Asymmetric Auctions with Discretely Distributed Valuations

We examine a two-bidder auction setting in which the distributions for the bidders’ valuations are asymmetric over a support consisting of three elements. For the first price auction, for each parameter values we derive the unique Bayes Nash Equilibrium in closed form. We rely on this result to compare the revenue in the first price auction with the revenue in the second price auction. The latter is often revenue superior to the former, and we determine precisely, given a distribution for the value of a bidder, when a distribution for the value of the other bidder exists such that the first price auction is superior to the second price auction.