A seller wishes to auction a 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 submits 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 payoff 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 valuations, regardless of the seller's valuation and bid.

(b) Let v(1) be the highest valuation among n buyers and v(2) be the second highest. Show 1 that Pr(x > v(1) ) = F (x ) n Pr(v(2)

A seller wishes to auction a single indivisible object among :1 potential buyers each of whom has values drawn independently from some absolutely continuous distribution F on [[1, 1] with the associated density f. Let y; be buyer is 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. lithe 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} Conrm 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 I'll] be the highest valuation among :1 buyers and L12] be the second highest. Show