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Banning bidders from all-pay auctions.

Article Excerpt
Abstract We consider an all-pay auction with complete information among the bidders; the seller does not observe the bidders' values. We show that for some information structures in which the seller has a small uncertainty about the valuations, it is profitable for him to exclude from the auction all but two (randomly selected) bidders even though the latter are ex ante identical from his point of view.

Keywords All-pay auction * Complete information * Exclusion principle

JEL Classification Numbers D44 * D82

1 Introduction

Consider an auction with n [greater than or equal to] 3 ex ante symmetric bidders. Is it possible that the seller gains by banning some bidders from participating in the auction? A positive answer to this question would be somewhat surprising, since traditional economic wisdom predicts that the stronger the competition on one side of the market (as measured by the number of competitors), the better off are the agents on the other side of the market. Therefore, more bidders should increase the seller's revenue. Indeed, according to Klemperer (2004), one of the most important issues in auction design is "to attract bidders, since an auction with too few bidders risks being unprofitable for the auctioneer" (Section 3.3, p 106).

In the basic auction setting with incomplete information and i.i.d. private values the following consequence of the revenue equivalence theorem1 is well known. In a wide class of auctions, including the first price auction, the Vickrey auction, the ascending auction, and the all-pay auction, the expected revenue is equal to the expected second highest valuation among the bidders who participate in the auction. In these cases, therefore, the seller wants to maximize the number of bidders in the auction. Conversely, in some auctions with common values the opposite result is obtained when the severity of the winner's curse increases quickly with the number of bidders: see Bulow and Klemperer (2002) and Matthews (1984), for instance.

In this short paper, we prove that more bidders reduce the revenue in a simple environment with private values; hence, no winner's curse is at work here. Specifically, we consider a (first price) all-pay auction in which the n bidders' valuations are common knowledge among bidders; the seller does not observe these values and regards them as i.i.d. random variables with a distribution F. Suppose for one moment that F is degenerate at a point [v.sub.L] > 0; this means that bidder i's valuation is equal to [v.sub.L] with probability one, for i = 1, 2,..., n. In this case, the seller has no incentive to exclude any bidder from the auction. Now suppose that F is slightly modified such that a bidder's value is equal to [v.sub.L] with probability close to one and equal to [v.sub.H] > [v.sub.L] with probability close to zero. In this setting the seller is almost certain that all valuations are equal to [v.sub.L] and we prove that the expected revenue is higher in an auction with...