problem-set-2

耶鲁大学公开课博弈论课习答案

ECON159aSolutionSet2

KeithWannamaker

keith@http://wendang.chazidian.com

December9,2009

1

1.1PenaltyShotsRevisited(a)

Neitherplayerhasapuredominantstrategy.Fors1,MdoesbetterthanLforl,yetLdoesbetterthanMforr;RdoesbetterthanMforl,yetMdoesbetterthanRforr;RdoesbetterthanLforl,yetLdoesbetterthanRforr.Asimilarargumentcanbemadefors2.

1.2(b)

s2(m)isabestresponse(BR)fors1(M).However,s1(M)isnotaBRforanybeliefabouts2(l’sBRisR;m’sbetterresponsesareLorR;r’sBRisL).

1.3(c)

No.Sinces1(M)isnotaBRtos2(m),s1willchooseLorR.However,misnotaBRforLorR,thuss2shouldneverplaym.

1.4(d)

ANashequilibriumexistswhenthereisasetofstrategieswiththepropertythatnoplayerbene?tsbychangingstrategieswhileotherplayerskeeptheirstrategiesunchanged.Thereisnosuchset:s(L,m)

s(L,r)

s(M,l)

s(M,m)

s(M,r)

s(R,l)

s(R,m)

s(R,r)1--RLorRL--L2llm-mrr-

1

耶鲁大学公开课博弈论课习答案

2

2.1

PartnershipsRevisited

(a)

Werepeattheprocessofdi?erentiatingui,giventhepartnersdothesameamountofwork,s,but?rstsummingbothu1andu2todeterminethetotalrevenue:

u1+2(s1,s2)

f(s)f′(s)s??

====

2

[4(s1+s2+bs1s2)]?s21?s24(s+s+bs2)?s2?s28+8bs?4s21

s??isbetween2to3timestherationalizablee?ort(with0≤b≤1/4).

Themidpointofthee?ortrangesi(2)providesthemaximumshareablepro?tandminimume?ortcost.Examiningtheextremesofs1ands2showsthatsi(4)isnobetterthansi(0).

s14044s20444banyany01/4u1-8808u28-808

2.2(b)

s2

BR1(s2)

s???1

4

=

===

1+bs2

2

1+b12Player 1 best response

s1*s1**s1***

3.5

3

s1

2.5

2

1.5

1

0 0.05 0.1

b

0.15 0.2 0.25

Player1willworkhalfasdiligentlyasplayer2,ifplayer1knowss2=s??.Thee?ortcostoutweighsthebene?tofworkingmore.

2

耶鲁大学公开课博弈论课习答案

2.3(c)

s1s2

4

==1?1?

ss==BR1(s2)BR2(s1)

BR2(s1)

best response, b=-1/2

3.5

3

2.5

s2

2

1.5

1

0.5

0 0.5 1 1.5

2s1

2.5 3 3.5 4

?

s?1=s2=2

Revenue Net of Total Effort, b=-1/2

5 0-5-10-15-20-25-30-35

Iftheplayerscontractedtoprovidethesameamountofwork,theywouldbothwork

s??s??

b

11?==

1

2

,b1=?Nashequilibrium(s?)

42”Equalcontract”(s??)

41

ui(s?)

4016ui(s??)82

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耶鲁大学公开课博弈论课习答案

3

3.1NashEquilibriaandIterativeDeletion(a)

BandCarestrictlydominated,leavingT,B,L,andR.

3.2(b)

NEare(M,L)and(T,R).

3.3(c)

Strictlydominatedstrategiesareneverabestresponsetoanything;byde?nition,Nashequilibriumareabestresponsetosomething,soiterativelydeletingstrictlydominatedstrategieswillnevereliminateaNashequilibrium.

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4.1SplittingtheDollar(a)

NE(s1,s2),wheres1+s2=$10,aswellasNE($10,$10).

4.2(b)

NE($5.00,$5.00),NE($5.00,$5.01),NE($5.01,$5.00),NE($5.01,$5.01).

4.3(c)

For(a)TheNEinvolvingstrategieswithoutwhole-dollarunitsareeliminated.

For(b)TheequilibriabecomeNE($5,$5),NE($5,$6),NE($6,$5),NE($6,$6).

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