Explanation Given Information: The provided payoff matrix is: [ − 1 1 2 2 − 1 − 2 1 2 0 ] Formula used: Formula to solve the matrix game: Step1: Reduce the matrix in the dominance form. Step2: Add a constant number k to the matrix. Step3: Now write 1 s to the right of the matrix and below the matrix and then write down the associated standard maximization problem. Step4: Now solve for the primal problem by use of the simplex method. Step5: The obtain the optical strategies and the expected value: Operation for the column Strategy: (1): Now express the dual problem as a column vector . (2): Normalize the matrix by dividing each entry of the solution vector by p (which is sum of all the entries). (3): Insert zeros in positions corresponding to the rows that where deleted in during the reduction to dominance form. Operation for the Row Strategy: (1): Now express the dual problem as a row vector. (2): Normalize the matrix by dividing each entry by p and once again be the add all the entries. (3): Insert zeros in positions corresponding to the rows that where deleted in during the reduction to dominance form. Formula for the value of the game can be calculated as e = 1 p − k . Calculation: Consider the provided payoff matrix is: [ − 1 1 2 2 − 1 − 2 1 2 0 ] Add k = 2 to each entry and put 1 s to the right and be below: Represent this above problem by the matrix as: [ 1 3 4 1 4 1 0 1 3 4 2 1 1 1 1 0 ] Now, write down the dual, which is the LP problem corresponding to the new matrix as: Maximize p = x + y + z . Subject to 3 x + 4 y + 2 z ≤ 1 , 4 x + y ≤ 1 , x + 3 y + 4 z ≤ 1 and x ≥ 0 , y ≥ 0 , z ≥ 0 . Solve the dual problem using the standard simplex method as: x y z s t u p s 1 3 4 1 0 0 0 1 t 4 1 0 0 1 0 0 1 u 3 4 2 0 0 1 0 1 p − 1 − 1 − 1 0 0 0 1 0 Apply row operation, 4 R 1 − R 2 , 4 R 3 − 3 R 2 and 4 R 4 + R 2 . x y z s t u p s 0 11 16 4 − 1 0 0 3 x 4 1 0 0 1 0 0 1 u 0 13 8 0 − 3 4 0 1 p 0 − 3 − 4 0 1 0

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Chapter 6.5, Problem 25E

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To calculate: The game with the given payoff matrix and provided payoff matrix is:

[−1122−1−2120]

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Chapter 6.5, Problem 24E

Chapter 6.5, Problem 26E

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The game with the given payoff matrix and provided payoff matrix is: [ − 1 1 2 2 − 1 − 2 1 2 0 ]

Solution Summary: The author explains the formula used to solve the matrix game. The solution to the primal problem is p=25.

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To calculate: The game with the given payoff matrix and provided payoff matrix is:

[−1122−1−2120]

Want to see the full answer?

Chapter 6 Solutions

The game with the given payoff matrix and provided payoff matrix is: [ − 1 1 2 2 − 1 − 2 1 2 0 ]

Solution Summary: The author explains the formula used to solve the matrix game. The solution to the primal problem is p=25.

Videos

To calculate: The game with the given payoff matrix and provided payoff matrix is: [−1122−1−2120]

Want to see the full answer?

Chapter 6 Solutions

To calculate: The game with the given payoff matrix and provided payoff matrix is:

[−1122−1−2120]