Question

Consider the following simultaneous-move game:

			Column
		L	M	N	P
Row	U	(1,1)	(2,2)	(3,4)	(9,3)
	D	(2,5)	(3,3)	(1,2)	(7,1)

(a) Find all pure-strategy Nash equilibria.

(b) Suppose Row mixes between strategies U and D in the proportions p and (1 − p). Graph the payoffs of Column’s four strategies as functions of p. What is Column’s best response to Row’s p-mix?

(c) Find the mixed-strategy Nash equilibrium. What are the players’ expected payoffs?

Answer 1

a) Given Player 2,

1) if player 1 chooses L, his best option is (2,5)

2) if player 1 chooses M, his best option is (3,3)

3) if player 1 chooses N, his best option is (3,4)

4) if player 1 chooses P, his best option is (9,3)

Giver Player 1,

1) if player 2 chooses U, his best option is (3,4)

2) if player 2 chooses D, his best option is (2,5)

Both (3,4) and (2,5) are chosen by both the players. both are pure strategy Nash Equilibria.

b) Row mixes between strategies U and D in proportions p and (1-p).

E(L)= p+5(1-p)

E(M)=2p+ 3(1-p)

E(N)= 4p + 2(1-p)

E(P)= 3p + (1-p)

The maximin point in the diagram gives us the highest payoff. It is given by the intersection of E(p) and E(M) at point A.

Solving E(P) and E(M) we get the probabilities as (2/3,1/3). Thus there is a mixed strategy Nash Equilibrium with probabilities (2/3, 1/3).