problem-set-1

耶鲁大学公开课博弈论课习答案

ECON159aSolutionSet1

KeithWannamakerkeith@http://wendang.chazidian.comNovember15,2009

1Solution1

i,jsiSi

ui(s1,...sN)s?i

player

strategyofaparticularplayeri

setofstrategiesforplayeri

payout(utile)

strategychoicesforeveryoneelsebutplayeri

1.1(a)

AstrictlydominatedstrategyA(s′i)meansthat,regardlessoftheopponent’sstrategy,thereisahigherpayo?for(strictlydominating)strategyB(si).Formallyplayeri’sstrategys′iisstrictlydominatedby

′

playeri’sstrategysiifui(si,s?i)>ui(si,s?i)foralls?i.

1.2(b)

AweaklydominatedstrategyA(s′i)meansthat,regardlessoftheopponent’sstrategy,thereisatleastasgoodofapayo?for(weaklydominating)strategyB(si),and,foratleastoneoftheopponent’sstrategies,thereisahigherpayo?forstrategyB.Formallyplayeri’sstrategys′iisweaklydominatedbyplayeri’s

′

strategysiifbothui(si,s?i)>=ui(si,s?i)foralls?iandalsoui(si,s?i)>ui(s′i,s?i)foratleastones?i.

1.3(c)

player

C2RTisstrictlydominatedbyMforplayer1;Cisweakly(butnotstrictly)domaintedbyLforplayer2.

Lplayer1

TMB

1

耶鲁大学公开课博弈论课习答案

2

2.1

Solution2

(a)

Therearenostrictlydominatedstrategies.MisweaklydominatedbyDforplayer1andcisweaklydominatedbyrforplayer2.

2.2(b)

Ifplayer2assumesplayer1willneverplayweaklydominatedM,andplayer1assumesplayer2willneverplayweaklydominatedc,thesestrategiescanbedeleted.AfterdeletingMandc,DisweaklydominatedbyTforplayer1andrisweaklydominatedbylforplayer2.AsecondroundofdeletionwouldleavejustTandl.

2.3(c)

Inthe?rstroundofdeletion,theworst-caseutilewas1forbothplayers.Inthesecondroundofdeletion,again,theworst-caseutilwas1forbothplayers.Iterativelydeletingtheweaklydominatedstrategiesleftnothingbuttheworst-caseutilof1forbothplayers.

3

3.1

Solution3

(a)

Thepayo?matrixislistedintable3.a.AgraphofuiisincludedinFigure1.si(1)isstrictlydominatedbysi(2,3..7).Becauseofsymmetry,si(10)iscertainlynotanybetterthansi(1).si(8)andsi(9)areworsethansi(1)whentheopponentpicksanadjacentornearlyadjacentposition.

Table1:

530,7035,6540,6050,5050,5045,5540,6035,6530,70

3.a635,6540,6045,5550,5050,5040,6035,6530,7025,75

2345678910

190,1085,1580,2075,2570,3065,3560,4055,4550,50250,5080,2075,2570,3065,3560,4055,4550,5045,55320,8050,5070,3065,3560,4055,4550,5045,5540,60425,7530,7050,5060,4055,4550,5045,5540,6035,65740,6045,5550,5055,4560,4050,5030,7025,7520,80845,5550,5055,4560,4065,3570,3050,5020,8015,85950,5055,4560,4065,3570,3075,2580,2050,5010,901055,4560,4065,3570,3075,2580,2085,1590,1050,50

3.2(b)

Strategy1isweaklydominatedbystrategy2.Theworstuiforsi(1)istoreceiveonly5votes(otherplayerschooses(1)ands(2)),butotherwiseuiforsi(1)>=10asplayer1’svoteisnotsplit.Theworstuifor

2

耶鲁大学公开课博弈论课习答案

si(2)istoreceive10voteswhenanotherplayerchoosess(3)andthethirdplayerchoosess(1)ors(2),butotherwiseuiforsi(2)>10.si(2)>si(1)becausesi(2)includesatleastaportionofs(1)’s10votes.

si(1,s?i)si(1,3,2)si(1,2,3)

<===

si(2,s?i),exceptsi(2,3,2)si(2,2,3)10

Strategy1isalsoweaklydominatedbystrategy3,forthesamereason.Theworstuiforsi(3)istoreceive10votes(sj(2),sk(4)),butotherwisesi(3)givestheopportunityofmorevotes.

si(1,s?i)si(1,4,2)si(1,4,3)si(1,2,4)si(1,3,4)

<====

si(3,s?i),si(3,4,2)si(3,4,3)si(3,2,4)si(3,3,4)

except=10=15=10=15

Afteriterativelydeletingstrategies1and10,strategy2isnotdominatedbyanyotherpurestrategysiinthereducedgame.Therearealwayscaseswhereuisi(2)doesbetterthans′iwhensjandskareadjacentto′

sibecauseinthesi(2)case,s?ican’tbe1byde?nition.

si(3,2,4)si(4,3,5)si(5,3,6)si(6,4,7)si(7,5,7)si(8,4,8)si(9,2,8)

<<<<<<<3

si(2,2,4)si(2,3,5)si(2,3,6)si(2,4,7)si(2,5,7)si(2,4,8)si(2,2,8)

耶鲁大学公开课博弈论课习答案

4

4.1Solution4(a)

(SeeFarquharson,Thetheoryofvoting,orMcKelvey/Niemi)

s1(a)strictlydominatess2(b)ands2(c).s1(a)winsa7/9ofthetimeandintheother2cases,s1isirreveant.Assumings1(a),thepayo?forplayer2and3is:Member

ab

1,00,2

1,01,03c1,02,1Member2bc

Fors2,s2(c)weaklydominatess2(b)ands2(a).Fors3,s3(a)isweaklydominatedbys3(b)ands3(c).

4.2(b)

Afterremovings2(a),s2(b),ands3(a),thepredictedvoteiss(a,c,c).cwillwin,whichisworstfor1,eventhough1hadtiebreakerpower.

4