Problem 2 - A Simple Auction A seller wishes to auction 3 single indivisible object among n potential buyers each of whom has values drawn independently from some absolutely continuous distribution F on [0, 1] with the associated density f. Let vi be buyer i's valuation. The auction mechanism used is a sealed-bid second price auction. Ties are broken by assigning the object randomly to one of the highest bidders. Suppose the seller's valuation is also drawn from F independently and that he also sub- mits a bid. If the seller's bid is the highest, he simply keeps the object and his payoff is equal to his valuation. If not, the highest bidder pays the second highest bid, and the seller's pay- off is the payment received from the winner. We thus have a sealed second-price auction with (n + 1) bidders. (a) Confirm that it is still a weakly dominant strategy for buyers to submit their own val- uations, regardless of the seller's valuation and bid. (b) Let Y, be the highest valuation among n buyers and 12 be the second highest. Show that Pr( Y, Y . The second term corresponds to the case Yz b. You will find it useful to define the distribution function of the second highest valuation conditional on Yz > b, which is given by G(x) = Pr(b b) for x 2 b and to compute its associated density g( x) = G'(x). Note: You can answer (d)-(f) correctly using a(b) as given above even if you cannot fully answer (c). (d) Argue that it is not optimal for the seller to bid his true valuation. (e) Derive the seller's optimal bid function when F is a uniform distribution. (f) Suppose that the seller cannot make a bid by himself, possibly because a law or some other rule prohibits her from doing so. But the seller can set a reserve price. That is, any bidder can either bid weakly above the reserve price or not participate in the auction. Revert to the assumption that the seller's valuation is zero. Use the results you have obtained above to suggest a modification of the second price mechanism which leads to an increase in the seller's expected payoff