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 consider a situation in which we have data from ascending auctions with symmetric bidders, independent private values, and exogenous entry in which the bidders’ value distribution is partially identified. Focusing on the case in which the seller intends to use a second price auction, we discuss how to determine an optimal reserve price. We justify the use of maximum entropy, explore the properties of the estimand, determine the asymptotic properties of our maximum entropy estimator, evaluate its behavior in a simulation study, and demonstrate its use in a modest application. As an extension, we propose a maxmin decision rule with entropy regularization, which includes Aryal and Kim (2013) and the maximum entropy solution as extreme cases.
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.
This paper studies federal auctions for wildcat leases on the Outer Continental Shelf from 1954 to 1970. These are leases where bidders privately acquire (at some cost) noisy, but equally informative, signals about the amount of oil and gas that may be present. We develop tests of rational and equilibrium bidding in a common values model that are implemented using data on bids and ex post values. We also use data on tract location and ex post values to test the comparative static prediction that bidders may bid less aggressively in common value auctions when they expect more competition. We find that bidders are aware of the “winner's curse” and their bidding is largely consistent with equilibrium.