Auctions in which losers set the price

We study auctions of a single asset among symmetric bidders with a¢ liated values. We show that the second-price auction minimizes revenue among all e¢ cient auction mechanisms in which only the winner pays, and the price only depends on the losers bids. In particular, we show that the k-th price auction generates higher revenue than the second-price auction, for all k > 2. If rationing is allowed, with shares of the asset rationed among the t highest bidders, then the (t + 1)-st price auction yields the lowest revenue among all auctions with rationing in which only the winners pay and the unit price only depends on the losers bids. Finally, we compute bidding functions and revenue of the k-th price auction, with and without rationing, for an illustrative example much used in the experimental literature to study rst-price, second-price and English auctions. Journal of Economic Literature Classi cation Numbe