Explanation Given Information: A payoff matrix, [ 4 − 2 1 2 3 1 ] . Formula used: Value of the game is: v = 1 z 1 + z 2 + ⋯ z n The optimal strategy for C is: [ c 1 c 2 ⋮ c n ] Where, c 1 = v z 1 , c 2 = v z 2 , … c n = v z n . Calculation: Consider the payoff matrix, [ 4 − 2 1 2 3 1 ] . As one entry is negative, add 3 to each entry to make all the entries positive. Thus, [ 4 − 2 1 2 3 1 ] = [ 4 + 3 − 2 + 3 1 + 3 2 + 3 3 + 3 1 + 3 ] = [ 7 1 4 5 6 4 ] So, the given payoff matrix, [ 4 − 2 1 2 3 1 ] is transformed to [ 7 1 4 5 6 4 ] . These two payoff matrices have the same optimal strategies. But the value of the game in new matrix is 2 more than that of the given matrix. Let z 1 and z 2 be chosen so as to maximize z 1 + z 2 subject to the constraints, { 7 z 1 + z 2 ≤ 1 4 z 1 + 5 z 2 ≤ 1 6 z 1 + 4 z 2 ≤ 1 z 1 ≥ 0 , z 2 ≥ 0 Hence, the linear programming problem corresponding to the provided payoff matrix is, Maximize z 1 + z 2 Subjects to the constraints { 7 z 1 + z 2 ≤ 1 4 z 1 + 5 z 2 ≤ 1 6 z 1 + 4 z 2 ≤ 1 z 1 ≥ 0 , z 2 ≥ 0 Now, solve the above problem as below: Introduce the slack variables s , t and u and write the problem in terms of a system of linear equations. { 7 z 1 + z 2 + s = 1 4 z 1 + 5 z 2 + t = 1 6 z 1 + 4 z 2 + u = 1 z 1 ≥ 0 , z 2 ≥ 0 The objective function z 1 + z 2 gives the equation, M = z 1 + z 2 . So, the corresponding system of linear equation with slack variables is, { 7 z 1 + z 2 + s = 1 4 z 1 + 5 z 2 + t = 1 6 z 1 + 4 z 2 + u = 1 − z 1 − z 2 + M = 0 Where, M is as large as possible. Thus, the initial simplex table corresponding to above system is: z 1 z 2 s t u M s 7 1 1 0 0 0 1 t 4 5 0 1 0 0 1 u 6 4 0 0 1 0 1 M − 1 − 1 0 0 0 1 0 As the left part of the bottom row has negative entries − 1 , the above table does not give the maximum value of the objective function. So, create a new simplex table by pivoting. Since, − 1 is the most negative entry in the last row, the pivot column can be chosen arbitrarily. Let first column becomes the pivot column. Now, to determine the pivot element, compute ratios as below: z 1 z 2 s t u M Ratios s 7 1 1 0 0 0 1 1 7 t 4 5 0 1 0 0 1 1 4 u 6 4 0 0 1 0 1 1 6 M − 1 − 1 0 0 0 1 0 Since, the smallest non-negative ratio is 1 7 , so the pivot element is 7. Now, the new simplex table is constructed by pivoting around the element 7. Therefore, the new simplex table is, z 1 z 2 s t u M z 1 1 1 7 1 7 0 0 0 1 7 t 0 31 7 − 4 7 1 0 0 3 7 u 0 22 7 − 6 7 0 1 0 1 7 M 0 − 6 7 1 7 0 0 1 1 7 The left part of the bottom row has a negative entry ( − 6 7 )

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Author: Larry Goldstein, David Schneider

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Chapter 9.3, Problem 20E

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To calculate: The value of the game and optimal strategy for C for the payoff matrix, [4−21231].

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Chapter 9.3, Problem 19E

Chapter 9.3, Problem 21E

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Chapter 9 Solutions

Pearson eText for Finite Mathematics & Its Applications -- Instant Access (Pearson+)

Ch. 9.1 - Solutions can be found following the section...Ch. 9.1 - Prob. 2CYU Ch. 9.1 - Prob. 3CYU Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - In Exercises 1–12, determine the optimal pure...Ch. 9.1 - In Exercises 1–12, determine the optimal pure...Ch. 9.1 - Prob. 7E

