Suppose two game-show contestants, Alex and Bob, each separately select one of three doors numbered 1, 2, and 3. Both players get dollar prizes if their choices match, as indicated in the following...

Suppose two game-show contestants, Alex and Bob, each separately select one of three doors numbered 1, 2, and 3. Both players get dollar prizes if their choices match, as indicated in the following table:

(a) What are the Nash equilibria of this game? Which, if any, is likely to emerge as the (focal) outcome? Explain.

(b) Consider a slightly changed game in which the choices are again just numbers, but the two cells with (15, 15) in the table become (25, 25). What is the expected (average) payoff to each player if each flips a coin to decide whether to play 2 or 3? Is this better than focusing on both of them choosing 1 as a focal equilibrium? How should you account for the risk that Alex might do one thing while Bob does the other?