Question

	x	y
a	7,3	5,4
b	1,4	6,2

Identify the set of Nash equilibria of this game (pure and mixed strategy).

Answer 1

Let us first look at pure strategy nash equilibrium.

The dominant strategy for player 1 is to choose a when player 2 chooses x ( 7>1) and to choose b when player 2 chooses y (6>5)

The dominant strategy for player 2 is to choose y when player 1 chooses a ( 4>3) and to choose x when player 1 chooses b (4>2)

Thus, it is clear that there no pure strategy nash equilibrium here.

Mixed strategy:

Let player 1 choose a with probability p and b with probability 1-p

Let player 2 choose x with probability q and y with probability 1-q

For player 1 to be indifferent between choosing a and b, expected value from both must be equal:

E(a) = E(b) => 7q + 5(1-q) = q + 6(1-q)

=> 2q + 5 = 6 - 5q => 7q =1 => q = 1/7

For player 2 to be indifferent between choosing x and y, expected value from both must be equal:

E(x) = E(y) => 3p + 4(1-p) = 4p + 2(1-p)

=> 4 -p = 2p + 2 => 3p =2 => p = 2/3

Thus, we have a mixed strategy equilibrium in which player 1 chooses a with probability 2/3 and player 2 chooses x with probability 1/7