A model of a two-stage all-pay auction

Abstract

The present paper studies a simple two-stage model of an all-pay auction under complete information. All-pay auctions are often used to model competition with irreversible investments such as political lobbying, and in the existing models, the equilibrium outcomes are quite different from the winner-pay auctions (under complete information): The unique equilibrium is in non-degenerate mixed strategies in the sealed-bid all-pay auction, and the highest value bidder wins at (virtually) no cost in the dollar auction. In sharp contrast with those existing models, the equilibrium outcome in the present setting is almost identical to the winner-pay auctions. That is, (a) the highest value bidder wins with probability one, and (b) the revenue of the seller is equal to the second highest value among the bidders. Also, from a mechanism-design point of view, the present game form is more robust than other all-pay mechanisms in that the seller does not need any information about the bidders’ valuations. Although the analysis focuses on the two-bidder two-stage case, the results extend to arbitrary numbers of bidders and stages.