  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.
