Module4GameTheoryToaccompany

QuantitativeAnalysisforManagement,EleventhEdition,byRender,Stair,andHannaPowerPointslidescreatedbyCherry1Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallLearningObjectivesUnderstandtheprinciplesofzero-sum,two-persongames.Analyzepurestrategygamesandusedominancetoreducethesizeofthegame.Solvemixedstrategygameswhenthereisnosaddlepoint.Aftercompletingthischapter,studentswillbeableto:2Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallChapterOutlineM4.1 IntroductionM4.2 LanguageofGamesM4.3TheMinimaxCriterionM4.4 PureStrategyGamesM4.5 MixedStrategyGamesM4.6 Dominance3Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallGame:acontestinvolvingtwoormoredecisionmakers,eachofwhomwantstowin.Gametheory:thestudyofhowoptimalstrategiesareformedinconflict.Gametheoryhasbeeneffectivelyusedfor:WarstrategiesUnionnegotiatorsCompetitivebusinessstrategiesIntroduction4Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallWhatisagame?ElementsPlayersRulesStrategiesPayoffs5Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallGamemodels-Gamesclassifiedby:NumberofplayersSumofallpayoffsNumberofstrategiesemployedIfcooperationispermitted?Azerosumgameimpliesthatwhatisgainedbyoneplayerislostfortheother.Introduction

(continued)6Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallLanguageofGamesConsideraduopolycompetitivebusinessmarketinwhichonecompanyisconsideringadvertisinginhopesofluringcustomersawayfromitscompetitor.Thecompanyisconsideringradioand/ornewspaperadvertisements.

Let’susegametheorytodeterminethebeststrategy.7Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallLanguageofGames(continued)STOREX’sPAYOFFsY’sstrategy1(useradio)Y’sstrategy2(usenewspaper)X’sstrategy1(useradio)35X’sstrategy2(usenewspaper)1-2Belowisthepayoffmatrix(asapercentofchangeinmarketshare)forStoreX.ApositivenumbermeansthatXwinsandYloses,whileanegativenumberimpliesYwinsandXloses.Note:AlthoughXisconsideringtheadvertisements(thereforetheresultsfavorX),Ymustplaythegame.8Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallLanguageofGames(continued)StoreX’sStrategyStoresY’sStrategyOutcome(%changeinmarketshare)X1:RadioY1:RadioXwins3Yloses3X1:RadioY2:NewspaperXwins5Yloses5X2:NewspaperY1:RadioXwins1Yloses1X2:NewspaperY2:NewspaperXloses2Ywins2Note:AlthoughXisconsideringtheadvertisements(thereforetheresultsfavorX),Ymustplaythegame.9Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallTheMinimaxCriterionTheminimaxcriterionisusedinatwo-personzero-sumgame.Eachpersonshouldchoosethestrategythatminimizesthemaximumloss.

Note:Thisisidenticaltomaximizingone’sminimumgains.10Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallTheMinimaxCriterion(continued)Theuppervalueofthegameisequaltotheminimumofthemaximumvaluesinthecolumns.

Thelowervalueofthegameisequaltothemaximumoftheminimumvaluesintherows.

11Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallMinmaxSolutionSTOREX’sPAYOFFsY1(radio)Y2(newspaper)MinimumX1(radio)353X2(newspaper)1-2-2Maximum35MinimumofmaximumsMaximumofminimums12Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallSTOREX’sPAYOFFsY1(radio)Y2(newspaper)MinimumX1(radio)353X2(newspaper)1-2-2Maximum35TheMinimaxCriterion(continued)Saddlepoint:Bothupperandlowervaluesare3.A

saddlepoint

conditionexistsiftheupperandlowervaluesareequal.Thisiscalledapurestrategybecausebothplayerswillfollowthesamestrategy.13Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallSTOREX’sPAYOFFsY1(radio)Y2(newspaper)MinimumX1(radio)1066X2(newspaper)-122-12Maximum106TheMinimaxCriterion(continued)SaddlepointLet’slookatasecondexampleofapurestrategygame.LowervalueUppervalueWheneverasaddlepointispresent,thestrategyaplayershouldfollowwillalwaysbethesame,regardlessofthestrategyoftheotherplayer.14Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallMixedStrategyGameAmixedstrategygameexistswhenthereisnosaddlepoint.Eachplayerwillthenoptimizetheirexpectedgainbydeterminingthepercentoftimetouseeachstrategy.

Note:Theexpectedgainisdeterminedusinganapproachverysimilartotheexpectedmonetaryvalueapproach.15Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallMixedStrategyGames(continued)Y1(P)Y2(1-P)ExpectedGainX1(Q)424P+2(1-P)X2(1-Q)1101p+10(1-P)ExpectedGain4Q+1(1-Q)2Q+10(1-Q)Eachplayerseekstomaximizehis/herexpectedgainbyalteringthepercentoftime(PorQ)thathe/sheuseeachstrategy.SetthesetwoequationsequaltoeachotherandsolveforQSetthesetwoequationsequaltoeachotherandsolveforP16Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallMixedStrategyGames(continued)4P+2(1-P)=1P+10(1-P)

4P–2P–1P+10P=10–2

P=8/11and1-P=3/11

Expectedpayoff:1P+10(1-P)

=1(8/11)+10(3/11)

=3.46

4Q+1(1-Q)=2Q+10(1-Q)

4Q–1Q–2Q+10Q=10–1

Q=9/11and1-Q=2/11

Expectedpayoff:2Q+10(1-Q)=2(9/11)+10(2/11)

=3.4617Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallDominanceDominanceisaprinciplethatcanbeusedtoreducethesizeofgamesbyeliminatingstrategiesthatwouldneverbeplayed.

Note:Astrategycanbeeliminatedifallitsgame’soutcomesarethesameorworsethanthecorrespondingoutcomesofanotherstrategy.18Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallDominance

(continued)Y1Y2X143X2220X311Y1Y2X143X2220InitialgameX3isadominatedstrategyGameafterremovalofdominatedstrategy19Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallDominance

(continued)Y1Y2Y3Y4X1-546-3X2-262-20InitialgameGameafterremovalofdominatedstrategiesY1Y4X1-5-3X2-2-2020Copyright?2012PearsonEducation,Inc.publishingasPrenticeHallProcedureforSolvingTwo-Person,ZeroSum