Question

	F	B	C
A	5,-1	1,3	10,0
B	3,3	3,-1	10,0
C	1,3	6,0	10,1

Consider the following normal form game:

Part a: Are there any pure strategy Nash equilibria?

Part b: Are there any mixed strategy Nash equilibria in which player 1 plays all three strategies, A, B, C with positive probability? Justify your answer.

Part c: Are there any mixed strategy Nash equilibria in which player1 plays only A, B with positive probability? Justify your answer.

Part d: Are there any mixed strategy Nash equilibria in which player 1 plays only B, C with positive

probability? Justify your answer.

Answer 1

Answer:

Given that:

Consider the following normal form game:

	F	B	C
A	5,-1	1,3	10,0
B	3,3	3,-1	10,0
C	1,3	6,0	10,1

Part a: Are there any pure strategy Nash equilibria?

There is no any pure strategy Nash Equilibrium

Part b: Are there any mixed strategy Nash equilibria in which player 1 plays all three strategies, A, B, C with positive probability? .

Let the probabilty of player 1 going for A,B,C be p1, p2, 1-p1-p2

Probabilty of player 2 going for F,B,C be q1, q2, 1-q1-q2

So, expected payoffs for A,B,C for player 1:-

From these three equations, we will get,

The expected payoffs for A,B,C for player 1 is 0 in each case. So there is no any mixed strategy Nash Equilibrium.

Part c: Are there any mixed strategy Nash equilibria in which player1 plays only A, B with positive probability?

Player 1 plays only A & B

This implies that, player 2 cannot mix the probability q1 and q2.

So, there is no incentive for player 1 to randomize betweern A and B.

So, mixed strategy Nash Equilibrium doesn't exist.

Part d: Are there any mixed strategy Nash equilibria in which player 1 plays only B, C with positive probability?

Player 1 only plays B & C

This tells that if player 2 is willing to mix the probability q1 and q2, player 1 can randomize between B & C.

So, mixed strategy Nash equilibrium exists.