(c) (10 points) Give an example of a two player game with at least two mixed Nash equilibria
(that are not pure Nash equilibria).
Solution: Any pair of mixed strategies in the game below is NE.
Player 2
a
b
A
(0,0)
(0,0)
Player 1
B
(0,0)
(0,0)
Problem 2 (30 points)
. Consider the following two player game:
Player 2
H
M
L
H
(0,0)
(3,4)
(6,0)
Player 1
M
(4,3)
(0,0)
(0,0)
L
(0,6)
(0,0)
(5,5)
(a) (5 points) Are any strategies strictly dominated?
Solution: L is strictly dominated by a mixture of H and M; e.g., suppose H is played with
probability 0.99, and M with probability 0.01. The resulting payoff is 0.04 if the opponent
plays H; 2.97 if the opponent plays M; and 5.94 if the opponent plays L. In all cases the
payoff is strictly higher than the payoff obtained by playing L.
(b) (5 points) Find all pure Nash equilibria of this game.
Solution: (H,M) and (M,H) are the pure NE.
(c) (10 points) Find a symmetric mixed Nash equilibrium of this game, i.e., where both players
play the same mixed strategy.
Solution: By (a), L cannot be played in an NE. So we assume Player 2 plays H with proba
bility
p
, and M with probability
1

p
. Player 1 must be indifferent between H and M, so we
require
4
p
= 3(1

p
)
, i.e.,
p
= 3
/
7
. Similarly, if player 1 plays H with probability 3/7, and
M with probability 4/7, then player 2 will be indifferent between H and M. So both players
using this mixed strategy is a symmetric mixed NE.
3