Consider a game between a police officer (player 3) and two drivers (players 1 and 2). Player 1 lives and drives in the Clairemont neighborhood of San Diego, whereas player 2 lives and drives in the Downtown area. On a given day, players 1 and 2 each have to decide whether or not to use their cell phones while driving. They are not friends, so they will not be calling each other. Thus, whether player 1 uses a cell phone is independent of whether player 2 uses a cell phone. Player 3 (the police officer) selects whether to patrol in Clairemont or Downtown. All of these choices are made simultaneously and independently.
Note that the strategy spaces are S1 = {U, N}, S2 = {U, N}, and S3 = {C, D}, where "U" stands for "use cell phone," "N" means "not use cell phone," "C" stands for "Clairemont," and "D" means "Downtown." Suppose that using a cell phone while driving is illegal. Furthermore, if a driver uses a cell phone and player 3 patrols in his or her area (Clairemont for player 1, Downtown for player 2), then this driver is caught and punished. A driver will not be caught if player 3 patrols in the other neighborhood. A driver who does not use a cell phone gets a payoff of zero. A driver who uses a cell phone and is not caught obtains a payoff of 2.
Finally, a driver who uses a cell phone and is caught gets a payoff of -y, where y > 0. Player 3 gets a payoff of 1 if she catches a driver using a cell phone, and she gets zero otherwise.
(a) Does this game have a pure-strategy Nash equilibrium? If so, describe it. If not, explain why
(b) Suppose that y = 1. Calculate and describe a mixed-strategy equilibrium of this game. Explain whether people obey the law.
(c) Suppose that y = 3. Calculate and describe a mixed-strategy equilibrium of this game. Explain whether people obey the law.