1) Here is a game that illustrates how the government balances the social cost of crime with law-enforcement costs and how criminals balance the value of illegal activity with the probability of arrest. The game has two players: a criminal (C) and the government (G). The government selects a level of law enforcement, which is a number x 0. The criminal selects a level of crime, y 0. These choices are made simultaneously and independently. The government's payoff is given by uG = x(c^4) (y^2)/x with the interpretation that (y^2)/x is the negative effect of crime on society (moderated by law enforcement) and c^4 is the cost of law enforcement, per unit of enforcement. The number c is a positive constant.

The criminal's payoff is given by uC = (y)/(1+xy) with the interpretation that y is the value of criminal activity when the criminal is not caught, whereas 1/(1+xy) is the probability that the criminal evades capture.

a) Compute the Nash equilibrium of this game.

b) Explain how the equilibrium levels of crime and enforcement change as c increases.

2) Consider the following three-player team production problem. Simultaneously and independently, each player chooses between exerting effort (E) or not exerting effort (N). Exerting effort imposes a cost of 2 on the player who exerts effort. If two or more of the players exert effort, each player receives a benefit of 4 regardless of whether she herself exerted effort. Otherwise, each player receives zero benefit. The payoff to each player is her realized benefit less the cost of her effort (if she exerted effort). For instance, if player 1 selects N and players 2 and 3 both select E, then the payoff vector is (4,2,2). If player 1 selects E and players 2 and 3 both select N , then the payoff vector is (2, 0, 0).

Find a symmetric mixed-strategy Nash equilibrium of this game. Let p denote the probability that an individual player selects N.