博弈论习题集

PROBLEMSETIOFGAMETHEORY

Statewhetherthefollowinggameshaveuniquepurestrategysolutions,andifsowhattheyareandhowtheycanbefound.

Player2

Left

Middle

Right

Up

4,3

2,7

0,4

Down

5,5

5,-1

-4,-2

Player1

Player2

Left

Middle

Right

Up

4,10

3,0

1,3

Down

0,0

2,10

1,3

Player1

Player2

Left

Middle

Right

Up

10,10

4,3

7,2

Down

5,6

8,10

6,12

Player1

Drawthenormalformgameforthefollowinggameandidentifyboththepure-andmixed-strategyequilibria.Inthemixed-strategyNashequilibriumdetermineeachfirm’sexpectedprofitlevelifitentersthemarket.

Therearetwofirmsthatareconsideringenteringanewmarket,andmustmaketheirdecisionwithoutknowingwhattheotherfirmhasdone.Unfortunatelythemarketisonlybigenoughtosupportoneofthetwofirms.Ifbothfirmsenterthemarket,thentheywilleachmakealossof£onlyonefirmentersthemarket,thatfirmwillearnaprofitof£50m,andtheotherfirmwilljustbreakeven.

Convertthefollowingextensiveformgameintoanormalformgame,andidentifytheNashequilibriaandsubgameperfectNashequilibria.Finally,whatistheNashequilibriumifbothplayersmaketheirmovessimultaneously

Consideraneconomyconsistingofonegovernmentandtwopeople.Letxibethechoiceofthepeople,wherexi∈X={xL,xM,xH},andi=1,2，andythechoiceofthegovernment,wherey∈Y={yL,yM,yH}.Thepayoffstothegovernment-householdaregivenbythevaluesofu1(x1,x2,y)andu2(x1,x2,y)=u1(x2,x1,y).Thesepayoffsareenteredinthefollowingtable:

One-periodpayoffstothegovernment-household(valuesofu(x1,x2,y))

(x1,x2,)

y

(xL,xL)

(xM,xM)

(xH,xH)

(xL,xH)

(xH,xL)

(xL,xM)

(xM,xL)

(xM,xH)

(xH,xM)

yL

3*

7

9

YM

5

10*

20

yH

0

4

15*

Suppose(*)denotesthat(x1,x2)isaNashequilibriumgiveny,thegovernment’spolicy.EntertheblankwithvaluerangessuchthattheNashequilibriaaresupported.

Supposethegovernmentmovesfirst,findNashEquilibria,thesubgameperfectNashequilibria,andthesubgameperfectoutcome.IstheoutcomeefficientWhy

ShowwhetherthereexistsNashequilibrium(inpurestrategies)fortheone-periodeconomywhenhouseholdsandthegovernmentmovesimultaneously.

Ifthehouseholdchoosefirst,doquestion(2)again.

AssumethattwoplayersarefacedwithRosenthal’scentipedegame.UseBayes’theoremtocalculatetheplayers’reputationforbeingco-operativeinthefollowingsituationsiftheyplayacross.

Atthebeginningofthegameeachplayerbelievesthatthereisa50/50chancethattheotherplayerisrationalorco-operative.Itisassumedthataco-operativeplayeralwaysplaysacross.Furthermoresupposethatarationalplayerwillplayacrosswithaprobabilityof

Attheirsecondmovetheplayersagainmoveacross.(Continuetoassumethattheprobabilitythatarationalplayerplaysacrossremainsequal.

Howwouldtheplayers’reputationhavechangedafterthefirstmovehadtheotherplayerbelievedthatrationalplayersalwaysplayacross.(Assumeallotherprobabilitiesremainthesame.)

Finally,howwouldtheplayers’reputationhavechangedafterthefirstmovehadtheotherplayerbelievedthatrationalplayersneverplayacross.(Againassumeallotherprobabilitiesremainthesame.)

AssumetherearemidenticalStackelbergleadersinanindustry,indexedj=1,…,m,andnidenticalStackelbergfollowers,indexedk=1,…,n.Allfirmshaveaconstantmarginalcostofcandnofixedcosts.Themarketprice,Q,isdeterminedaccordingtotheequationP=ɑ-Q,whereQistotalindustryoutput,andɑisaconstant.FindthesubgameperfectNashequilibriumsupplyfortheleadersandthefollowers.ConfirmtheduopolyresultsforbothCournotcompetitionandStackelbergcompetition,andthegeneralizedCournotresultfornfirmsderivedinExercise.

Assumethattherearei=1,…,nidenticalfirmsinanindustry,eachwithconstantmarginalcostsofcandnofixedcosts.Ifthemarketprice,P,isdeterminedbytheequationP=ɑ-Q,whereQistotalindustryoutputandɑisaconstant,determinetheCournot-Nashequilibriumoutputlevelforeachfirm.Wherehappensasn→∞

Findtheseparatingequilibriumbehaviourofthelow-costincumbentinthefollowingtwo-periodmodel.Theincumbenthasmarginalcostsequaltoeither£4or£2.Onlytheincumbentinitiallyknowsitsexactcosts.Theentrantobservestheincumbent’soutputdecisioninthefirstperiodandonlyentersthemarketinthesecondperiodifitbelievesthattheincumbenthashighmarginalcosts.Ifentrydoesoccur,thetwofirmsCournotcompete,andweassumethatatthisstageinthegametheincumbent’struecostsarerevealed.Price,P,isdeterminedbythefollowingequationP=10-Q,whereQisthecombinedoutputofthetwofirms.Finally,itisassumedthatthefirms’discountfactorisequalto.

Inthetextwearguedthataweakgovernmentcanexploittheprivatesector’suncertaintyaboutthegovernment’spreferencestopartiallyavoidtheinflationarybiasassociatedwithtime-inconsistentmonetarypolicy.Inthisexerciseweprovidedasimplemodelthatillustratesthisresult.

Assumethatthegovernment,viaitsmonetarypolicy,canperfectlycontrolinflation.Furthermorethegovernmentcanbeoneoftwotypes.Eitheritisstrongoritisweak.Astronggovernmentisonlyconcernedabouttherateofinflation,andsoneverinflatestheeconomy.Aweakgovernment,however,isconcernedaboutbothinflationandunemployment.Specially,itswelfareintime-periodtisgivenbythefollowingequation：

VtG=c-dOPt2-eUt,

whereoPtandUtaretheratesofinflationandunemploymentintime-periodtrespectively,andc,dandeareallpositiveparameters.Itisassumedthegovernmentdoesnotdiscountfuturewelfare,andsoaweakgovernmentattemptstomaximizethesumofitsper-periodwelfareoverallcurrentandfutureperiods.Theconstraintfacingthegovernmentisgivenbytheexpectations-augmentedPhillipscurve.Thisiswrittenas

OPt=a-bUt+OePt,

whereoPteistheexpecte