Revenue Comparison in Asymmetric Auctions with Discrete Valuations

Abstract: We consider an asymmetric auction setting with two bidders such that the valuation of each bidder has a binary support. First we characterize the unique equilibrium outcome in the first price auction for any values of parameters. Then we compare the first price auction with the second price auction in terms of expected revenue. Under the assumption that the probabilities of low values are the same for the two bidders, we obtain two main results: (i) the second price auction yields a higher revenue unless the distribution of a bidder's valuation first order stochastically dominates the distribution of the other bidder's valuation "in a strong sense"; (ii) introducing reserve prices implies that the first price auction is never superior to the second price auction. In addition, in some cases the revenue in the first price auction decreases when all the valuations increase.