Ch. 9.1 - In Exercises 1–12, determine the optimal pure...Ch. 9.1 - In Exercises 112, determine the optimal pure...Ch. 9.1 - In Exercises 1–12, determine the optimal pure...Ch. 9.1 - In Exercises 1–12, determine the optimal pure...Ch. 9.1 - Prob. 12E Ch. 9.1 - Prob. 13E Ch. 9.1 - Prob. 14E Ch. 9.1 - Prob. 15E Ch. 9.1 - Prob. 16E Ch. 9.1 - Prob. 17E Ch. 9.1 - Prob. 18E Ch. 9.1 - Prob. 19E Ch. 9.1 - For each of the games that follow, give the payoff...Ch. 9.1 - Prob. 21E Ch. 9.1 - Prob. 22E Ch. 9.1 - Prob. 23E Ch. 9.2 - Solutions can be found following the section...Ch. 9.2 - Prob. 2CYU Ch. 9.2 - Prob. 1E Ch. 9.2 - Suppose that a game has payoff matrix [102120011]...Ch. 9.2 - Prob. 3E Ch. 9.2 - Prob. 4E Ch. 9.2 - Prob. 5E Ch. 9.2 - Flood Insurance A small business owner must decide...Ch. 9.2 - 7. Two players, Robert and Carol, play a game with...Ch. 9.2 - Rework Exercise 7 with [.7.3] as Roberts strategy.Ch. 9.2 - Two players, Robert and Carol, play a game with...Ch. 9.2 - 10. Rework Exercise 9 with as Robert’s...Ch. 9.2 - 11. Assume that two players, Renée and Carlos,...Ch. 9.2 - Prob. 12E Ch. 9.2 - Prob. 13E Ch. 9.2 - Prob. 14E Ch. 9.2 - Prob. 15E Ch. 9.2 - 16. Three-Finger Morra Reven and Coddy play a game...Ch. 9.3 - Prob. 1CYU Ch. 9.3 - Prob. 2CYU Ch. 9.3 - Prob. 1E Ch. 9.3 - Prob. 2E Ch. 9.3 - Prob. 3E Ch. 9.3 - Prob. 4E Ch. 9.3 - Prob. 5E Ch. 9.3 - Prob. 6E Ch. 9.3 - Prob. 7E Ch. 9.3 - In Exercises 5–12, determine the value of the game...Ch. 9.3 - In Exercises 512, determine the value of the game...Ch. 9.3 - Prob. 10E Ch. 9.3 - Prob. 11E Ch. 9.3 - Prob. 12E Ch. 9.3 - In Exercises 13–16, determine the value of the...Ch. 9.3 - Prob. 14E Ch. 9.3 - Prob. 15E Ch. 9.3 - Prob. 16E Ch. 9.3 - Prob. 17E Ch. 9.3 - Prob. 18E Ch. 9.3 - Prob. 19E Ch. 9.3 - Prob. 20E Ch. 9.3 - Prob. 21E Ch. 9.3 - Prob. 22E Ch. 9.3 - Football Suppose that, when the offense calls a...Ch. 9.3 - Prob. 24E Ch. 9.3 - Prob. 25E Ch. 9.3 - Three-Finger Mor ra Reven and Coddy play a game in...Ch. 9.3 - Advertising Strategies The Carter Company can...Ch. 9 - 1. What do the individual entries of a payoff...Ch. 9 - Prob. 2FCCE Ch. 9 - Prob. 3FCCE Ch. 9 - Prob. 4FCCE Ch. 9 - Prob. 5FCCE Ch. 9 - Prob. 6FCCE Ch. 9 - Prob. 7FCCE Ch. 9 - What is meant by the optimal mixed strategies of R...Ch. 9 - In Exercises 14, state whether or not the games...Ch. 9 - Prob. 2RE Ch. 9 - Prob. 3RE Ch. 9 - Prob. 4RE Ch. 9 - Prob. 5RE Ch. 9 - Prob. 6RE Ch. 9 - Prob. 7RE Ch. 9 - Prob. 8RE Ch. 9 - Prob. 9RE Ch. 9 - Prob. 10RE Ch. 9 - Prob. 11RE Ch. 9 - Prob. 12RE Ch. 9 - Prob. 13RE Ch. 9 - Prob. 14RE Ch. 9 - Prob. 15RE Ch. 9 - Prob. 16RE Ch. 9 - Prob. 17RE Ch. 9 - Prob. 18RE Ch. 9 - Prob. 1P Ch. 9 - Prob. 2P Ch. 9 - Prob. 3P Ch. 9 - Prob. 4P Ch. 9 - Prob. 5P Ch. 9 - Prob. 6P

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The value of the game and optimal strategy for C for the payoff matrix, [ 4 − 2 1 2 3 1 ] .

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To calculate: The value of the game and optimal strategy for C for the payoff matrix, [4−21231].

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Chapter 9 Solutions