Question

Consider a game between a police officer (player 3) and two drivers (players 1 and 2). Player 1 lives and drives in Wynwood, whereas player 2 lives and drives in Sweetwater. 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 Wynwood or Sweetwater. All of these choices are made simultaneously and independently. We can describe player 1 and 2’s choices as each choosing between U and N, where “U” stands for “use cell phone” and “N” means “not use cell phone.” The police officer chooses between W and S, where “W” stands for “Wynwood” and “S” means “Sweetwater.”

Suppose that using a cell phone while driving is illegal. Furthermore, if a driver uses a cell phone and player 3 patrols in his/her area (Wynwood for player 1, Sweetwater 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 3. 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.

Does this game have a pure-strategy Nash equilibrium? (By pure strategy, we mean that player 1 chooses either U or N with probability 1, player 2 chooses either U or N with probability 1, and player 3 chooses either W or S with probability 1. That is none of the players randomizes with positive probability on both of his or her possible action choices.) If so, describe it. If not, explain why in a two or three sentences.

Answer 1