Question

1. Find ALL Nash Equilibria (pure strategy and mixed) of the following games. Also identify the efficient outcome of each game:

a)

		Player 2
		Left	Right
Player 1	Up	5,6	4,9
	Down	7,7	3,8

b)

		Player 2
		Left	Right
Player 1	Up	4,3	1,10
	Down	2,5	4,2

c)

		Player 2
		Left	Right
Player 1	Up	5,5	4,4
	Down	4,2	7,3

2. Find all of the PURE STRATEGY Nash Equilibria to the following game:

		Player 2
		Left	Center	Right
Player 1	Up	5,5	7,7	4,4
	Middle	9,1	3,7	7,8
	Down	4,2	7,3	7,3

Answer 1

Q1) a)

PLAYER 1/PLAYER 2	LEFT	RIGHT
UP	5, 6	4, 9
DOWN	7, 7	3, 8

PURE STRATEGY EQUILIBRIUM

If player 1 chooses up, then it is profitable for player 2 to choose right due to a higher pay-off of 9.

If player 1 chooses down, then player 2 will choose right due to to a higher pay off of 8 as compared to 7.

If player 2 chooses left, then player 1 will choose down to get a higher pay off of 7 as compared to 5.

If player 2 chooses right, then player 1 will choose up to get a higher pay-off of 4 as compared to 3.

So, 4, 9 is the pure strategy nash equilibrium.

b)

PLAYER 1/PLAYER 2	LEFT	RIGHT
UP	4 , 3	1, 10
DOWN	2 , 5	4 , 2

PURE STRATEGY EQUILIBRIUM

If player 1 chooses up, then player 2 will choose right to get a higher pay off of 10.

If player 1 chooses down, then player 2 will choose left to get a higher pay off of 5 as compared to 2.

If player 2 chooses left, then player 1 will choose up.

If player 2 chooses right, then player 1 will choose down.

So, in this case there is no pure strategy nash equilibrium.

MIXED STRATEGY EQUILIBRIUM

Let player 1 choose up with probability q and down with probability 1 - q.

Let player 2 choose left with probability p and right with probability 1 - p.

PAY-OFF PLAYER 1

If player 2 choose left, then pay off 1 will be 4q + 2(1 - q)

If player 2 chooses right, then pay off player 1 will be q + 4(1 - q)

In equilibrium both these pay offs will be equal.

So, 4q + 2(1 - q) = q + 4(1 - q)

4q + 2 - 2q = q + 4 - 4q

2q + 2 = 4 - 3q

5q = 2

q = 2/5 and 1-q = 3/5 is the answer.

PAY-OFF PLAYER 2

If player 1 chooses up, then pay off of player 2 will be 3p + 10(1 - p)

If player 1 chooses down, then pay off of player 2 will be 5p + 2(1 - p)

In equilibrium both these pay offs will be equal.

So, 3p + 10(1 - p) = 5p + 2(1 - p)

10 - 7p = 2 + 3p

10p = 8

p = 4/5 and 1-p = 1/5 is the answer.

c)

PLAYER 1/PLAYER 2	LEFT	RIGHT
UP	5 , 5	4, 4
DOWN	4, 2	7, 3

PURE STRATEGY EQUILIBRIUM

If player 1 chooses up, then player 2 will choose left to get a higher pay off of 5.

If player 1 chooses down, then player 2 will choose right to get a higher pay off of 3 as compared to 2.

If player 2 chooses left, then player 1 will choose up.

If player 2 chooses right, then player 1 will choose down.

So, this game has 2 pure strategy nash equilibrium at 5,5 and 7,3.

MIXED STRATEGY EQUILIBRIUM

Let player 1 choose up with probability q and down with probability 1 - q.

Let player 2 choose left with probability p and right with probability 1 - p.

PAY-OFF PLAYER 1

If player 2 choose left, then pay off 1 will be 5q + 2(1 - q)

If player 2 chooses right, then pay off player 1 will be 4q + 3(1 - q)

In equilibrium both these pay offs will be equal.

So, 5q + 2(1 - q) = 4q + 3(1 - q)

3q + 2 = q + 3

2q = 1

q = 1/2 and 1-q = 1/2 is the answer.

PAY-OFF PLAYER 2

If player 1 chooses up, then pay off of player 2 will be 5p + 4(1 - p)

If player 1 chooses down, then pay off of player 2 will be 4p + 7(1 - p)

In equilibrium both these pay offs will be equal.

So, 5p + 4(1 - p) = 4p + 7(1 - p)

p + 4 = 7 - 3p

4p = 3

p = 3/4 and 1-p = 1/4 is the answer.