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.