1. Two players, Sauron (player 1) and Saruman (player 2), each own a house. Each player values his own house at vi. The value of player i 's house to the other player, i.e. to player j=i, is vic where >1. Each player knows the value vi of his own house to himself, but not the value of the opponent's house. Both players know . The values vi are distributed uniformly on the interval [0,1] and are independent across players. (a) Suppose players announce simultaneously whether they want to exchange their houses (without paying each other). If both players agree to an exchange, the exchange takes place. Otherwise, they stay in their own houses. Find a Bayesian Nash equilibirum of this game in pure strategies. (b) How does this equilibrium depend on ? In particular how does the probability of exchage depends on in this equilibrium? Is the equilibrium outcome always efficient? (c) Give an intuition about why we should focus on these threshold strategies when looking for an equilibrium. 1. Two players, Sauron (player 1) and Saruman (player 2), each own a house. Each player values his own house at vi. The value of player i 's house to the other player, i.e. to player j=i, is vic where >1. Each player knows the value vi of his own house to himself, but not the value of the opponent's house. Both players know . The values vi are distributed uniformly on the interval [0,1] and are independent across players. (a) Suppose players announce simultaneously whether they want to exchange their houses (without paying each other). If both players agree to an exchange, the exchange takes place. Otherwise, they stay in their own houses. Find a Bayesian Nash equilibirum of this game in pure strategies. (b) How does this equilibrium depend on ? In particular how does the probability of exchage depends on in this equilibrium? Is the equilibrium outcome always efficient? (c) Give an intuition about why we should focus on these threshold strategies when looking for an equilibrium