We introduce several new estimation methods that leverage shape constraints in auction models to estimate various objects of interest, including the distribution of a bidder’s valuations, the bidder’s ex ante expected surplus, and the seller’s counterfactual revenue. The basic approach applies broadly in that (unlike most of the literature) it works for a wide range of auction formats and allows for asymmetric bidders. Though our approach is not restrictive, we focus our analysis on first–price, sealed–bid auctions with independent private valuations. We highlight two nonparametric estimation strategies, one based on a least squares criterion and the other on a likelihood criterion. We establish several theoretical properties of our methods to guide empirical analysis and inference. In addition to providing the asymptotic distributions of our estimators, we identify ways in which methodological choices should be tailored to the objects of interest. For objects like the bidders’ ex ante surplus and the seller’s counterfactual expected revenue with an additional symmetric bidder, we show that our input–parameter–free estimators achieve the semiparametric efficiency bound. For objects like the bidders’ inverse strategy function, we provide an easily implementable boundary–corrected kernel smoothing and transformation method in order to ensure the squared error is integrable over the entire support of the valuations. An extensive simulation study illustrates our analytical results and demonstrates the respective advantages of our least–squares and maximum likelihood estimators in finite samples. Compared to estimation strategies based on kernel density estimation, the simulations indicate that the smoothed versions of our estimators enjoy a relatively large degree of robustness to the choice of an input parameter.
We study the monotonicity of the equilibrium bid with respect to the number of bidders n in affiliated private-value models of first-price sealed-bid auctions and prove the existence of a large class of such models in which the equilibrium bid function is not increasing in n. We moreover decompose the effect of a change in n on the bid level into a competition effect and an affiliation effect. The latter suggests to the winner of the auction that competition is less intense than she had thought before the auction. Since the affiliation effect can occur in both private- and common-value models, a negative relationship between the bid level and n does not allow one to distinguish between the two models and is also not necessarily (only) due to bidders taking account of the winner's curse.