Page 385, Question 1a

Using the definition provided for the movement diagram, determine whether the following zero-zum games have a pure strategy Nash equilibrium. If the game does have one, state the Nash Equilibrium. Assume the row player is maximizing his payoffs which are shown in the matrices below.

Page 385, Answer 1a

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Please note that at outcome (R1,C1) or (R1, C2) all arrows point to that point, and no arrow exits the outcome This indicates that neither player can unilaterally improve a stable situation that is Nash equilibrium.
Also, no matter what Colin does (choosing C1 or C2), as long as Rose choses R1, Rose will maximize the outcome. Hence it is Nash equilibrium with pure strategy.

Page 385, Question 1c

Using the definition provided for the movement diagram, determine whether the following zero-zum games have a pure strategy Nash equilibrium. If the game does have one, state the Nash Equilibrium. Assume the row player is maximizing his payoffs which are shown in the matrices below.

Page 385, Answer 1c

alt text

The batter can either anticipate the pitcher to throw a fast or knuckle bal. if the pitcher anticipate fast, he will whether hit . 400 or .100 depending on whether the pitcher throws fats or knuckle. If he guesses knuckle, he will either hit .300 or .250 depending on whether the pitcher throws fats or knuckle. In this scenario the pitcher attempts to minimize the batting average, while the batter attempts to maximize his batting average. Please note that at the oucome.250 all arrows point to that point, and no arrow exits the outcome. This indicates that neither player can unilaterally improve a stable situation that is Nash equilibrium with mixed strategy.

Page 404, Question 2a

Build a linear programming model for each player’s decisions and solve it both geometrically and algebraically. Assume the row player is maximizing their payoffs which are shown in the matrices below.

alt text

Page 404, Answer 2a  

Let consider Rose’s decision. Rose wants to choose the mix of R1 and R2 to maximize her average. Let define our variables as follow: A: rose average x: Portion for Rose to guess R1 1- x : Portion for Rose to guess R2 Our objective function: Rose objective is to : Maximize A Constraints What constraints Rose ability to maximize A? Colin is free to choose all C1’s or all C2’s. That is Collin can employ one of his pure strategies against Rose’s mixed strategy.

If Collin pure C1 strategy, we will have :   A<= 10x + 5 (1-x) <= 10x +5 -5x <= 5x +5
A<= 10x + 0 (1-x)
x>= 0   x<=1

Hence our linear system is as follow: Rose objective is to : Maximize
A
Constraints
A<= 5x + 5
A <= 10x
x>=0
x<=1

- Algebraically, the max happens when x = 1 since 5x +5 = 10x when x = 1. - Geometrically, as per below the max happens when x = 1.

plot(c(0, 5), c(0, 20), type='n')
abline(5, 5, col = "red")
text(3,19,"y=5x+5", pos=3)
abline(0,10, col = "blue")
text(2,19,"y = 10x", pos=3)
abline(v=1)
text(1,19,"x<=1", pos=3)
abline(v=0)
text(0,19,"x>=0", pos=3)

Let consider Colin’s decision. Colin wants to choose the mix of C1 and C2 to minimize the average. Let define our variables as follow: A: Colin average y: Portion for Colin to guess C1 1- y : Portion for Colin to guess C2

Our objective function: Colin’ objective is to :

Minimize:
A
Constraints
A<= 10 y + 10(1-y) <= 10
A<= 5y + 0 (1-y) <= 5y
y>=0
y<=1

- Algebraically, the minimum happens when y = 0. - Geometrically, as per below the minimum happens at 5x=0 when x = 0.

plot(c(0, 5), c(0, 20), type='n')
abline(10,0,col = "red")
text(5,10,"y=10", pos=3)
abline(0,5, col = "blue")
text(4,19,"y=5x", pos=3)
abline(v=1,  col = "darkgreen")
text(1,19,"x<=1", pos=3)
abline(v=0,col = "black")
text(0,19,"x>=0", pos=3)

Page 420, Question 1

In the following problems, use the maximin and minimax method and movement diagram to determine if any pure strategy solutions exist. Assume the row player is maximizing his payoffs which are shown in the matrix below:

alt text

Page 420, Answer 1

alt text

From the above diagram, the maximum of the row minimum is 10 and represents Rose max value.

alt text

From the above diagram, the minimum of the row maxis 10 and represents Colin minimax value.

Note:

Please note that the maximin and minimax values are the same; the resulting outcome is called saddle point. If a game has a saddle point (10 is this case), it gives the value of the game. Rose and Colin can guarantee at least this value by choosing their maximin and minimax strategies.