Game Theory and Business Strategy

Solutions to Self Test

Finding Mixed Strategy Equilibria


Question 1

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

What is the mixed strategy equilibrium of the above game?
Let p be the probability of "Run" for the Wumpus, and q the probability of "Run" for the hunter.

SOLUTION:

Consider the equilibrium strategy for the Hunter. First, we must calculate the expected payoff for the Hunter from each strategy:
Run: 60p + 0(1-p)= 60p
Hide: 0p + 20(1-p)= 20-20p
Next, we set these equal, since the Hunter must be indifferent between his strategies, in equilibrium:
  • 60p = 20-20p
  • 20=80p
  • p = 1/4
Calculate the same for Wumpus:
Run: 20q + 0(1-q)
Hide: 0q + 60(1-q)
  • 20q = 60-60q
  • 60=80q
  • q = 3/4
p=1/4, q=1/4
p=1/2, q=1/2
p=1/4, q=3/4
p=3/4, q=1/4


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?

SOLUTION:

Doubling all of the payoffs does not change the equilibrium. Since we find the equilibrium by setting the payoffs from one strategy equal to the other, doubling both does not change the equality.

For example, if all of the payoffs were doubled, the payoffs for Hunter would become:
Run: 120p + 0(1-p)= 100p
Hide: 0p + 40(1-p)= 40-40p
Next, we set these equal, since the Hunter must be indifferent between his strategies, in equilibrium:
  • 120p=40-40p
dividing both sides by two:
  • 60p = 20-20p
is exactly the equation we solved earlier.
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
  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.

SOLUTION:

Consider the equilibrium strategy for Mars. First, we must calculate the expected payoff for Venus from each strategy:
Warm: 90(1-p)
Chill: 10p
Next, we set these equal, since Venus must be indifferent between her strategies, in equilibrium:
  • 90(1-p) = 10p
  • 90-90p=10p
  • 90=100p
  • p = 9/10
Calculate the same for Venus using the payoffs for Mars:
Warm: 20q
Chill: 20-20q
  • 20q = 20-20q
  • 20=40q
  • q = 1/2
p=9/10, q=1/2
p=1/2, q=1/10
p=1/2, q=1/2
p=1/10, q=1/10


Question 4

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

SOLUTION:

Equilibrium payoffs for Mars:
20 (1/2) + 0(1/2) = 10
Equilibrium payoffs for Venus:
90 (1/10) + 0(9/10) = 9

Mars
Venus
Same
It depends