  First price auctions are widely used in government contracts and industrial
auctions. In this paper, we consider the Bayesian Nash Equilibrium (BNE) in
first price auctions with discrete value distributions. We study the
characterization of the BNE in the first price auction and provide an algorithm
to compute the BNE at the same time. Moreover, we prove the existence and the
uniqueness of the BNE. Some of the previous results in the case of continuous
value distributions do not apply to the case of discrete value distributions.
In the meanwhile, the uniqueness result in discrete case cannot be implied by
the uniqueness property in the continuous case.
  Unlike in the continuous case, we do not need to solve ordinary differential
equations and thus do not suffer from the solution errors therein. Compared to
the method of using continuous distributions to approximate discrete ones, our
experiments show that our algorithm is both faster and more accurate.
  The results in this paper are derived in the asymmetric independent private
values model, which assumes that the buyers' value distributions are common
knowledge.
