Explanation Given Information: A payoff matrix, [ 5 − 8 3 6 ] . 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, [ 5 − 8 3 6 ] . As only two entries are positive, add 9 to each entry to make all the entries positive. Thus, [ 5 − 8 3 6 ] = [ 5 + 9 − 8 + 9 3 + 9 6 + 9 ] = [ 14 1 12 15 ] So, the given payoff matrix, [ 5 − 8 3 6 ] is transformed to [ 14 1 12 15 ] . These two payoff matrices have the same optimal strategies. But the value of the game in new matrix is 9 more than that of the given matrix. Let z 1 , z 2 be chosen so as to maximize z 1 + z 2 subject to the constraints, { 14 z 1 + z 2 ≤ 1 12 z 1 + 15 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 { 14 z 1 + z 2 ≤ 1 12 z 1 + 15 z 2 ≤ 1 z 1 ≥ 0 , z 2 ≥ 0 Now, solve the above problem as below: Introduce the slack variables t and u and write the problem in terms of a system of linear equations. { 14 z 1 + z 2 + t = 1 12 z 1 + 15 z 2 + u = 1 z 1 ≥ 0 , z 2 ≥ 0 , t ≥ 0 , u ≥ 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, { 14 z 1 + z 2 + t = 1 12 z 1 + 15 z 2 + u = 1 − 1 z 1 − 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 t u M t 14 1 1 0 0 1 u 12 15 0 1 0 1 M − 1 − 1 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 second column becomes the pivot column. Now, to determine the pivot element, compute ratios as below: z 1 z 2 t u M Ratios t 14 1 1 0 0 1 1 u 12 15 0 1 0 1 1 15 M − 1 − 1 0 0 1 0 Since, the smallest non-negative ratio is 1 15 , so the pivot element is 15. Now, the new simplex table is constructed by pivoting around the element 15. Therefore, the new simplex table is, z 1 z 2 t u M t 66 5 0 1 − 1 15 0 14 15 z 2 4 5 1 0 1 15 0 1 15 M − 1 5 0 0 1 15 1 1 15 The left part of the bottom row has a negative entry ( − 1 5 )

12th Edition

ISBN: 9781323788707

Author: Goldstein

Publisher: PEARSON

See similar textbooks

Concept explainers

Correlation

Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.

Linear Correlation

A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.

Regression Analysis

Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as p…

Videos

Question

Chapter 9.3, Problem 10E

To determine

To calculate: The value of the game and optimal strategy for R and C for the payoff matrix, [5−836].

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.3, Problem 9E

Chapter 9.3, Problem 11E

Students have asked these similar questions

find the value of each variable

Consider the following system of equations, Ax=b : x+2y+3z - w = 2 2x4z2w = 3 -x+6y+17z7w = 0 -9x-2y+13z7w = -14 a. Find the solution to the system. Write it as a parametric equation. You can use a computer to do the row reduction. b. What is a geometric description of the solution? Explain how you know. c. Write the solution in vector form? d. What is the solution to the homogeneous system, Ax=0?

2. Find a matrix A with the following qualities a. A is 3 x 3. b. The matrix A is not lower triangular and is not upper triangular. c. At least one value in each row is not a 1, 2,-1, -2, or 0 d. A is invertible.

Chapter 9 Solutions

FINITE MATHEMATICS & ITS APPLICATIONS

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

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The value of the game and optimal strategy for R and C for the payoff matrix, [ 5 − 8 3 6 ] .

Solution Summary: The author calculates the value of the game and optimal strategy for R and C for the payoff matrix, left[cc5&

Concept explainers

Videos

To calculate: The value of the game and optimal strategy for R and C for the payoff matrix, [5−836].

Want to see the full answer?

Chapter 9 Solutions