Question

3. Consider the following game:

	A	B
A	5,8	5,10
B	6,14	4,8

a. List all of the Pure Strategy Nash Equilibria of this game.

b. There is also a mixed strategy Nash Equilibrium to this game. In that equilibrium, with what probabilities will each player play action A? Action B?

Answer 1

a. Given that player 1 choose A, player 2's best response is B(10).
Given that player 1 choose B, player 2's best response is A(14).

Given that player 2 choose A, player 2's best response is B(6).
Given that player 2 choose B, player 2's best response is A(5).

So, there are two Nash equilibria in pure strategy. They are (A,B) and (B,A) or (5,10) and (6,14) as best response of both players occur simultaneously when they choose these strategies.

b. Let player 1 choose A with probability, p and B with probability, 1-p. If player 2 best responds with a mixed strategy then player 1 must make him indifferent between his strategies such that his expected payoff from his two strategies is equal. That is,
E2(A) = E2(B)
Or, 8p + 14(1-p) = 10p + 8(1-p)
So, 8p + 14 - 14p = 10p + 8 - 8p
So, 10p - 8p - 8p + 14p = 14 - 8
So, 8p = 6
So, p = 6/8 = 3/4
And, 1-p = 1 - (3/4) = 1/4

Let player 2 choose A with probability, q and B with probability, 1-q. If player 1 best responds with a mixed strategy then player 2 must make him indifferent between his strategies such that his expected payoff from his two strategies is equal. That is,
E1(A) = E1(B)
Or, 5q + 5(1-q) = 6q + 4(1-q)
So, 5q + 5 - 5q = 6q + 4 - 4q
So, 6q - 4q = 5 - 4
So, 2q = 1
So, q = 1/2
And, 1-q = 1 - (1/2) = 1/2

Player 1 will play A with probability 3/4 and B with probability 1/4, and Player 2 will play A with probability 1/2 and B with probability 1/2.