Trading Places: Two players, 1 and 2, each own a house. Each player i values his own house at vi. The value of player i's house to the other player, i.e., to player j = i, is 23vi. Each player i knows the value vi of his own house to himself, but not the value of the other player's house. The values vi are drawn independently from the interval [0, 1] with uniform distribution.

1. Suppose players announce simultaneously whether they want to ex- change their houses. If both players agree to an exchange, the exchange takes place. Otherwise no exchange takes place. Find a Bayesian Nash equilibrium of this game in pure strategies in which each player i ac- cepts an exchange if and only if the value vi does not exceed some threshold i.
2. How would your answer to (a) change if player j 's valuation of player i's house were 25vi?
3. Try to explain why any Bayesian Nash equilibrium of the game de- scribed in (a) must involve threshold strategies of the type postulated in (a).