Explanation Given Information: The payoff matrix for the game is given to be [ 2 14 6 12 8 6 ] . Formula used: Optimal mixed strategy for C is calculated as follows: Let the payoff matrix of a game be [ a 11 a 12 ⋯ a 1 n a 21 a 22 ⋯ a 2 n ⋮ ⋮ ⋱ ⋮ a m 1 a m 2 ⋯ a m n ] where all entries of the matrix are positive numbers. Let z 1 , z 2 , … , z n be chosen so as to maximize z 1 + z 2 + … + z n subject to the constraints { z 1 ≥ 0 , z 2 ≥ 0 , … z n ≥ 0 a 11 z 1 + a 12 z 2 + ⋯ + a 1 n z n ≤ 1 a 21 z 1 + a 22 z 2 + ⋯ + a 2 n z n ≤ 1 ⋮ a m 1 z 1 + a m 2 z 2 + ⋯ + a m n z n ≤ 1 Let v = 1 z 1 + z 2 + ⋯ + z n Then an 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 given payoff matrix, [ 2 14 6 12 8 6 ] Follow the steps to find the optimal strategy for C . The associated linear programming problem is as follows: Let z 1 , z 2 be chosen so as to maximize z 1 + z 2 subject to constraints, { z 1 ≥ 0 , z 2 ≥ 0 2 z 1 + 14 z 2 ≤ 1 6 z 1 + 12 z 2 ≤ 1 8 z 1 + 6 z 2 ≤ 1 Consider the system of inequalities: z 1 ≥ 0 , z 2 ≥ 0 2 z 1 + 14 z 2 ≤ 1 6 z 1 + 12 z 2 ≤ 1 8 z 1 + 6 z 2 ≤ 1 The graph of each individual system of inequalities are as follows: Consider the inequality z 1 ≥ 0 . The graph of the equation z 1 = 0 is the axes z 2 . The inequality z 1 ≥ 0 means the region lying to the right of the axes z 2 . Therefore, the resulting region is shown below: Consider the inequality z 2 ≥ 0 . The graph of the equation z 2 = 0 is the axes z 1 . The inequality z 2 ≥ 0 means the region lying above of the axes z 1 . Therefore, the resulting region is shown below: Consider the inequality, 2 z 1 + 14 z 2 ≤ 1 Take convenient points to plot the corresponding equation 2 z 1 + 14 z 2 = 1 as follows: z 1 Equation 2 z 1 + 14 z 2 = 1 z 2 = 1 − 2 z 1 14 z 2 0 z 2 = 1 − 2 ( 0 ) 14 = 1 14 1 14 1 2 z 2 = 1 − 2 ( 1 2 ) 14 = 1 − 1 14 = 0 0 1 30 z 2 = 1 − 2 ( 1 30 ) 14 = 1 − 1 15 14 = 1 15 1 15 Put test point ( 0 , 0 ) in the inequality 2 z 1 + 14 z 2 ≤ 1 , 1 ≥ ? 2 ( 0 ) + 14 ( 0 ) 1 ≥ ? 0 Above statement is true. Therefore, shade the region of the graph which contains the test point ( 0 , 0 ) . The graph for 2 z 1 + 14 z 2 ≤ 1 is as follows: Consider the inequality, 6 z 1 + 12 z 2 ≤ 1 Take convenient points to plot the corresponding equation 6 z 1 + 12 z 2 = 1 as follows: z 1 Equation 6 z 1 + 12 z 2 = 1 z 2 = 1 − 6 z 1 12 z 2 0 z 2 = 1 − 6 ( 0 ) 12 = 1 12 1 12 1 10 z 2 = 1 − 6 ( 1 10 ) 12 = 1 − 3 5 12 = 1 30 1 30 1 30 z 2 = 1 − 6 ( 1 30 ) 12 = 1 − 1 5 12 = 1 15 1 15 Put test point ( 0 , 0 ) in the inequality 6 z 1 + 12 z 2 ≤ 1 , 6 ( 0 ) + 12 ( 0 ) ≤ ? 1 0 ≤ ? 1 Above statement is true. Therefore, shade the region of the graph which contains the test point ( 0 , 0 ) . The graph for 6 z 1 + 12 z 2 ≤ 1 is as follows: Consider the inequality, 8 z 1 + 6 z 2 ≤ 1 Take convenient points to plot the corresponding equation 8 z 1 + 6 z 2 = 1 as follows: z 1 Equation 8 z 1 + 6 z 2 = 1 z 2 = 1 − 8 z 1 6 z 2 0 z 2 = 1 − 8 ( 0 ) 6 = 1 6 1 6 1 8 z 2 = 1 − 8 ( 1 8 ) 6 = 1 − 1 6 = 0 0 1 10 z 2 = 1 − 8 ( 1 10 ) 6 = 1 − 8 10 6 = 1 30 1 30 Put test point ( 0 , 0 ) in the inequality 8 z 1 + 6 z 2 ≤ 1 , 8 ( 0 ) + 6 ( 0 ) ≤ ? 1 0 ≤ ? 1 Above statement is true

Finite Mathematics & Its Applications (12th Edition)

12th Edition

ISBN: 9780134437767

Author: Larry J. Goldstein, David I. Schneider, Martha J. Siegel, Steven Hair

Publisher: PEARSON

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Chapter 9.3, Problem 1CYU

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To calculate: The optimal strategy for C for the game having payoff matrix [21461286].

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Chapter 9.2, Problem 16E

Chapter 9.3, Problem 2CYU

Chapter 9 Solutions

Finite Mathematics & Its Applications (12th Edition)

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 optimal strategy for C for the game having payoff matrix [ 2 14 6 12 8 6 ] .

Solution Summary: The author explains how to calculate the optimal strategy for C for the game having payoff matrix left[cc2& 14 6& 12 8& 6end

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To calculate: The optimal strategy for C for the game having payoff matrix [21461286].

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Chapter 9 Solutions