Explanation Given Information: The provided payoff matrix is: [ − 1 1 2 − 1 2 − 2 ] 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 − 1 2 − 2 ] 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 4 1 0 1 1 1 1 1 0 ] Now, write down the dual, which is the LP problem corresponding to the new matrix as: Maximize p = x + 3 y + z . Subject to x ≥ 0 , y ≥ 0 , z ≥ 0 , 4 x + y ≤ 1 and x + 3 y + 4 z ≤ 1 . Solve the dual problem using the standard simplex method as: x y z s t p s 1 3 4 1 0 0 1 t 4 1 0 0 1 0 1 p − 1 − 1 − 1 0 0 1 0 Apply row operation, 4 R 1 − R 2 and 4 R 3 + R 2

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ISBN: 9781337280426

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 5.5, Problem 23E

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

[−112−1 2−2 ]

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Chapter 5.5, Problem 22E

Chapter 5.5, Problem 24E

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

Solution Summary: The author calculates the matrix game using the simplex method. The solution to the primal problem is p=511 and the column player's optimal strategy is

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

[−112−1 2−2 ]

Want to see the full answer?

Chapter 5 Solutions

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

Solution Summary: The author calculates the matrix game using the simplex method. The solution to the primal problem is p=511 and the column player's optimal strategy is

Videos

To calculate: The game with the given payoff matrix and the provided payoff matrix is: [−112−1 2−2 ]

Want to see the full answer?

Chapter 5 Solutions

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

[−112−1 2−2 ]