  While auction theory views bids and valuations as continuous variables,
real-world auctions are necessarily discrete. In this paper, we use a
combination of analytical and computational methods to investigate whether
incorporating discreteness substantially changes the predictions of auction
theory, focusing on the case of uniformly distributed valuations so that our
results bear on the majority of auction experiments. In some cases, we find
that introducing discreteness changes little. For example, the first-price
auction with two bidders and an even number of values has a symmetric
equilibrium that closely resembles its continuous counterpart and converges to
its continuous counterpart as the discretisation goes to zero. In others,
however, we uncover discontinuity results. For instance, introducing an
arbitrarily small amount of discreteness into the all-pay auction makes its
symmetric, pure-strategy equilibrium disappear; and appears (based on
computational experiments) to rob the game of pure-strategy equilibria
altogether. These results raise questions about the continuity approximations
on which auction theory is based and prompt a re-evaluation of the experimental
literature.
