtheorem.

Theorem2.11(Kuhn,1953).Everyfiniteextensiveformgamewithperfectinformationhasasubgameperfectequilibrium.

GivenagameGwithallowedhistoriesH,denotebyl(G)themaximallengthofanyhistoryinH.

ProofofTheorem

2.11.

Weprovetheclaimbyinductiononl(G).Forl(G)=0theclaimisimmediate,sincethetrivialstrategyprofileisanequilibrium,andtherearenopropersubgames.AssumewehaveprovedtheclaimforallgamesGwithl(G)<n.

Letl(G)=n,anddenotei=P(Φ).Foreachb∈A(Φ),letsG(b)besomesubgameperfectequilibriumofG(b).Theseexistbyourinductiveassumption,asl(G(b))<n.

Leta*∈A(Φ)beamaximizerofui(sa*).ThenbytheBackwardInductionLemmasa*isasubgameperfectequilibriumofG,andourproofisconcluded.

GivenKuhn’sTheorem,Zermelo’sTheorem,asstatedbelow,admitsasimpleproof.

Theorem2.12(Zermelo).LetGbeafiniteextensiveformgamewithtwoplayersandwhereu1=−u2andu1(h)∈{−1,0,1}.Thenexactlyoneofthefollowingthreehold:

1.Thereexistsastragegysforplayer1suchthatu1(s,s2)=1forallstrategiess2of

player2.

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2.Thereexistsastragegysforplayer2suchthatu2(s1,s)=1forallstrategiess1of

player2.

3.Thereexiststrategiess,sforplayers1and2suchthatu1(s,s2)≥0andu2(s1,s)≥0

forallstrategiess1,s2ofplayers1and2.

Proof.Lets*beasubgameperfectequilibriumofanyfiniteextensiveformgamewithtwoplayersandwhereu1=−u2andu1(h)∈{−1,0,1}.Considerthesethreecases.Ifu1(s*)=1thenforanysB

u2(s,s2)≤u2(s*)=−1.

Butu2≥−1,andsou2(s,s2)=−1.Thatis,player1canforcevictorybyplayings.The

sameargumentshowsthatifu2(s*)=1thenblackcanforcevictory.Finally,ifu1(s*)=0thenforanys2

u2(s,s2)≤u2(s*)=0,

sou2(s,s2)iseither0or−1.Bythesameargumentu1(s1,s)iseither0or−1foranys1,

andsowehaveproventheclaim.

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3Strategicformgames

3.1Definition

Agameinstrategicform(ornormalform)isatupleG=(N,{Si}i∈N,{ui}i∈N)where

•Nisthesetofplayers.

•Siisthesetofactionsorstrategiesavailabletoplayeri.WedenotebyS=niSithesetofstrategyprofiles.

•Thefunctionui:S→Risplayeri’sutility(orpayoff)foreachstrategyprofile.

Wewillassumethatplayershavetheobviouspreferencesoverutility:moreispreferedtoless.WesaythatGisfiniteifNisfiniteandSisfinite.

3.2Nashequilibria

Givenastrategyprofiles,aprofitabledeviationforplayeriisastrategytisuchthat

ui(s−i,ti)>ui(s−i,si).

AstrategyprofilesisaNashequilibriumifnoplayerhasaprofitabledeviation.ThesearealsocalledpureNashequilibria,forreasonsthatwewillseelater.Theyareoftenjustcalledequilibria.

3.3Classicalexamples

•Extensiveformgamewithperfectinformation.LetG=(N,A,H,P,{ui}i∈N)beanextensiveformgamewithperfectinformation,where,insteadoftheusualout-comesandpreferences,eachplayerhasautilityfunctionui:Z→Rthatassignsherautilityateachterminalnode.LetG′bethestrategicformgamegivenbyG′=(N′,{Si}i∈N,{ui}i∈N),where

–N′=N.

–SiisthesetofG-strategiesofplayeri.

–Foreverys∈S,ui(s)istheutilityplayerigetsinGattheterminalnodeatwhichthegamearrivewhenplayersplaythestrategyprofiles.

Wehavethusdonenothingmorethanhavingwrittenthesamegameinadifferentform.Note,however,thatnoteverygameinstrategicformcanbewrittenasanextensiveformgamewithperfectinformation.

Exercise3.1.Showthats∈SisaNashequilibriumofGiffitisaNashequilibriumofG′.

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Notethatadisadvantageofthestrategicformisthatthereisnonaturalwaytodefinesubgamesorsubgameperfectequilibria.

•Matchingpennies.Inthisgame,andinthenextfew,therewillbetwoplayers:arowplayer(R)andacolumnplayer(C).Wewillrepresentthegameasapayoffmatrix,showingforeachstrategyprofiles=(sR,sC)thepayoffsuR(s),uC(s)oftherowplayerandthecolumnplayer.

H

T

HT

1,0

0,1

0,1

1,0

Inthisgameeachplayerhastochooseeitherheads(H)ortails(T).Therowplayerwantsthechoicestomatch,whiletherowplayerwantsthemtomismatch.

Exercise3.2.ShowthatmatchingpennieshasnopureNashequilibria.

•Prisoners’dilemma.

Twoprisonersarefacedwithadilemma.Acrimewascommittedintheprison,andtheyaretheonlytwowhocouldhavedoneit.Eachprisonerhastomakeachoicebetweentestifyingagainsttheother(andthusbetrayingtheother)andkeepinghermouthshut.Intheformercasewesaythattheprisonerdefected(i.e.,betrayedtheother),andinthelattershecooperated(withtheotherprisoner,notwiththepolice).

Ifbothcooperate(i.e.,keeptheirmouthsshut),theywillhavetoservetheremainderoftheirsentences,whichare2yearseach.Ifbothdefect(i.e.,agreetotestifyagainsteachother),eachwillserve3years.Ifonedefectsandtheothercooperates,thedefectorwillbereleasedimmediately,andthecooperatorwillserve10yearsforthecrime.

Assumingthataplayer’sutilityisminusthenumberofyearsserved,thepayoffmatrix