Explanation Given: The payoff matrix of a game is given by, [ 12 10 6 14 ] 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 = 12 b = 10 c = 6 d = 14 Write the expression to calculate the first term of optional mixed strategy for the row. p 1 = d − c a + d − b − c Substitute 12 for a , 10 for b , 6 for c and 14 for d in the above expression. p 1 = 14 − 6 12 + 14 − 10 − 6 = 8 10 = 4 5 Write the expression to calculate the second term of optional mixed strategy for the row. p 2 = 1 − p 1 Substitute 4 5 for p 1 in the above expression. p 2 = 1 − 4 5 = 5 − 4 5 = 1 5 Write the expression for the optional mixed strategy for the row. P = [ p 1 p 2 ] Substitute 4 5 for p 1 and 1 5 for p 2 in the above expression

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Chapter 9.CRE, Problem 28CRE

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Chapter 9.CRE, Problem 27CRE

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