Explanation Given Information: The payoff matrix provided below shows the number of votes gained or lost for each one-day campaign trip. For election, the candidates start campaigning in the three cities. Each candidate in each day decides in which city to spend the day. T .N . Spend T .L Down Littileville Metropolis Urbantown Littileville − 200 − 300 300 Metropolis − 500 500 − 100 Urbantown − 500 0 0 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 payoff matrix, T .N . Spend T .L Down Littileville Metropolis Urbantown Littileville − 200 − 300 300 Metropolis − 500 500 − 100 Urbantown − 500 0 0 From the observation of the payoff matrix, the first column of the matrix dominates the third column. Eliminate the third column of the matrix. So, the matrix is, [ − 200 − 300 − 500 500 − 500 0 ] The second row dominates the third row of the matrix, Eliminate the third row of the matrix. [ − 200 − 300 − 500 500 ] The matrix cannot be reduced further. Add k = 500 in each entry of the matrix to make the entries nonnegative. [ 300 200 0 1000 ] Write 1’s on the right side of matrix and below the matrix, [ 300 200 1 0 1000 1 1 1 0 ] So, the standard maximization problem is, Maximize p = x + y subject to 300 x + 200 y ≤ 1 , 1000 y ≤ 1 , x ≥ 0 , y ≥ 0

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ISBN: 9781337280426

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 5.5, Problem 41E

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To calculate: The percentage of time each candidate must spend in each city in order to maximize votes gained. Also determine the expected vote if both candidates use their optimal strategies.

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The percentage of time each candidate must spend in each city in order to maximize votes gained. Also determine the expected vote if both candidates use their optimal strategies.

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To calculate: The percentage of time each candidate must spend in each city in order to maximize votes gained. Also determine the expected vote if both candidates use their optimal strategies.

Chapter 5 Solutions