Suspect A (A), suspect B. (B), and a police officer (P) interact in the following way. A has just robbed a store and is fleeing. He has a choice of two routes to take to try to escape: X and Y. Route Y goes by a club that plays loud music that A does not like so it entails a cost of 20 to him. The music is not payoff relevant for the other two players.

P would like to catch A. B would also like to catch A to settle a score from their previous criminal dealings. Being caught by either P or B yields an immediate payoff of -100 to A. If A is not caught (by either P or B), his payoff is 0.

The three players simultaneously and independently choose between routes X and Y. Assume that if either A or B (or both) chooses the same route as P, P catches that suspect (or both). If A and B select the same route and P selects a different route, B catches A.

If P catches A, P's payoff is 130. P cares only about catching A. So if P catches both A and B, P's payoff is also 130. If P catches only B, P's payoff is 0. If P catches no one, P's payoff is 0.

If B catches A, B's payoff is 70. If B is caught by P (regardless of whether A is also caught by P), B's payoff is -70. If B does not catch A and B is not caught by P, B's payoff is 0.

Describe a Nash equilibrium of this game. Please show any calculations used.