Tournament Auctions

We examine ``tournament'' second-price auctions in which N bidders compete for the right to participate in a second stage and contend against bidder $N+1$. When the first N bidders are committed so that their bids cannot be changed in the second stage, the analysis yields some unexpected results. The first N bidders consistently bid above their values in equilibrium. When bidder $N+1$ is sufficiently stronger than the first N, overbidding leads to an increase in expected revenue in comparison to the standard second-price auction when N is large.