(a) Explanation Given information: The payoff matrix, P = [ − 1 2 0 − 1 ] Formula used: Steps to find the expected payoff from mixed strategies R and C. Step 1: As a row matrix R write the mixed strategy for row player. Step 2: As a column matrix C write the mixed strategy for column player. Step 3: Calculate the product RPC , it is 1 × 1 matrix and its entry is e (expected payoff), where P represents the payoff matrix. Calculation: Consider the general strategy as R = [ x y ] . Let x and y are the probability of winning the game and losing the game respectively. As, the total probability of win or lose the game is one. So, x + y = 1 . Take the value of y as 1 − x , in the general strategy and consider R = [ x 1 − x ] , C = [ 1 0 ] T and compute the value of e by the equation e = R P C , where P is the payoff matrix for the game, Consider, R = [ x 1 − x ] , C = [ 1 0 ] T Thus, e = R P C = [ x 1 − x ] [ − 1 2 0 − 1 ] [ 1 0 ] For matrix R , since, number of rows in [ x 1 − x ] is 1 and number of columns is 2 , thus the dimension of [ x 1 − x ] is 1 × 2 . For matrix P , since, number of rows in [ − 1 2 0 − 1 ] is 2 and number of columns is 2 , thus the dimension of [ − 1 2 0 − 1 ] is 2 × 2 . For matrix C , since, number of rows in [ 1 0 ] is 2 and number of columns is 1, thus the dimension of [ 1 0 ] is 2 × 1 . Since, number of columns in R is equal to the number of rows in P , the product RP is defined. Since dimension of first matrix is 1 × 2 and dimension of second matrix is 2 × 2 , the dimension of the product is 1 × 2 . Simplify the product [ x 1 − x ] [ − 1 2 0 − 1 ] , [ x 1 − x ] [ − 1 2 0 − 1 ] = [ x ⋅ ( − 1 ) + 0 ( 1 − x ) 2 ⋅ x + ( − 1 ) ( 1 − x ) ] = [ − x 2 x − 1 + x ] = [ − x 3 x − 1 ] Now, compute the product [ − x 3 x − 1 ] [ 1 0 ] . Since dimension of first matrix is 1 × 2 and dimension of second matrix is 2 × 1 , the dimension of the product is 1 × 1 . [ − x 3 x − 1 ] [ 1 0 ] = [ ( − x ) ⋅ 1 + ( 3 x − 1 ) ⋅ 0 ] = [ − x + 0 ] = [ − x ] Thus, e = − x . Again, take R = [ x 1 − x ] , C = [ 0 1 ] T and compute the value of f by the equation f = R P C , where P is the payoff matrix for the game. Consider, R = [ x 1 − x ] , C = [ 0 1 ] T Thus, f = R P C = [ x 1 − x ] [ − 1 2 0 − 1 ] [ 0 1 ] For matrix R , since, number of rows in [ x 1 − x ] is 1 and number of columns is 2 , thus the dimension of [ x 1 − x ] is 1 × 2 (b) To determine To calculate: The optimal mixed column strategy of the payoff matrix, P = [ − 1 2 0 − 1 ] (c) To determine To calculate: The expected value of the game of the payoff matrix, P = [ − 1 2 0 − 1 ]

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Question

Chapter 5.4, Problem 21E

(a)

To determine

To calculate: The optimal mixed row strategy of the payoff matrix, P=[−120−1].

(b)

To determine

To calculate: The optimal mixed column strategy of the payoff matrix,

P=[−120−1]

(c)

To determine

To calculate: The expected value of the game of the payoff matrix,

P=[−120−1]

Expert Solution & Answer

Chapter 5 Solutions

Finite Mathematics and Applied Calculus (MindTap Course List)

Ch. 5.1 - In Exercises 110, find the dimensions of the given...Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - In Exercises 110, find the dimensions of the given...Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E

