Explanation Given Information: The table provided below shows the Pat’s strategies through which he can expect to earn scores in case the entire test is in a specific subject. He has no knowledge in linear programming or matrix algebra. Even if he studies game theory all night, he will be unable to prepare for the topic. Game Theory Linear Programming Matrix Algebra Study Game Theory 30 0 20 Study Linear Programming 0 70 0 Study Matrix Algebra 0 0 70 Formula used: Steps to solve a matrix game, 1. Reduce the payoff matrix with the help of dominance. 2. Add a fixed number k in each entry of the matrix to make the entries nonnegative. No column in the matrix can contain all zero. 3. Write 1’s on the right side of matrix and below the matrix. Write the associated standard maximization problem and then solve the primal problem by simplex method. 4. Determine the optimal strategies and the expected value e = 1 p − k . Steps to follows the Column Strategy, 1. Represent the solution to the primal problem by a column vector. 2. Divide each entry in the column vector by p which is the sum of all entries. 3. Insert the zeros in position to the columns which are deleted during the reduction by dominance. Steps to follows the Row Strategy, 1. Represent the solution to the dual problem by a row vector. 2. Divide each entry in the row vector by p which is the sum of all entries. 3. Insert the zeros in position to the columns which are deleted during the reduction by dominance. Calculation: Consider the table, Game Theory Linear Programming Matrix Algebra Study Game Theory 30 0 20 Study Linear Programming 0 70 0 Study Matrix Algebra 0 0 70 There exist no dominated rows or column due to which this game cannot be reduced by dominance. All the terms are positive due to which there is no need to add k or 0 . So, the matrix is, [ 30 0 20 0 70 0 0 0 70 ] All the entries in the matrix is nonnegative and no column is all zero. So, the standard maximization problem is, Maximize p = x + y + z subject to 30 x + 20 z ≤ 1 , 70 y ≤ 1 , 70 z ≤ 1 , x ≥ 0 , y ≥ 0 , z ≥ 0 . Use simplex method to solve the game, So, the tableaus are, x y z s t u p s 30 0 20 1 0 0 0 1 t 0 70 0 0 1 0 0 1 u 0 0 70 0 0 1 0 1 p − 1 − 1 − 1 0 0 0 1 0 Apply the row operation 30 R 4 + R 1 as, x y z s t u p x 30 0 20 1 0 0 0 1 t 0 70 0 0 1 0 0 1 u 0 0 70 0 0 1 0 1 p 0 − 30 − 10 1 0 0 30 1 Apply the row operation 7 R 4 + 3

7th Edition

ISBN: 9781337280426

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 5, Problem 46RE

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To calculate: The percentage of the night which Pat must spend on each topic. Also determine the score he can expect to get. Consider the principles of game theory.

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Chapter 5 Solutions

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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 percentage of the night which Pat must spend on each topic. Also determine the score he can expect to get. Consider the principles of game theory.

Solution Summary: The author calculates the percentage of the night which Pat must spend on each topic and the score he can expect to get.

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To calculate: The percentage of the night which Pat must spend on each topic. Also determine the score he can expect to get. Consider the principles of game theory.

Chapter 5 Solutions