(a) Consider a second-price, sealed-bid auction with a seller who has one unit of the object which he values at s and two buyers 1, 2 who have values of v1 and v2 for the object. The values s, v1, v2 are all independent, private values. Suppose that both buyers know that the seller will submit his own sealed bid of s (and will keep the item if bid s wins), but they do not know the value of s. The buyers know that the seller must submit his bid before seeing the buyer’s bids and they know that the seller will actually run a second price auction with the three bids he has: his own bid and the two buyer’s bids. Each buyer knows his own value but not the other buyer’s value.
(b) Now suppose that the seller opens the bids from the buyers and then submits his own bid after seeing the bids from the two buyers. The seller runs a second price auction with these bids in the sense that the object is awarded to the highests bidder (one of the two buyers or the seller) and that bidder pays the second highest bid. Now is it optimal for the buyers to bid truthfully; that is, should they each bid their true value? Give a brief explanation for your answer.
