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In a zero-sum game, the pure strategies of two players constitute a saddle point if the corresponding entry of the payoff matrixis simultaneously a maximum ... Home QuantitativeTechniquesForManagementTutorial PURESTRATEGIES:GAMEWITHSADDLEPOINT-QuantitativeTechniquesformanagement PURESTRATEGIES:GAMEWITHSADDLEPOINT-QuantitativeTechniquesformanagement «PreviousTopics Two-personZero-sumGameDangerLevelAverageStockLevel NextTopics» MixedStrategies:GamesWithoutSaddlePointDominancePropertySolvingProblemOnTheComputerWithTora InterviewQuestions PracticeTest AllTutorials JobSearch InterviewQuestion AllTutorials JobSearch PracticeTest Inazero-sumgame,thepurestrategiesoftwoplayersconstituteasaddlepointifthecorrespondingentryofthepayoffmatrixissimultaneouslyamaximumofrowminimaandaminimumofcolumnmaxima.Thisdecision-makingisreferredtoastheminimax-maximinprincipletoobtainthebestpossibleselectionofastrategyfortheplayers. Inapay-offmatrix,theminimumvalueineachrowrepresentstheminimumgainforplayerA.PlayerAwillselectthestrategythatgiveshimthemaximumgainamongtherowminimumvalues.TheselectionofstrategybyplayerAisbasedonmaximinprinciple.Similarly,thesamepay-offisalossforplayerB.ThemaximumvalueineachcolumnrepresentsthemaximumlossforPlayerB.PlayerBwillselectthestrategythatgiveshimtheminimumlossamongthecolumnmaximumvalues. TheselectionofstrategybyplayerBisbasedonminimaxprinciple.Ifthemaximinvalueisequaltominimaxvalue,thegamehasasaddlepoint(i.e.,equilibriumpoint).ThusthestrategyselectedbyplayerAandplayerBareoptimal. GameTheorySaddlePointProblem Thegametheoryforsaddlepointproblemisshownbelow Example:Considertheexampletosolvethegamewhosepay-offmatrixisgiveninthefollowingtableasfollows: GameProblem Thegameisworkedoutusingminimaxprocedure.Findthesmallestvalueineachrowandselectthelargestvalueofthesevalues.Next,findthelargestvalueineachcolumnandselectthesmallestofthesenumbers.Theprocedureisshowninthefollowingtable. MinimaxProcedure IfMaximumvalueinrowisequaltotheminimumvalueincolumn,thensaddlepointexists. MaxMin=MinMax 1=1 Therefore,thereisasaddlepoint. Thestrategiesare, PlayerAplaysStrategyA1,(AA1). PlayerBplaysStrategyB1,(BB1). Valueofgame=1. Example:Solvethegamewiththepay-offmatrixforplayerAasgivenintable. GameProblem Solution:Findthesmallestelementinrowsandlargestelementsincolumnsasshownintable. MinimaxProcedure Selectthelargestelementinrowandsmallestelementincolumn.Checkfortheminimaxcriterion, MaxMin=MinMax 1=1 Therefore,thereisasaddlepointanditisapurestrategy. OptimumStrategy: PlayerAA2Strategy PlayerBB1Strategy Thevalueofthegameis1. Example:CheckwhetherthefollowinggameisgiveninTable,determinableandfair. GameProblem Solution:ThegameissolvedusingmaximincriteriaasshowninTable. MaximinProcedure Thegameisstrictlyneitherdeterminablenorfair. Example:IdentifytheoptimalstrategiesforplayerAandplayerBforthegame,givenbelowinTable.Alsofindifthegameisstrictlydeterminableandfair. GameProblem Thegameisstrictlydeterminableandfair.Thesaddlepointexistsandthegamehasapurestrategy.Theoptimalstrategiesaregiveninthefollowingtable. OptimalStrategies Example:SolvethegamewiththepayoffmatrixgivenintableanddeterminethebeststrategiesforthecompaniesAandBandfindthevalueofthegameforthem. GameProblem Solution:Thematrixissolvedusingmaximincriteria,asshownintablebelow. MaximinProcedure Therefore,thereisasaddlepoint. OptimumstrategyforcompanyAisA1and OptimumstrategyforcompanyBisB1orB3. «PreviousTopics Two-personZero-sumGameDangerLevelAverageStockLevel NextTopics» MixedStrategies:GamesWithoutSaddlePointDominancePropertySolvingProblemOnTheComputerWithTora QuantitativeTechniquesformanagementRelatedTutorials BusinessAnalystTutorial ProjectManagementTutorial BusinessEnvironmentTutorial MarketingResearchTutorial BusinessManagementforFinancialAdvisersTutorial ResearchMethodologyTutorial BusinessEthicsTutorial InternationalBusinessManagementTutorial QuantitativeTechniquesformanagementRelatedInterviewQuestions BusinessAnalystInterviewQuestions ProjectManagementInterviewQuestions BusinessEnvironmentInterviewQuestions MarketingResearchInterviewQuestions BusinessManagementforFinancialAdvisersInterviewQuestions ResearchMethodologyInterviewQuestions BusinessEthicsInterviewQuestions BusinessManagementInterviewQuestions InternationalBusinessManagementInterviewQuestions 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