Problem
Consider the private value auction model from class, with n bidders. Each bidder's value is drawn independently from a uniform distribution on [0,100].
1. Suppose the seller runs an ascending auction. What is the expected price, and the expected value of the winning bidder?
2. From an ex ante perspective, that is, before the bidders have learned their private values, what is the expected profit of any given bidder?
Suppose in order to learn its value and participate in the auction, a bidder must invest an amount k=5. Imagine there are a large number of potential entrants. They play the following game. Each bidder decides whether to enter. Each entrant learns the number of other entrants, draws its value, and bids in the auction.
3. How many bidders will enter the auction? (Note: in a pure strategy entry equilibrium with n entrants, each entrant expects at least zero profit from entering, and each non-entrant expects a negative profit it were to enter.)
4. How would each of the following changes affect equilibrium entry: (i) buyer values are drawn from U[100,200]; (ii) buyer values are drawn from U[0,200]; (iii) instead of an ascending auction, the seller uses a first-price sealed bid auction (Hint: the last one should not take much work).