大学博弈论讲义Word下载.docx

大学博弈论讲义Word下载.docx

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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

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

-6,-6

0,-9

-9,0

-1,-1

thenwhatistheNashequilibrium?

Table3

0,-1

1,0

-1,3

IfthepayoffsarechangedasshowninTable3

2.Prisoners’dilemma

ParetoEfficiency:

AnallocationisParetoefficientifgoodscannotbereallocatedtomakesomeonebetteroffwithoutmakingsomeoneelseworseoff.

Twoprisonerswhowerepartnersinacrimewerebeingquestionedinseparaterooms.Eachprisonerhasachoiceofconfessingtothecrime（implicatingtheother）ordenying.Ifonlyoneconfesses,thenhewouldgofreeandhispartnerwillspend6monthsinprison.Ifbothprisonersdeny,thenbothwouldbeintheprisonfor1month.Ifbothconfess,theywouldbothbeheldforthreemonths.ThepayoffmatrixforthisgameisdepictedinTable4.

Table4

Confess

Deny

Theequilibriumoutcome

LetN=1to4asetofpossiblestatesofnature,andletS=1to4beasetofstrategydecidedbyyou.Nowyouhavetodecidewhichstrategyyouhavetochoosegiventhefollowingpayoffmatrix.

Table5

Nature

S=

1

2

3

4

S=You

N=Opponent

Inthiscaseyoudon'

tcarethepayoffofyouropponenti.e.nature.

3.1TheMaximinDecisionRuleorWaldcriterion

Welookfortheminimumpay-offsineachchoiceandthenmaximisingtheminimumpay-off

Letushighlightthemimimumpayoffforeachstrategy.

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.

Table6:

RegretMatrix

Maximum

Regret

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