(Auction Design) A seller hopes to sell an item, and has two bidders, A and B. The seller knows that the item is either ideal for A, ideal for B or ideal for neither. The probability of each of the first two events is z, and there is a 1-2z probability the object will be ideal for neither. If the item is ideal for a bidder, the bidder will value winning at 100. Otherwise, a bidder will value winning at 50. The two bidders can recognize whether an item is ideal for them, and hence know their individual values.

a. If the seller runs a 2nd price auction, what is the optimal strategy for each bidder, and the Nash equilibrium (mathematically)?

b. What is the expected revenue from the 2nd price auction (mathematically)?

c. Can the publisher raise revenue by setting a reserve price? By how much? (mathematically)

d. Suppose instead that the seller announces that it will first offer to sell at price p_H, and then if no one offers to buy at that price, offer price at p_L, and that if more than one bidder wants to buy at a given price, it will randomize between them in deciding the winner. Find the choices of p_H and p_L (mathematically) that maximize the seller's revenue, assuming fully understand the mechanism and respond strategically. e. Compare the seller's revenue (mathematically) in b, c, and d. Which is the revenue-maximizing option for the seller?