Question

Consider a Hawk-Dove game which has been used by evolutionary biologists to model animal conflicts:

	H	D
H	(0, 0)	(4, 1)
D	(1, 4)	(2, 2)

Find all pure strategy Nash equilibria in this game and a one mixed-strategy Nash equilibrium.

Answer 1

When player 2 selects H, player 1 selects D and when player 2 selects D player 1 selects H. The game is symmetric so the same is true for player 2. The two NE in pure strategies are when players select opposing strategies (H, D) and (D, H)

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. For Firm 2, assume that the probability with which he selects ‘H’ is p and the probability of choosing ‘D’ is 1-p. Given this, Firm 1’s expected payoff from choosing ‘H’ is 0p + 4(1 – p) = 4 - 4p and the expected payoff from choosing ‘D’ is 1p + 2(1 – p) = 2 – P. Firm 1 must be indifferent between these two payoffs if he is to randomize. This implies:

4 - 4p = 2 - p

2 = 3p

p = 2/3

The game is symmetric so we have a mixed strategy NE where each player selects H with a probability of 2/3 and D with a probability of 1/3.