Question

3. Player 1 and Player 2 are going to play the following stage game twice:

		Player 2
		Left	Middle	Right
Player 1	Top	4, 3	0, 0	1, 4
	Bottom	0, 0	2, 1	0, 0

There is no discounting in this problem and so a player’s payoff in this repeated game is the sum of her payoffs in the two plays of the stage game.

(a) Find the Nash equilibria of the stage game. Is (Top, Left) a Nash of the stage game?

(b) Find a subgame perfect Nash equilibrium of the repeated game where the first time they play the stage game Player 1 chooses Top and Player 2 chooses Left.

3. Player 1 and Player 2 are going to play the following stage game twice: Player 2 Middle 0,0 2,1 Left 4,3 0,0 Right 1,4 Top Player 1 Bottom 0,0 There is no discounting in this problem and so a player's payoff in this repeated game is the sum of her payoffs in the two plays of the stage game. (a) Find the Nash equilibria of the stage game. Is (Top, Left) a Nash of the stage game? (b) Find a subgame perfect Nash equilibrium of the repeated game where the first time they play the stage game Player 1 chooses Top and Player 2 chooses Left.

Answer 1

	Left	middle	right
Top	(4*,3)	(0,0)	(1*,4•)
Bottom	(0,0)	(2*,1•)	(0,0)

A) NE :

( Bottom, middle ) & ( top , right )

(Top , left) is not NE of game

.

b)now player 1, will never deviate from top, if P2 is playing left, bcoz P1 is already earning the Maximum Payoff

So when P1 plays top, then P2 has Incentive to deviate to right, bcoz he gets more payoff than left

So if P2 deviates, then in next period they will play NE of the game , which is (Bottom, middle)

So total deviation payoff to P2 = 4+1 = 5

If P2 maintains Cooperation, then in second stage of game, then they play (top, right) , otherwise they play (bottom , middle )

.

Bcoz Cooperation payoff to P2 = 3+4 = 7

so the SPNE strategy

for P2

play Left in first period , then play right in second stage

• if P2 cheats, deviates, then play middle in second stage

For P1

• play Top in first period ,  

• play Top in second period , if P2 plays right, otherwise play bottom