Question

In the following games indicate any pure strategy Nash equilibria that exist. Then find the mixed strategy Nash equilibrium for each game.

Player 2

Left Colum | Center Column | Right Column

U 100,0	25,75	0, 100
M 25,75	100, 0	25, 75
D 0,100	25, 75	100, 0

Answer 1

a) There is no pure strategy Nash equilibrium exists for the game. This is shown by the choices made by each player which are circled below. There is no single strategy set where both the startegies meet. Player 1 as well as player 2 both have no dominant strategies.

b) Let player 1 selects U with a probability p, M with a probability of q and D with a probability 1 - p - q. In the mixed-strategy Nash equilibrium, players are found to be randomizing and each player must be indifferent in choosing the available set of strategies. Player 2 randomizes when he is indifference between selecting left, cente or right. Then the expected payoff from these must be same.

Evaluate the expected profit from selecting Left, cen-t-r-e and right

E(L) = 75q + 100(1 - p - q) or 100 - 100p - 25q, E(C) = 75p + 75(1 - p - q) = 75 - 75q and E(R) = 100q + 75p

Solve these equations to get p = 3/8 and q = 1/4

The game is symmetric and so the probabilities are same for both players. This is the mixed strategy Nash Equilibrium.