Explanation Given: The payoff matrix with a 2 × 2 matrix game is given by, [ a 11 a 12 a 21 a 22 ] Approach: Let [ a b c d ] be the payoff matrix for a not 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: Consider the matrix, A = [ a 11 a 12 a 21 a 22 ] Write the expression to calculate the first term of the optional mixed strategy for the row player. p 1 = d − c a + d − b − c Substitute a 11 for a , a 12 for b , a 21 for c and a 22 for d in the above expression. p 1 = a 22 − a 21 a 11 + a 22 − a 12 − a 21 Write the expression to calculate the second term of the optional mixed strategy for the row player. p 2 = 1 − p 1 Substitute ( a 22 − a 21 a 11 + a 22 − a 12 − a 21 ) for p 1 in the above expression. p 2 = 1 − a 22 − a 21 a 11 + a 22 − a 12 − a 21 The optional mixed strategy for the row player is given by P = [ p 1 p 2 ] Write the expression to calculate the second term of the optional mixed strategy for the column player. q 1 = d − b a + d − b − c Substitute a 11 for a , a 12 for b , a 21 for c and a 22 for d in the above expression. q 1 = a 22 − a 12 a 11 + a 22 − a 12 − a 21 Write the expression to calculate the second term of the optional mixed strategy for the column player. q 2 = 1 − q 1 Substitute a 22 − a 12 a 11 + a 22 − a 12 − a 21 for q 1 in the above expression. q 2 = 1 − a 22 − a 12 a 11 + a 22 − a 12 − a 21 Write the expression to calculate the expected value of the game. E = a d − b c a + d − b − c Substitute a 11 for a , a 12 for b , a 21 for c and a 22 for d in the above expression. E = a 11 a 22 − a 12 a 21 a 11 + a 22 − a 12 − a 21 ... ( 1 ) Write the expression to calculate the expected value of the game

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Chapter 9.5, Problem 22E

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The expected value of the game is given by E=PAQ.

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Chapter 9.5, Problem 21E

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The expected value of the game is given by E = P A Q .

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The expected value of the game is given by E=PAQ.