On the “Exclusion Principle” in All-Pay Auctions with Incomplete Information

The number, and types, of participants in an auction can meaningfully affect the outcome of the competition. In a complete-information, all-pay context, previous studies proved that in first-price auctions, if the auctioneer maximizes the total bid received, then under appropriate conditions on the bidders’ values, it may be optimal for them to exclude some participants, in particular the best ones. In this paper, we investigate whether the same principle can still hold with incomplete information. In the simplest context of independent and identically distributed values, by considering the symmetric Bayes–Nash Bidding Equilibrium, we show that this principle does not hold in all-pay auctions, whether in first- or second-price auctions, as the auctioneer would want to admit as many bidders as possible when maximizing the total bid. Exclusion of some bidders instead may be optimal if, in a first-price all-pay auction, the auctioneer maximizes the highest bid. Such findings exhibit an interesting duality with the complete-information case, since the opposite conclusions follow. Moreover, in a first-price, winner-only pays auction, including as many bidders as possible would be convenient for the auctioneer in terms of both maximizing the total bid and the first bid. Finally, a comparison of first-price all-pay auctions with winner-only pays auctions with incomplete information shows the following interesting relation: The total bid in all-pay auctions is equal to the highest bid in winner-only pays auctions and, by Revenue Equivalence, is also equal to the second-highest bid. Therefore, whether or not exclusion is profitable for the auctioneer depends upon the type of the auction, the goal of the auctioneer, and the information distribution among bidders and the auctioneer.