1. Alice and Bob have inherited a factory from their parents. The value of the factory is vA for Alice and vB for Bob, where vA and vB are independently and uniformly distributed over [0, 1], and each of them knows his or her own value. Simultaneously, Alice and Bob bid bA and bB, respectively, and the highest bidder wins the factory and pays the other sibling's bid. (If the bids are equal, the winner is determined by a coin toss.) Find a symmetric, linear Bayesian Nash equilibrium of this game.

2. An object is to be sold to one of n bidders through a first-price auction. The value of object for bidder i is vi 0, which is privately known by the bidder i. The values (v1, . . . , vn) are independently and identically distributed with probability density function f and cumulative distribution function F. formally as a Bayesian game and compute the symmetric Bayesian Nash equilibria in increasing differentiable strategies.

3. There are n hunters. Simultaneously each hunter chooses between Stag and Rabbit. If a hunter chooses Rabbit, his payoff is regardless of what other hunters do. If he chooses Stag, he gets v > 0 if at least k other hunters also choose Stag and gets 0 otherwise. The possible values for is [a, b] where a < 0 and b > v. (a) Assume that is known and compute the set of pure strategy Nash equilibria. (b) Assume that is not known. Each player i observes a signal xi = + i where 0 < min {a, b v} and (1, . . . , ) are independently and identically dis- n tributed with uniform distribution on [1, 1]. Find the set of symmetric, monotone Bayesian Nash equilibria. (That is, find the possible values for a cutoff x such that each player i chooses Stag if and only if xi x in equilibrium.) Hint: By symmetry, the probability that xi is the mth highest among (x1, . . . , xn) is 1/n. (c) Briefiy discuss your answers.

4. Alice owns a car, and Bob may buy it. The car can be a Lemon or a Peach. Alice knows whether it is Lemon or Peach, while Bob assigns probability 1/2 to each case. The value of a Lemon is $1,000 for Alice and $3,000 for Bob, and the value of a Peach is $5,000 for Alice and $10,000 for Bob. Alice sets a price p P, and Bob decides whether to accept the price. If the price is accepted, they trade the car at price p; no trade occurs otherwise. For each set P below find a sequential equilibrium and verify that it is indeed a sequential equilibrium.

(a) P = {2000, 7000} (b) P = [0, ).