First Price Auctions in the Independent Private Value Model

Within the Independent Private Value or IPV Model, the estimation of the valuation distributions from the bids submitted at first price auctions is receiving growing attention (see the surveys by Laffont 1997 and Perrigne and Vuong 1999). Such a structural estimation is important in order, for example, to assess potential improvements in the auction design (see, for example, Elyakime, |Laffont, Loisel, and Vuong 1994). In parametric studies, the valuation distributions are supposed to belong to a particular parametric family. In nonparametric studies, the valuation distributions are sometimes assumed to be identical across bidders or at least their supports are often assumed to be identical and they are always assumed to be absolutely continuous with density functions strictly positive and sometimes continuous. In both kinds of studies restrictions are also placed on the type of equilibrium. In this paper, we examine the first price auction in the IPV Model without any restriction on the valuation distributions. We give characterizations of all Nash equilibria with and without the natural restriction that bidders not use weakly dominant strategies involving bids strictly larger than their valuations. These characterizations are similar to those obtained in Lebrun (1999a) for the more particular case of absolutely continuous distributions with identical supports. Here, what play the role of the inverse bid functions in Lebrun (1999a) are the extremities of the inverse of the best bid correspondence within which the bidders bid with probability one. From our characterizations, we give necessary and sufficient conditions for the observable bid distributions to come from a Nash equilibrium, show precisely to what extent the valuation distributions are identified from the bid distributions, and prove that the identified parts of the valuation distributions depend continuously on the bid distributions thereby bringing some robustness to the estimates obtained in empirical studies.