基础博弈论大学英文讲义.docx
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基础博弈论大学英文讲义.docx
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基础博弈论大学英文讲义
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.Replacetheprecedingnodesbytheuncrossedpayoffvectorsuntilyoureach
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