Question

Problem 1

In each of the three games shown below, let p be the probability that player 1 plays cooperates (and 1- p the probability that player 1 defects), and let q be the probability that Player 2 plays cooperates (and 1- q the probability that player 2 defects).

Prisoner’s Dilemma

	Player 2
Player 1		cooperate	defect
	cooperate	70,70	10,80
	defect	80,10	40,40

Stag Hunt

	Player 2
Player 1		cooperate	defect
	cooperate	70,70	5,40
	defect	40,5	40,40

Chicken

	Player 2
Player 1		cooperate	defect
	cooperate	70,70	50,80
	defect	80,50	40,40

1. For each game, draw a graph with player 1’s best response function (choice of p as a function of q), and player 2’s best response function (choice of q as a function of p), with p on the horizontal axis and q on the vertical axis.

2. Using this graphs, find all the Nash equilibriums for the game, both pure and mixed strategy Nash equilibriums (if any). Label these equilibriums on the corresponding graph.

3. In those games that have multiple pure strategy Nash equilibriums, how do the expected payoffs from playing the mixed strategy Nash equilibrium compare with the payoffs from playing the pure strategy Nash equilibriums? Which type of strategy (mixed or pure) would players prefer to play in these games?

Problem 2

Two people are involved in a dispute. Player 1 does not know whether player 2 is strong or weak; she assigns probability α to player 2 being strong. Player 2 is fully informed. Each player can either fight or yield. Each player obtains a payoff of 0 is she yields (regardless of the other person’s action) and a payoff of 1 if she fights and her opponent yields. If both players fight, then their payoffs are (-1; 1) if player 2 is strong and (1;-1) if player 2 is weak. The Bayesian game is the following, depending on the type of player 2:

	Y	F			Y	F
Y	0, 0	0, 1		Y	0, 0	0, 1
F	1, 0	-1, 1		F	1, 0	1, -1
Player 2 is strong (α)				Player 2 is weak (1-α)					Player 2 is strong (α)

After writing all the strategies and payoffs in the same matrix, find the Bayesian Nash equilibriums, depending on the value of α (α ≤ 1/2 or α ≥1/2).