  In a two-round auction, a subset of bidders is selected (probabilistically),
according to their bids in the first round, for the second round, where they
can increase their bids. We formalize the two-round auction model, restricting
the second round to a dominant strategy incentive compatible (DSIC) auction for
the selected bidders. It turns out that, however, such two-round auctions are
not directly DSIC, even if the probability of each bidder being selected for
the second round is monotonic to its first bid, which is surprisingly
counter-intuitive. We also illustrate the necessary and sufficient conditions
of two-round auctions being DSIC. Besides, we characterize the Nash equilibria
for untruthful two-round auctions. One can achieve better revenue performance
by setting proper probability for selecting bidders for the second round
compared with truthful two-round auctions.
