Explanation Given: The payoff matrix of a game is given by, [ 2 5 3 − 6 ] Approach: Let [ a b c d ] be the payoff matrix for a non-strictly determined game. Then the optional mixed strategy for the row player is given by P = [ p 1 p 2 ] Where, p 1 = d − c a + d − b − c p 2 = 1 − p 1 And the optimal mixed strategy for the column player is given by, Q = [ q 1 q 2 ] Where, q 1 = d − b a + d − b − c q 2 = 1 − q 1 Furthermore, the value of the game is given by the expected value of the game, E = P A Q , where P and Q are the optional mixed strategies for the row and column players, respectively. Thus, E = P A Q = a d − b c a + d − b − c Calculation: Compare the given matrix with a standard payoff matrix. a = 2 b = 5 c = 3 d = − 6 Write the expression to calculate the first term of optional mixed strategy for the row. p 1 = d − c a + d − b − c Substitute 2 for a , 5 for b , 3 for c and − 6 for d in the above expression. p 1 = − 6 − 3 2 − 6 − 5 − 3 = − 9 12 = − 3 4 Write the expression to calculate the second term of optional mixed strategy for the row. p 2 = 1 − p 1 Substitute − 3 4 for p 1 in the above expression. p 2 = 1 − ( − 3 4 ) = 4 + 3 4 = 7 4 Write the expression for the optional mixed strategy for the row. P = [ p 1 p 2 ] Substitute − 3 4 for p 1 and 7 4 for p 2 in the above expression. P = [ − 3 4 7 4 ] Write the expression to calculate the first term of optional mixed strategy for the column

Finite Mathematics for the Managerial, Life, and Social Sciences-Custom Edition

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ISBN: 9781305283831

Author: Tan

Publisher: Cengage Learning

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Chapter 9.5, Problem 12E

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The optimal strategies P and Q for the row and column players and expected payoff E of each matrix game.

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Chapter 9.5, Problem 11E

Chapter 9.5, Problem 13E

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Finite Mathematics for the Managerial, Life, and Social Sciences-Custom Edition

Ch. 9.1 - What is a finite stochastic process? What can you...Ch. 9.1 - Prob. 2CQ Ch. 9.1 - Consider a transition matrix T for a Markov chain...Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E

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To find:

The optimal strategies P and Q for the row and column players and expected payoff E of each matrix game.