Explanation Given Information: The provided payoff matrix is: [ 1 − 1 2 1 2 0 0 1 1 ] Formula used: Formula for the matrix game as: Step1: First reduce the payoff matrix into dominance form. Step2: Add a Constant number k to each of the entries to make the each entries non-negative and no column have an all zero. Step3: Write 1 s to the right of and below the matrix, and then write down the associated standard maximization problem. Step4: Now solve this primal problem using the simplex method. Step5: Find the optical strategies and the expected value as follow: Operation for the column Strategy: (1): Now express the solution to the dual problem as a column vector . (2): Normalize by dividing each entry of the solution vector by p (which is also the sum of all the entries). (3): Insert zeros in position corresponding to the columns deleted during reduction by dominance. Operation for the Row Strategy: (1): Now express the solution to the dual problem as a row vector. (2): For normalize, dividing each entry by p and once again be the add all the entries. (3): Now insert zeros in positions corresponding to the rows deleted during reduction by dominance. 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 0 0 1 1 ] Add k = 2 to each entry and put 1 s to the right and be below: Represent this above problem by the matrix as: [ 2 0 3 1 2 3 1 1 1 2 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 x + 2 y + 2 z ≤ 1 , 2 x + 3 y + z ≤ 1 , 4 x + y ≤ 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 2 0 3 1 0 0 0 1 t 2 3 1 0 1 0 0 1 u 1 2 2 0 0 1 0 1 p − 1 − 1 − 1 0 0 0 1 0 Apply row operation, R 2 − R 1 , 2 R 3 − R 1 and 2 R 4 + R 1 . x y z s t u p s 2 0 3 1 0 0 0 1 t 2 3 1 0 1 0 0 0 u 1 2 2 0 0 1 0 1 p − 1 − 1 − 1 0 0 0 1 1 Apply row operation, 3 R 3 − 4 R 2 and 3 R 4 + 2 R 2

7th Edition

ISBN: 9781337280426

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 5.5, Problem 26E

To determine

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

[1−12120011]

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Chapter 5 Solutions

Finite Mathematics

Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - Prob. 6E Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - Prob. 9E Ch. 5.1 - In Exercises 1-26, sketch the region that...

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

Solution Summary: The author explains the formula for the matrix game as follows: reduce the payoff matrix into dominance form, add a Constant number k to each entry, and write down the associated standard maximization problem.

Videos

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

[1−12120011]

Chapter 5 Solutions

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

Solution Summary: The author explains the formula for the matrix game as follows: reduce the payoff matrix into dominance form, add a Constant number k to each entry, and write down the associated standard maximization problem.

Videos

To calculate: The game with the given payoff matrix and the provided payoff matrix is: [1−12120011]

Chapter 5 Solutions

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

[1−12120011]