We examine the relevance of an auction format in a competitive environment by comparing uniform and discriminatory price auctions with many bidders in a private values setting. We show that if the number of objects for sale is small relative to the number of bidders, then all equilibria of both auctions are approximately efficient and lead to approximately the same revenue. When the number of objects for sale is proportional to the number of bidders, then the particulars of the auction format matter. All equilibria of the uniform auction are efficient, while all of the equilibria of the discriminatory auction are inefficient. The relative revenue rankings of the auction formats can go in either direction, depending on the specifics of the environment. These conclusions regarding the efficiency and revenue ranking are in contrast to the previous literature, which focused on the case of independent information across agents.
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To see that indeed t∗(si) is increasing on (s∗,1] note first s∗(x) is increasing in x. Moreover, Assumption 4 implies that the distribution of X conditional on Si = si is stochastically dominated by the distribution of X conditional on Si = si′, where si′ > si. The result then follows from the stochastic dominance. ‖ Acknowledgements. We are grateful for financial support from the Guggenheim Foundation, the Center for Advanced Studies in Behavioral Sciences, and the National Science Foundation under grants SES-9986190 and SES-0316493. We thank Jeremy Bulow, Laurent Mathevet, Tom Palfrey, and Jeroen Swinkels for helpful conversations and suggestions on earlier drafts, as well as Juuso Välimäki and two anonymous referees.