大学博弈论讲义.docx
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大学博弈论讲义
Week11:
GameTheory
RequiredReading:
Schotterpp.229-260
LecturePlan
1.TheStaticGameTheory
NormalFormGames
SolutionTechniquesforSolvingStaticGames
DominantStrategy
NashEquilibrium
2.Prisoner’sDilemma
3.DecisionAnalysis
MaximimCriteria
MinimaxCriteria
4.DynamicOne-OffGames
ExtensiveFormGames
TheSub-GamePerfectNashEquilibrium
1.ThestaticGameTheory
Staticgames:
theplayersmaketheirmoveinisolationwithoutknowingwhatotherplayershavedone
1.1NormalFormGames
Ingametheorytherearetwowaysinwhichagamecanberepresented.
1st）Thenormalformgameorstrategicformgame
2nd）Theextensiveformgame
Anormalformgameisanygamewherewecanidentitythefollowingthreethings:
1.Players:
2.Thestrategiesavailabletoeachplayer.
3.ThePayoffs.Apayoffiswhataplayerwillreceiveattheendofthegamecontingentupontheactionsofallplayersinthegame.
Supposethattwopeople（AandB）areplayingasimplegame.Awillwriteoneoftwowordsonapieceofpaper,“Top”or“Bottom”.Atthesametime,Bwillindependentlywrite“left”or“right”onapieceofpaper.Aftertheydothis,thepaperswillbeexaminedandtheywillgetthepayoffdepictedinTable1.
Table1
B
A
Left
Right
Top
2,1
0,0
Bottom
0,0
1,2
IfAsaystopandBsaysleft,thenweexaminethetop-leftcornerofthematrix.Inthismatrix,thepayofftoA（B）isthefirst（Second）entryinthebox.Forexample,ifAwrites“top”andBwrites“left”payoffofA=1payoffofB=2.
Whatis/aretheequilibriumoutcome（s）ofthisgame?
1.2
NashEquilibriumApproachtoSolvingStaticGames
NashequilibriumisfirstdefinedbyJohnNashin1951basedontheworkofCournotin1893.
ApairofstrategyisNashequilibriumifA'schoiceisoptimalgivenB'schoice,andB'schoiceisoptimalgivenA'schoice.Whenthisequilibriumoutcomeisreached,neitherindividualwantstochangehisbehaviour.
FindingtheNashequilibriumforanygameinvolvestwostages.
1）identifyeachoptimalstrategyinresponsetowhattheotherplayersmightdo.
GivenBchoosesleft,theoptimalstrategyforAis
GivenBchoosesright,theoptimalstrategyforAis
GivenAchoosestop,theoptimalstrategyforBis
GivenAchoosesbottom,theoptimalstrategyforBis
Weshowthisbyunderlyingthepayoffelementforeachcase.
2）aNashequilibriumisidentifiedwhenallplayersareplayertheiroptimalstrategiessimultaneously
InthecaseofTable2,
IfAchoosestop,thenthebestthingforBtodoistochooseleftsincethepayofftoBfromchoosingleftis1andthepayofffromchoosingrightis0.IfBchoosesleft,thenthebestthingforAtodoistochoosetopasAwillgetapayoffof2ratherthan0.
ThusifAchoosestopBchoosesleft.IfBchoosesleft,Achoosestop.ThereforewehaveaNashequilibrium:
eachpersonismakingoptimalchoice,giventheotherperson'schoice.
Ifthepayoffmatrixchangesas:
Table2
B
A
Left
Right
Top
-6,-6
0,-9
Bottom
-9,0
-1,-1
thenwhatistheNashequilibrium?
Table3
B
A
Left
Right
Top
0,0
0,-1
Bottom
1,0
-1,3
IfthepayoffsarechangedasshowninTable3
2.Prisoners’dilemma
ParetoEfficiency:
AnallocationisParetoefficientifgoodscannotbereallocatedtomakesomeonebetteroffwithoutmakingsomeoneelseworseoff.
Twoprisonerswhowerepartnersinacrimewerebeingquestionedinseparaterooms.Eachprisonerhasachoiceofconfessingtothecrime（implicatingtheother）ordenying.Ifonlyoneconfesses,thenhewouldgofreeandhispartnerwillspend6monthsinprison.Ifbothprisonersdeny,thenbothwouldbeintheprisonfor1month.Ifbothconfess,theywouldbothbeheldforthreemonths.ThepayoffmatrixforthisgameisdepictedinTable4.
Table4
B
A
Confess
Deny
Confess
Deny
Theequilibriumoutcome
3.DecisionAnalysis
LetN=1to4asetofpossiblestatesofnature,andletS=1to4beasetofstrategydecidedbyyou.Nowyouhavetodecidewhichstrategyyouhavetochoosegiventhefollowingpayoffmatrix.
Table5
Nature
S=
1
2
3
4
1
2
2
0
1
2
1
1
1
1
3
0
4
0
0
4
1
1
3
0
S=You
N=Opponent
Inthiscaseyoudon'tcarethepayoffofyouropponenti.e.nature.
3.1TheMaximinDecisionRuleorWaldcriterion
Welookfortheminimumpay-offsineachchoiceandthenmaximisingtheminimumpay-off
Letushighlightthemimimumpayoffforeachstrategy.
