Question 1

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first, and identify the submatrices R and S. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 2

The sum of the entries in each column of a transition matrix must not exceed 1.

Accepted Answer

A) True 
 B)False

Question 3

Roland's Barber Shop and Charley's Barber Shop are both located in the business district of a certain town. Roland estimates that if he raises the price of a haircut by $1, he will increase his market share by 7% if Charley raises his price by the same amount; he will decrease his market share by 2% if Charley holds his price at the same level; and he will decrease his market share by 7% if Charley lowers his price by $1. If Roland keeps his price the same, he will increase his market share by 4% if Charley raises his price by $1; he will keep the same market share if Charley holds the price at the same level; and he will decrease his market share by 4% if Charley lowers his price by $1. Finally, if Roland lowers the price he charges by $1, his market share will increase by 8% if Charley raises his prices by the same amount; he will increase his market share by 4% if Charley holds his price at the same level; and he will increase his market share by 2% if Charley lowers his price by $1. ​
If neither party is willing to lower the price he charges for a haircut, what strategy should they choose?
​

Accepted Answer

A)  Roland should hold the price. Charley should hold the price. 
B)  Roland should hold the price. Charley should raise the price. 
C)  Roland should raise the price. Charley should raise the price. 
D)  Roland should raise the price. Charley should hold the price. 
E)  The game is not strictly determined. 
A)  Roland should hold the price. Charley should hold the price. 
B)  Roland should hold the price. Charley should raise the price. 
C)  Roland should raise the price. Charley should raise the price. 
D)  Roland should raise the price. Charley should hold the price. 
E)  The game is not strictly determined.

Question 4

Rewrite the absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices   and   . ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 5

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given.
​

Accepted Answer

The answer of Find the expected payoff E of the...

Question 6

Determine whether the given matrix is stochastic.
​

Accepted Answer

The answer of Determine whether the given matrix is stochastic.
​...

Question 7

Because of the proliferation of more affordable automated office equipment, more and more companies are turning to them as replacements for obsolete equipment. The following transition matrix describes the Markov process. Here,   stands for electric typewriters,   stands for electric typewriters with some form of word processing capabilities, and   stands for computers with word processing software. E W C   ​
Rewrite it so that the absorbing state appears first. Partition the resulting matrix and identify the submatrices   and   .
​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 8

Determine whether the given matrix is an absorbing stochastic matrix.
​

Accepted Answer

The answer of Determine whether the given matrix is an...

Question 9

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first and identify the submatrices R and S. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 10

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 11

Find   (the probability distribution of the system after two observations) for the given distribution vector   and the given transition matrix   .   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 12

Consider the two-person, zero-sum matrix, strictly determined game.
​   ​
Find the saddle point(s) of the game.
​
Find the optimal strategy for each player.
​
R: row __________ C: column __________
​
Find the value of the game.
​
Determine whether the game favors one player over the other.

Accepted Answer

The answer of Consider the two-person, zero-sum matrix, strictly determined...

Question 13

Is the matrix regular?
​

Accepted Answer

The answer of Is the matrix regular?
​  ...

Question 14

Determine whether the matrix is an absorbing stochastic matrix. ​   ​

Accepted Answer

A)  no 
B)  yes 
A)  no 
B)  yes

Question 15

Consider the two-person, zero-sum matrix, strictly determined game. ​   ​
Find the value of the game.
​

Accepted Answer

A)  6 
B)  2 
C)  4 
D)  10 
E)  1 
A)  6 
B)  2 
C)  4 
D)  10 
E)  1

Question 16

Determine whether the given matrix is stochastic.

Accepted Answer

The answer of Determine whether the given matrix is stochastic....

Question 17

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 18

Determine whether the given two-person, zero-sum matrix game is strictly determined.
​

Accepted Answer

The answer of Determine whether the given two-person, zero-sum matrix...