Ch. 5.1 - Solve for x,y,z,andw. [HINT: See Example 1.]...Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - In Exercises 1320, evaluate the given expression....Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - In Exercises 2128, evaluate the given expression....Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - In Exercises 2128, evaluate the given expression....Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - In Exercises 2936, use technology to evaluate the...Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Sales The following table shows the number of Mac...Ch. 5.1 - Home Prices The following table shows median home...Ch. 5.1 - Inventory The Left Coast Bookstore chain has two...Ch. 5.1 - Inventory The Left Coast Bookstore chain discussed...Ch. 5.1 - Prob. 41E Ch. 5.1 - Revenue The following spreadsheet gives annual...Ch. 5.1 - Prob. 43E Ch. 5.1 - Population Movement In 1990 the U.S. population,...Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - Prob. 49E Ch. 5.1 - Inventory Microbucks Computer Company, besides...Ch. 5.1 - Tourism in the 1990s The following table gives the...Ch. 5.1 - Prob. 52E Ch. 5.1 - Prob. 53E Ch. 5.1 - Prob. 54E Ch. 5.1 - Prob. 55E Ch. 5.1 - Prob. 56E Ch. 5.1 - Prob. 57E Ch. 5.1 - Prob. 58E Ch. 5.1 - Prob. 59E Ch. 5.1 - Prob. 60E Ch. 5.1 - Prob. 61E Ch. 5.1 - Prob. 62E Ch. 5.1 - Prob. 63E Ch. 5.1 - Prob. 64E Ch. 5.1 - Prob. 65E Ch. 5.1 - Prob. 66E Ch. 5.2 - Prob. 1E Ch. 5.2 - Prob. 2E Ch. 5.2 - Prob. 3E Ch. 5.2 - Prob. 4E Ch. 5.2 - Prob. 5E Ch. 5.2 - Prob. 6E Ch. 5.2 - Prob. 7E Ch. 5.2 - Prob. 8E Ch. 5.2 - Prob. 9E Ch. 5.2 - Prob. 10E Ch. 5.2 - Prob. 11E Ch. 5.2 - Prob. 12E Ch. 5.2 - Prob. 13E Ch. 5.2 - Prob. 14E Ch. 5.2 - Prob. 15E Ch. 5.2 - Prob. 16E Ch. 5.2 - Prob. 17E Ch. 5.2 - Prob. 18E Ch. 5.2 - Prob. 19E Ch. 5.2 - Prob. 20E Ch. 5.2 - Prob. 21E Ch. 5.2 - Prob. 22E Ch. 5.2 - Prob. 23E Ch. 5.2 - Prob. 24E Ch. 5.2 - Prob. 25E Ch. 5.2 - Prob. 26E Ch. 5.2 - Prob. 27E Ch. 5.2 - Prob. 28E Ch. 5.2 - Prob. 29E Ch. 5.2 - Prob. 30E Ch. 5.2 - Exercises 3138 should be done in two ways: by hand...Ch. 5.2 - Prob. 32E Ch. 5.2 - Prob. 33E Ch. 5.2 - Prob. 34E Ch. 5.2 - Prob. 35E Ch. 5.2 - Exercises 3138 should be done in two ways: by hand...Ch. 5.2 - Prob. 37E Ch. 5.2 - Prob. 38E Ch. 5.2 - Prob. 39E Ch. 5.2 - Prob. 40E Ch. 5.2 - Prob. 41E Ch. 5.2 - Prob. 42E Ch. 5.2 - Prob. 43E Ch. 5.2 - Prob. 44E Ch. 5.2 - Prob. 45E Ch. 5.2 - Prob. 46E Ch. 5.2 - Prob. 47E Ch. 5.2 - In Exercises 4548, translate the given matrix...Ch. 5.2 - Prob. 49E Ch. 5.2 - Prob. 50E Ch. 5.2 - Prob. 51E Ch. 5.2 - Prob. 52E Ch. 5.2 - Prob. 53E Ch. 5.2 - Revenue Karen Sandberg, your competitor in...Ch. 5.2 - Prob. 55E Ch. 5.2 - Prob. 56E Ch. 5.2 - Revenue Recall the Left Coast Bookstore chain from...Ch. 5.2 - Prob. 58E Ch. 5.2 - Prob. 59E Ch. 5.2 - Income Exercises 5962 are based on the following...Ch. 5.2 - Income Exercises 5962 are based on the following...Ch. 5.2 - Prob. 62E Ch. 5.2 - Prob. 63E Ch. 5.2 - Prob. 64E Ch. 5.2 - Foreclosure Crisis Starting in 2010, on the heels...Ch. 5.2 - Prob. 66E Ch. 5.2 - Prob. 67E Ch. 5.2 - Prob. 68E Ch. 5.2 - Foreclosure Crisis Starting in 2010, on the heels...Ch. 5.2 - Foreclosure Crisis Starting in 2010, on the heels...Ch. 5.2 - Prob. 71E Ch. 5.2 - Prob. 72E Ch. 5.2 - Tourism in the 1990s The following table gives the...Ch. 5.2 - Tourism in the 1990s Referring to the tourism...Ch. 5.2 - Prob. 75E Ch. 5.2 - Prob. 76E Ch. 5.2 - Give an example of two matrices A and B such that...Ch. 5.2 - Prob. 78E Ch. 5.2 - Prob. 79E Ch. 5.2 - Prob. 80E Ch. 5.2 - Prob. 81E Ch. 5.2 - Prob. 82E Ch. 5.2 - Prob. 83E Ch. 5.2 - Prob. 84E Ch. 5.2 - Prob. 85E Ch. 5.2 - Prob. 86E Ch. 5.3 - In Exercises 16, determine whether or not the...Ch. 5.3 - Prob. 2E Ch. 5.3 - Prob. 3E Ch. 5.3 - Prob. 4E Ch. 5.3 - Prob. 5E Ch. 5.3 - Prob. 6E Ch. 5.3 - In Exercises 726, use row reduction to find the...Ch. 5.3 - Prob. 8E Ch. 5.3 - Prob. 9E Ch. 5.3 - In Exercises 726, use row reduction to find the...Ch. 5.3 - Prob. 11E Ch. 5.3 - Prob. 12E Ch. 5.3 - Prob. 13E Ch. 5.3 - Prob. 14E Ch. 5.3 - Prob. 15E Ch. 5.3 - Prob. 16E Ch. 5.3 - Prob. 17E Ch. 5.3 - Prob. 18E Ch. 5.3 - Prob. 19E Ch. 5.3 - Prob. 20E Ch. 5.3 - Prob. 21E Ch. 5.3 - Prob. 22E Ch. 5.3 - Prob. 23E Ch. 5.3 - Prob. 24E Ch. 5.3 - Prob. 25E Ch. 5.3 - Prob. 26E Ch. 5.3 - Prob. 27E Ch. 5.3 - Prob. 28E Ch. 5.3 - Prob. 29E Ch. 5.3 - Prob. 30E Ch. 5.3 - Prob. 31E Ch. 5.3 - Prob. 32E Ch. 5.3 - In Exercises 2734, compute the determinant of the...Ch. 5.3 - Prob. 34E Ch. 5.3 - Prob. 35E Ch. 5.3 - Prob. 36E Ch. 5.3 - Prob. 37E Ch. 5.3 - Prob. 38E Ch. 5.3 - Prob. 39E Ch. 5.3 - In Exercises 3542, use technology to find the...Ch. 5.3 - Prob. 41E Ch. 5.3 - Prob. 42E Ch. 5.3 - Prob. 43E Ch. 5.3 - Prob. 44E Ch. 5.3 - Prob. 45E Ch. 5.3 - Prob. 46E Ch. 5.3 - In Exercises 4348, use matrix inversion to solve...Ch. 5.3 - Prob. 48E Ch. 5.3 - Prob. 49E Ch. 5.3 - Prob. 50E Ch. 5.3 - Some of the following exercises are similar or...Ch. 5.3 - Prob. 52E Ch. 5.3 - Some of the following exercises are similar or...Ch. 5.3 - Some of the following exercises are similar or...Ch. 5.3 - Investing: Inverse ETFs (Exchange Traded Funds)...Ch. 5.3 - Prob. 56E Ch. 5.3 - Investing: Lesser-Known Stocks Exercises 5758 are...Ch. 5.3 - Prob. 58E Ch. 5.3 - Population Movement In 2009 the population of the...Ch. 5.3 - Prob. 60E Ch. 5.3 - Prob. 61E Ch. 5.3 - Rotations If a point (x,y) in the plane is rotated...Ch. 5.3 - Encryption Matrices are commonly used to encrypt...Ch. 5.3 - Encryption Matrices are commonly used to encrypt...Ch. 5.3 - Prob. 65E Ch. 5.3 - Prob. 66E Ch. 5.3 - Multiple choice: If A and B are square matrices...Ch. 5.3 - Prob. 68E Ch. 5.3 - What can you say about the inverse of a 22 matrix...Ch. 5.3 - Prob. 70E Ch. 5.3 - Prob. 71E Ch. 5.3 - Derive the formula in Exercise 71 using row...Ch. 5.3 - Prob. 73E Ch. 5.3 - Prob. 74E Ch. 5.3 - Prob. 75E Ch. 5.3 - Solve the matrix equation A(B+CX)=D for X. (You...Ch. 5.3 - Prob. 77E Ch. 5.3 - Prob. 78E Ch. 5.4 - Prob. 1E Ch. 5.4 - Prob. 2E Ch. 5.4 - Prob. 3E Ch. 5.4 - Prob. 4E Ch. 5.4 - Prob. 5E Ch. 5.4 - Prob. 6E Ch. 5.4 - Prob. 7E Ch. 5.4 - Prob. 8E Ch. 5.4 - Prob. 9E Ch. 5.4 - In Exercises 914, reduce the given payoff matrix...Ch. 5.4 - In Exercises 914, reduce the given payoff matrix...Ch. 5.4 - In Exercises 914, reduce the given payoff matrix...Ch. 5.4 - Prob. 13E Ch. 5.4 - Prob. 14E Ch. 5.4 - Prob. 15E Ch. 5.4 - Prob. 16E Ch. 5.4 - Prob. 17E Ch. 5.4 - Prob. 18E Ch. 5.4 - Prob. 19E Ch. 5.4 - Prob. 20E Ch. 5.4 - Prob. 21E Ch. 5.4 - Prob. 22E Ch. 5.4 - Prob. 23E Ch. 5.4 - Prob. 24E Ch. 5.4 - Prob. 25E Ch. 5.4 - Prob. 26E Ch. 5.4 - Prob. 27E Ch. 5.4 - Prob. 28E Ch. 5.4 - In Exercises 2532, set up the payoff matrix....Ch. 5.4 - In Exercises 2532, set up the payoff matrix....Ch. 5.4 - In Exercises 2532, set up the payoff matrix....Ch. 5.4 - Prob. 32E Ch. 5.4 - Prob. 33E Ch. 5.4 - More Retail Discount Wars Your Abercrom B mens...Ch. 5.4 - Factory Location22 A manufacturer of electrical...Ch. 5.4 - Crop Choice23 A farmer has a choice of growing...Ch. 5.4 - Study Techniques Your mathematics test is tomorrow...Ch. 5.4 - Prob. 38E Ch. 5.4 - Prob. 39E Ch. 5.4 - Prob. 40E Ch. 5.4 - Prob. 41E Ch. 5.4 - Prob. 42E Ch. 5.4 - Prob. 43E Ch. 5.4 - Prob. 44E Ch. 5.4 - Prob. 45E Ch. 5.4 - Prob. 46E Ch. 5.4 - Prob. 47E Ch. 5.4 - Prob. 48E Ch. 5.4 - Campaign Strategies27 Florida and Ohio are swing...Ch. 5.4 - Prob. 50E Ch. 5.4 - Prob. 51E Ch. 5.4 - Prob. 52E Ch. 5.4 - Prob. 53E Ch. 5.4 - Prob. 54E Ch. 5.4 - Prob. 55E Ch. 5.4 - Exercises 37 and 38 seem to suggest that studying...Ch. 5.4 - Prob. 57E Ch. 5.4 - Prob. 58E Ch. 5.5 - Let A be the technology matrix A=[0.20.050.80.01],...Ch. 5.5 - Let A be the technology matrix...Ch. 5.5 - Prob. 3E Ch. 5.5 - Prob. 4E Ch. 5.5 - Prob. 5E Ch. 5.5 - In Exercises 512, you are given a technology...Ch. 5.5 - In Exercises 512, you are given a technology...Ch. 5.5 - In Exercises 512, you are given a technology...Ch. 5.5 - In Exercises 512, you are given a technology...Ch. 5.5 - In Exercises 512, you are given a technology...Ch. 5.5 - In Exercises 512, you are given a technology...Ch. 5.5 - In Exercises 512, you are given a technology...Ch. 5.5 - Given A=[0.10.40.20.5], find the changes in...Ch. 5.5 - Prob. 14E Ch. 5.5 - Prob. 15E Ch. 5.5 - Prob. 16E Ch. 5.5 - Prob. 17E Ch. 5.5 - Prob. 18E Ch. 5.5 - Campus Food The two campus cafeterias, the Main...Ch. 5.5 - Plagiarism Two student groups at Enormous State...Ch. 5.5 - Prob. 21E Ch. 5.5 - Wood and Paper Two sectors of the U.S. economy are...Ch. 5.5 - Prob. 23E Ch. 5.5 - Prob. 24E Ch. 5.5 - Prob. 25E Ch. 5.5 - Prob. 26E Ch. 5.5 - Prob. 27E Ch. 5.5 - Prob. 28E Ch. 5.5 - Prob. 29E Ch. 5.5 - Prob. 30E Ch. 5.5 - Exercises 3134 require the use of technology....Ch. 5.5 - Exercises 3134 require the use of technology....Ch. 5.5 - Prob. 33E Ch. 5.5 - Prob. 34E Ch. 5.5 - Prob. 35E Ch. 5.5 - Prob. 36E Ch. 5.5 - What would it mean if the total output figure for...Ch. 5.5 - Prob. 38E Ch. 5.5 - Prob. 39E Ch. 5.5 - Prob. 40E Ch. 5.5 - Prob. 41E Ch. 5.5 - Prob. 42E Ch. 5 - For Exercises 110, let...Ch. 5 - Prob. 2RE Ch. 5 - Prob. 3RE Ch. 5 - Prob. 4RE Ch. 5 - For Exercises 110, let...Ch. 5 - Prob. 6RE Ch. 5 - Prob. 7RE Ch. 5 - Prob. 8RE Ch. 5 - Prob. 9RE Ch. 5 - Prob. 10RE Ch. 5 - Prob. 11RE Ch. 5 - Prob. 12RE Ch. 5 - Prob. 13RE Ch. 5 - Prob. 14RE Ch. 5 - Prob. 15RE Ch. 5 - Prob. 16RE Ch. 5 - Prob. 17RE Ch. 5 - Prob. 18RE Ch. 5 - In Exercises 1720, write the given system of...Ch. 5 - Prob. 20RE Ch. 5 - Prob. 21RE Ch. 5 - Prob. 22RE Ch. 5 - Prob. 23RE Ch. 5 - Prob. 24RE Ch. 5 - In Exercises 2528, find the production vector X...Ch. 5 - Prob. 26RE Ch. 5 - Prob. 27RE Ch. 5 - Prob. 28RE Ch. 5 - Prob. 29RE Ch. 5 - Prob. 30RE Ch. 5 - Prob. 31RE Ch. 5 - Prob. 32RE Ch. 5 - It is now July 1, and online sales of romance,...Ch. 5 - Prob. 34RE Ch. 5 - Prob. 35RE Ch. 5 - Prob. 36RE Ch. 5 - Prob. 37RE Ch. 5 - Prob. 38RE Ch. 5 - OHaganBooks.com has two main...Ch. 5 - Prob. 40RE Ch. 5 - OHaganBooks.com has two main...Ch. 5 - Prob. 42RE Ch. 5 - Prob. 43RE Ch. 5 - Prob. 44RE Ch. 5 - Prob. 45RE Ch. 5 - Prob. 46RE Ch. 5 - Prob. 47RE Ch. 5 - Prob. 48RE Ch. 5 - Prob. 49RE Ch. 5 - Some of the books sold by OHaganBooks.com are...Ch. 5 - Some of the books sold by OHaganBooks.com are...Ch. 5 - Some of the books sold by OHaganBooks.com are...

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The optimal mixed row strategy of the payoff matrix, P = [ − 1 2 0 − 1 ] .

Solution Summary: The author calculates the optimal mixed row strategy of the payoff matrix, P=left[cc-1& 2 0& -1end

Videos

To calculate: The optimal mixed row strategy of the payoff matrix, P=[−120−1].

To calculate: The optimal mixed column strategy of the payoff matrix,

P=[−120−1]

To calculate: The expected value of the game of the payoff matrix,

P=[−120−1]

Chapter 5 Solutions

The optimal mixed row strategy of the payoff matrix, P = [ − 1 2 0 − 1 ] .

Solution Summary: The author calculates the optimal mixed row strategy of the payoff matrix, P=left[cc-1& 2 0& -1end

Videos

To calculate: The optimal mixed row strategy of the payoff matrix, P=[−120−1].

To calculate: The optimal mixed column strategy of the payoff matrix, P=[−120−1]

To calculate: The expected value of the game of the payoff matrix, P=[−120−1]

Chapter 5 Solutions

To calculate: The optimal mixed column strategy of the payoff matrix,

P=[−120−1]

To calculate: The expected value of the game of the payoff matrix,

P=[−120−1]