Two suspects in a major crime are held in separate cells

Question

Econ 521 - Week 3:

1. Prisoner's Dilemma-

Two suspects in a major crime are held in separate cells. There is enough evidence to convict each of them of a minor offense, but not enough evidence to convict either of them of the major crime unless one of them acts as an informer against the other (finks). If they both stay quiet, each will be convicted of the minor offense and spend one year in prison (equivalent to a payoff of 2 for each). If one and only one of them finks, she will be freed (payoff 3) and used as a witness against the other, who will spend four years in prison (payoff 0). If they both fink, each will spend three years in prison (payoff 1).

(a) Define this situation formally as a game. Represent the game in a matrix.

(b) Solve the game using iterated removal of strictly dominated actions.

Suppose now players have altruistic preferences. Now, each of the two players have the same possible actions, but each action pair results in the player's receiving a payoff equal to m_i(a) + αm_j(a), where m_i(a) is the payoff for player i when action profile is a, according to the matrix in (a).

(c) Find the range of values of α for which the resulting game is the Prisoner's Dilemma. For values of a for which the game is not the Prisoner's Dilemma, solve the game using iterated removal of strictly dominated actions.

2. Matching Pennies-

Two people choose, simultaneously, whether to show the head or the tail of a coin. If they both show head, each gets one dollar. If player 1 shows tail and player 2 shows head, then player 1 gets the two dollars. If both shows tail, then both get 1 and if player 1 shows head and player 2 tail the payoffs are (x, 1 - x).

(a) Define this situation formally as a game. Represent the game in a matrix.

(b) Is this a Prisoners Dilemma game? Explain.

3. Consider the normal-form game pictures here:

		2
		X	Y	Z
1	A	2,0	1,3	6,x
	B	6,4	1,3	5,2

All of the payoff numbers are specified, with the exception of that denoted by x. Find a number for x such that the following three statements are all true: (A, Z) is a Nash equilibrium, (A, Z) is the social optimum profile, and, for the belief θ₁ = (a ≥ ½, b ≤ ½), Z is a best response for player 2; that is Z ∈ BR₂(θ₁).