Revenue Ranking of Discriminatory and Uniform Auctions with an Unknown Number of Bidders

An important managerial question is the choice of the pricing rule. We study whether this choice depends on the uncertainty about the number of participating bidders by comparing expected revenues under discriminatory and uniform pricing within an auction model with affiliated values, stochastic number of bidders, and linear bidding strategies. We show that if uncertainty about the number of bidders is substantial, then the discriminatory pricing generates higher expected revenues than the uniform pricing. In particular, the first-price auction might generate higher revenues than the second-price auction. Therefore, uncertainty about the number of bidders is an important factor to consider when choosing the pricing rule. We also study whether eliminating this uncertainty, i.e., revealing the number of bidders, is in the seller's interests, and discuss the existence of an increasing symmetric equilibrium.