UConn
University of Connecticut

Self Test

Mixed Strategy Equilibria

Question 1.

  Wumpus
  RunHide
HunterRun 60 , 20  0 ,  0
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?

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?

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
  WarmChill
MarsWarm 20 ,  0  0 , 10
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.)

Question 4.

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

Question 5.

  Player 2
  ABC
Player 1X 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?)
Be careful. These are tricky!
  Player 2
  LeftRight
Player 1Left 20 , -20 10 , -10
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?

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?

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?

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?

Question 10.

  Player 2
  XY
Player 1A  6 ,  6 24 ,  2
B 30 ,  1  0 , 50
What is player 1's expected equilibrium payoff?