Nature
S=
1
2
3
4
1
2
2
0
1
2
1
1
1
1
3
0
4
0
0
4
1
1
3
0
3.2TheMinimaxDecisionRuleorSavagecriterion
Onthisruleweneedtocomputethelossesorregretmatrixfromthepayoffmatrix.Thelossesaredefinedasthedifferencebetweentheactualpayoffandwhatthatpayoffwouldhavebeenhadthecorrectstrategybeenchosen.
Regret/Loss=Max.payoffineachcolumn–actualpayoff
ForexampleofN=1occursandS=1ischosen,theactualgain=2fromTable3.However,thebestactiongivenN=1isalsotochooseS=1whichgivesthebestgain=2.For（N=1,S=1）regret=0.
IfN=1occursbutS=2ischosen,theactualgain=1.However,thebestactiongivenN=1isalsotochooseS=1whichgivesthebestgain=2.For（N=1,S=2）regret=2-1.
Followingthesimilaranalysis,wecancomputethelossesforeachNandSandsocancomputetheregretmatrix.
Nature
S=
1
2
3
4
1
2
2
0
1
2
1
1
1
1
3
0
4
0
0
4
1
1
3
0
Table6:
RegretMatrix
Nature
Maximum
S=
1
2
3
4
Regret
1
2
3
4
Aftercomputingtheregretmatrix,welookforthemaximumregretofeachstrategyandthentakingtheminimumofthese.
Minimaxisstillverycautiousbutlesssothanthemaximin.
4.Dynamicone-offGames
Agamecanbedynamicfortworeasons.First,playersmaybeabletoobservetheactionsofotherplayersbeforedecidingupontheiroptimalresponse.Second,one-offgamemayberepeatedanumberoftimes.
4.1ExtensiveFormGames
Dynamicgamescannotberepresentedbypayoffmatriceswehavetouseadecisiontree（extensiveform）torepresentadynamicgame.
Startwiththeconceptofdynamicone-offgamethegamecanbeplayedforonlyonetimebutplayerscanconditiontheiroptimalactionsonwhatotherplayershavedoneinthepast.
Supposethattherearetwofirms（AandB）thatareconsideringwhetherornottoenteranewmarket.Ifbothfirmsenterthemarket,thentheywillmakealossof$10mil.Ifonlyonefirmentersthemarket,itwillearnaprofitof$50mil.SupposealsothatFirmBobserveswhetherFirmAhasenteredthemarketbeforeitdecideswhattodo.
Anyextensiveformgamehasthefollowingfourelementsincommon:
Nodes:
Thisisapositionwheretheplayersmusttakeadecision.Thefirstpositioniscalledtheinitialnode,andeachnodeislabelledsoastoidentifywhoismakingthedecision.
Branches:
Theserepresentthealternativechoicesthatthepersonfacesandsocorrespondtoavailableactions.
PayoffVectors:
Theserepresentthepayoffsforeachplayer,withthepayoffslistedintheorderofplayers.Whenwereachapayoffvectorthegameends.
Inperiod1,FirmAmakesitsdecisions.ThisisobservedbyFirmBwhichdecidestoenterorstayoutofthemarketinperiod2.Inthisextensiveformgame,FirmB’sdecisionnodesarethesub-game.ThismeansthatfirmBobservesFirmA’sactionbeforemakingitsowndecision.
4.2SubgamePerfectNashEquilibrium
SubgameperfectNashequilibriumisthepredictedsolutiontoadynamicone-offgame.Fromtheextensiveformofthisgame,wecanobservethattherearetwosubgames,onestartingfromeachofFirmB’sdecisionnodes.
Howcouldweidentifytheequilibriumoutcomes?
Inapplyingthisprincipletothisdynamicgame,westartwiththelastperiodfirstandworkbackwardthroughsuccessivenodesuntilwereachthebeginningofthegame.
Startwiththelastperiodofthegamefirst,wehavetwonodes.Ateachnode,FirmBdecideswhetherornotenteringthemarketbasedonwhatFirmAhasalreadydone.
Forexample,atthenodeof“FirmAenters”,FirmBwilleithermakealossof–$10mil（ifitenters）orreceive“0”payoff（ifitstaysout）;theseareshownbythepayoffvectors（-10,-10）and（50,0）.IfFirmBisrational,itwillstaysout
Thenode“FirmAenters”canbereplacedbythevector（50,0）.
Atthesecondnode“FirmAstaysout”,FirmAhasnotenteredthemarket.Thus,FirmBwilleithermakeaprofitof$50mil（ifitenters）orreceive“0”payoff（ifitstaysout）;theseareshownbythepayoffvectors（0,50）and（0,0）.IfFirmBisrational,itwillenterthuswecouldruleoutthepossibilityofbothfirmsstayout
Wecannowmovebacktotheinitialnode.HereFirmAhastodecidewhetherornottoenter.IfFirmBisrational,itisknownthatthegamewillneverreachthepreviously“crossed”vectors.FirmAalsoknowsthatifitenters,thegamewilleventuallyendat（Aenters,Bstaysout）whereAgets50andBgets0.Ontheotherhand,ifFirmAstaysout,thegamewillendat（Astaysout,Benters）whereAgets0andBgets50FirmAshouldenterthemarketatthefirststage.Theeventualoutcomeis（Aenters,Bstaysout）
Howtofindasubgameperfectequilibriumofadynamicone-offgame?
1.Startwiththelastperiodofthegamecrossouttheirrelevantpayoffvectors.
2.Replacetheprecedingnodesbytheuncrossedpayoffvectorsuntilyoureachtheinitialnode.
3.Theonlyuncrossedpayoffvector（s）is
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