Self Test

Mixed Strategy Equilibria

Question 1.

		Wumpus
		Run	Hide
Hunter	Run	60 , 20	0 , 0
Hunter	Hide	0 , 0	20 , 60

Some games can have both pure strategy Nash equilibria and a mixed strategy Nash equilibrium. What is the mixed strategy Nash equilibrium of the above game?

	Hunter Runs 1/4 of the time and Hides 3/4 of the time; Wumpus Runs 1/4 of the time and Hides 3/4 of the time
	Hunter Runs 1/2 of the time and Hides 1/2 of the time; Wumpus Runs 1/2 of the time and Hides 1/2 of the time
	Hunter Runs 3/4 of the time and Hides 1/4 of the time; Wumpus Runs 1/4 of the time and Hides 3/4 of the time
	Hunter Runs 1/4 of the time and Hides 3/4 of the time; Wumpus Runs 3/4 of the time and Hides 1/4 of the time

Question 2.

Your accounting department announces that due to an error in its procedures, the numbers in the game from question 1 are wrong, and actually each number should be multiplied by 2. Does this change the equilibrium?

	Yes
	No
	Maybe
	It depends

Question 3.

The deities Mars and Venus often do battle to create the weather conditions on Earth. Venus prefers extreme temperatures (especially heat), while Mars prefers temperate conditions. The payoffs (expressed in Points of Wrath) are given below.

		Venus
		Warm	Chill
Mars	Warm	20 , 0	0 , 10
Mars	Chill	0 , 90	20 , 0

What is the unique mixed-strategy equilibrium of the above game?

(Let p be the probability of "Warm" for Mars, and q the probability of "Warm" for Venus.)

	Mars selects Warm with probability 9/10 and Chill with probability 1/10; Venus selects Warm with probability 1/2 and Chill with probability 1/2
	Mars selects Warm with probability 1/2 and Chill with probability 1/2; Venus selects Warm with probability 1/10 and Chill with probability 9/10
	Mars selects Warm with probability 1/2 and Chill with probability 1/2; Venus selects Warm with probability 1/2 and Chill with probability 1/2
	Mars selects Warm with probability 1/10 and Chill with probability 9/10; Venus selects Warm with probability 1/10 and Chill with probability 9/10

Question 4.

In the above game, who earns more Points of Wrath (on average) in equilibrium, Mars or Venus?

	Mars
	Venus
	Same
	It depends

Question 5.

		Player 2
		A	B	C
Player 1	X	8 , 4	4 , 8	1 , 1
	Y	4 , 8	8 , 4	1 , 1
	Z	1 , 1	1 , 1	0 , 0

What is the unique equilibrium of the above game? (HINT: Are any strategies dominated?)

	(X,A)
	Pr{A}=Pr{B}=Pr{C}=Pr{X}=Pr{Y}=Pr{Z}=1/3 (i.e., each player plays each strategy with equal probability)
	Pr{A}=Pr{B}=Pr{X}=Pr{Y}=1/2 (i.e., players play A and B and X and Y with equal probability)
	There is no equilibrium

Be careful. These are tricky!

		Player 2
		Left	Right
Player 1	Left	20 , -20	10 , -10
Player 1	Right	5 , -5	40 , -40

Question 6.

If the payoffs in the top-left box change from [20 , -20] to [40 , -40], how would Player 1's equilibrium strategy change?

	Player 1 would play Left more often
	Player 1 would play Left less often
	Player 1 would play Left just as often
	It depends

Question 7.

If the payoffs in the top-left box change from [20 , -20] to [40 , -40], how would Player 2's equilibrium strategy change?

	Player 2 would play Left more often
	Player 2 would play Left less often
	Player 2 would play Left just as often
	It depends

Question 8.

If the payoffs in the bottom-left box change from [5 , -5] to [10 , -10], how would Player 1's equilibrium strategy change?

	Player 1 would play Left more often
	Player 1 would play Left less often
	Player 1 would play Left just as often
	It depends

Question 9.

If the payoffs in the bottom-left box change from [5 , -5] to [10 , -10], how would Player 2's equilibrium strategy change?

	Player 2 would play Left more often
	Player 2 would play Left less often
	Player 2 would play Left just as often
	It depends

Question 10.

		Player 2
		X	Y
Player 1	A	6 , 6	24 , 2
Player 1	B	30 , 1	0 , 50

What is player 1's expected equilibrium payoff?

	0
	6
	15
	20
	30