Question 19

Find the steady-state vector for the transition matrix. ​   ​

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 20

Determine the maximin and minimax strategies for the two-person, zero-sum matrix game. ​   ​

Accepted Answer

A)  R: row 2; C: column 1 
B)  R: row 1; C: column 2 
C)  R: row 2; C: column 2 
D)  R: row 1; C: column 1 
A)  R: row 2; C: column 1 
B)  R: row 1; C: column 2 
C)  R: row 2; C: column 2 
D)  R: row 1; C: column 1

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first, and identify the submatrices R and S.

The sum of the entries in each column of a transition matrix must not exceed 1.

Rewrite the absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices and .

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given.

Determine whether the given matrix is stochastic.

Determine whether the given matrix is an absorbing stochastic matrix.

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first and identify the submatrices R and S.

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given.

Find (the probability distribution of the system after two observations) for the given distribution vector and the given transition matrix .

Consider the two-person, zero-sum matrix, strictly determined game. Find the saddle point(s) of the game. Find the optimal strategy for each player. R: row C: column Find the value of the game. Determine whether the game favors one player over the other.

Is the matrix regular?

Determine whether the matrix is an absorbing stochastic matrix.

Consider the two-person, zero-sum matrix, strictly determined game. Find the value of the game.

Determine whether the given matrix is stochastic.

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given.

Determine whether the given two-person, zero-sum matrix game is strictly determined.

Find the steady-state vector for the transition matrix.

Determine the maximin and minimax strategies for the two-person, zero-sum matrix game.

Straight Lines and Linear Functions

Systems of Linear Equations and Matrices

Linear Programming: a Geometric Approach

Linear Programming: an Algebraic Approach

Mathematics of Finance

Sets and Counting

Probability

Probability Distributions and Statistics

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Exam 9: Markov Chains and the Theory of Games

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first, and identify the submatrices R and S. ​ ​

The sum of the entries in each column of a transition matrix must not exceed 1.

Rewrite the absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices and . ​ ​

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given. ​

Determine whether the given matrix is stochastic. ​

Determine whether the given matrix is an absorbing stochastic matrix. ​

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first and identify the submatrices R and S. ​ ​

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given. ​ ​

Find (the probability distribution of the system after two observations) for the given distribution vector and the given transition matrix . ​

Is the matrix regular? ​

Determine whether the matrix is an absorbing stochastic matrix. ​ ​

Consider the two-person, zero-sum matrix, strictly determined game. ​ ​ Find the value of the game. ​

Determine whether the given matrix is stochastic.

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given. ​ ​

Determine whether the given two-person, zero-sum matrix game is strictly determined. ​

Find the steady-state vector for the transition matrix. ​ ​

Determine the maximin and minimax strategies for the two-person, zero-sum matrix game. ​ ​

Straight Lines and Linear Functions

Systems of Linear Equations and Matrices

Linear Programming: a Geometric Approach

Linear Programming: an Algebraic Approach

Mathematics of Finance

Sets and Counting

Probability

Probability Distributions and Statistics

Filters

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first, and identify the submatrices R and S.

Rewrite the absorbing stochastic matrix so that the absorbing states appear first, partition the resulting matrix, and identify the submatrices and .

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given.

Determine whether the given matrix is stochastic.

Determine whether the given matrix is an absorbing stochastic matrix.

Rewrite the given absorbing stochastic matrix so that the absorbing states appear first and identify the submatrices R and S.

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given.

Find (the probability distribution of the system after two observations) for the given distribution vector and the given transition matrix .

Is the matrix regular?

Determine whether the matrix is an absorbing stochastic matrix.

Consider the two-person, zero-sum matrix, strictly determined game. Find the value of the game.

Find the expected payoff E of the game whose payoff matrix and strategies P and Q (for the row and column players, respectively) are given.

Determine whether the given two-person, zero-sum matrix game is strictly determined.

Find the steady-state vector for the transition matrix.

Determine the maximin and minimax strategies for the two-person, zero-sum matrix game.