博弈论基础练习2

1.(CH.7.U9.)Considerthefollowinggame:

COLUMNyesno

yesx;x0;1no1;01;1ROW

(a)ForwhatvaluesofxdoesthisgamehaveauniqueNashequilibrium?Whatisthis

equilibrium?(b)Forwhatvaluesofxdoesthisgamehaveamixed-strategyNashequilibrium?With

whatprobability,expressedintermsofx,doeseachplayerplayYESinthismixed-strategyequilibrium?(c)Forthevaluesofxfoundinthepreviouspart,isthegameanexampleofanassurance

game,agameofchicken,oragamesimilartotennis?Explain.

2.(CH.10.U1.)InascenefromthemovieManhattanMurderMystery,WoodyAllenand

DianeKeatonareatahockeygameinMadisonSquareGarden.Sheisobviouslynotenjoyingherself,buthetellsher:“Rememberourdeal.Youstayherewithmefortheentirehockeygame,andnextweekIwillcometotheoperawithyouandstayuntiltheend.”Later,weseethemcomingoutoftheMetintothedesertedLincolnCenterPlazawhileinsidethemusicisstillplaying.Keatonisvisiblyupset:“Whataboutourdeal?Istayedtotheendofthehockeygame,andsoyouweresupposedtostaytilltheendoftheopera.”Allenanswers:“YouknowIcan’tlistentotoomuchWagner.Attheendofthefirstact,IalreadyfelttheurgetoinvadePoland.”

Commentonthestrategicchoicesmadeherebyusingyourknowledgeofthetheoryofstrategicmovesandcredibility.

3.(CH.11.U3.)Atowncouncilconsistsofthreememberswhovoteeveryyearontheirown

salaryincreases.TwoYesvotesareneededtopasstheincrease.Eachmemberwouldlikeahighersalarybutwouldliketovoteagainstitherselfbecausethatlooksgoodtothevoters.Specifically,thepayoffsofeachareasfollows:

Raisepasses,ownvoteisNo:10Raisefails,ownvoteisNo:5Raisepasses,ownvoteisYes:4Raisefails,ownvoteisYes:0

Votingissimultaneous.Writedownthe(three-dimensional)payofftable,andshowthatintheNashequilibriumtheraisefailsunanimously.Examinehowarepeatedrelationshipamongthememberscansecurethemsalaryincreaseseveryyearif(1)everymemberservesa3-yearterm,(2)everyyearinrotationoneofthemisupforreelection,and(3)thetownspeoplehavershortmemories,rememberingonlythevotesonthesalary-increasemotionofthecurrentyearandnotthoseofpastyears.

IntroductiontoGameTheory-WasedaUniversityR´obertVeszteg

Exercises2

4.(CH.12.U5.)Agroupof12countriesisconsideringwhethertoformamonetaryunion.

Theydifferintheirassessmentsofthecostsandbenefitsofthismove,buteachstandstogainmorefromjoining,andlosemorefromstayingout,whenmoreoftheothercountrieschoosetojoin.Thecountriesarerankedinorderoftheirlikingforjoining,1havingthehighestpreferenceforjoiningand12theleast.Eachcountryhastwoactions,INandOUT.Let

B(i,n)=2.2+n−i

bethepayofftocountrywithrankingiwhenitchoosesINandnothershavechosenIN.Let

S(i,n)=i−n

bethepayofftocountrywithrankingiwhenitchoosesOUTandnothershavechosenIN.(a)Showthatforcountry1,INisthedominantstrategy.

(b)HavingeliminatedOUTforcountry1,showthatINbecomesthedominantstrategy

forcountry2.(c)Continuinginthisway,showthatallcountrieswillchooseIN.

(d)ContrastthepayoffsinthisoutcomewiththosewhereallchooseOUT.Howmany

countriesaremadeworseoffbytheformationoftheunion?

IntroductiontoGameTheory-WasedaUniversityR´obertVeszteg