Nonparametric Identification of First-Price Auction with Unobserved Competition: A Density Discontinuity Framework

We consider nonparametric identification of independent private value first-price auction models, in which the analyst only observes winning bids. Our benchmark model assumes an exogenous number of bidders $N$. We show that, if the bidders observe $N$, the resulting discontinuities in the winning bid density can be used to identify the distribution of $N$. The private value distribution can be nonparametrically identified in a second step. This extends, under testable identification conditions, to the case where $N$ is a number of potential buyers, who bid with some unknown probability. Identification also holds in presence of additive unobserved heterogeneity drawn from some parametric distributions. A parametric Bayesian estimation procedure is proposed. An application to Shanghai Government IT procurements finds that the imposed three bidders participation rule is not effective. This generates loss in the range of as large as $10\%$ of the appraisal budget for small IT contracts.