Course: Game Theory

Please solve as soon as possible.

a Exercise 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 -xct with the interpretation that - is the negative effect of crime on society (moderated by law enforcement) and c4 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 with the interpretation that y1/2 is the value of criminal activity when the criminal is not caught, whereas is the probability that the criminal evades capture. - 31/2 1+ xy 1 1+ xy (a) Write the first-order conditions that define the players' best-response functions and solve them to find the best-response functions. (b) Compute the Nash equilibrium of this game. (c) Explain how the equilibrium levels of crime and enforcement change as c increases