[Paper Review] Nonparametric Identification of First-Price Auction with Unobserved Competition: A Density Discontinuity Framework

Emmanuel Guerre, Yao Luo|Kent Academic Repository (University of Kent)|Aug 15, 2019

Auction Theory and Applications12 references4 citations

TL;DR

This paper proposes a nonparametric identification method for first-price auction models using only winning bid data, leveraging density discontinuities in the bid distribution to recover the unobserved number of bidders and private value distributions. The key contribution is that discontinuities at the upper bounds of bid distributions—induced by increasing bid quantile functions with bidder count—enable identification of both the distribution of active bidders and the private value distribution under testable conditions.

ABSTRACT

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.

Motivation & Objective

• To address the challenge of identifying auction primitives when only winning bids are observed, particularly in settings with unobserved competition.
• To develop a nonparametric identification framework that does not require observing all bids or the number of bidders.
• To extend identification to cases with unobserved heterogeneity, bidder uncertainty, and collusion, under testable conditions.
• To provide a robust econometric tool for detecting participation anomalies such as collusion or non-competitive bidding.
• To apply the method to real-world data, such as USFS timber auctions, to analyze bidder behavior and competition levels.

Proposed method

• The method exploits discontinuities in the density of winning bids that arise when the upper bound of the bid distribution increases with the number of bidders.
• It uses the strict monotonicity of the bid quantile function with respect to the number of bidders to generate observable jumps in the density at the upper boundary of each bidder count's bid distribution.
• The distribution of the unobserved number of bidders is nonparametrically identified from the locations and sizes of these density discontinuities.
• The private value distribution is then nonparametrically identified in a second step using the identified bid distribution and the structural relationship between bids and values.
• The framework allows for additive unobserved heterogeneity drawn from known parametric families, and extends to models with collusion where cartel membership is unobserved but winner identities are observed.
• The identification relies on solving a system of differential equations derived from the first-order conditions of the first-price auction model, ensuring uniqueness under regularity conditions.

Experimental results

Research questions

• RQ1 Can the distribution of unobserved bidders be nonparametrically identified from only winning bid data in first-price auctions?
• RQ2 How can density discontinuities in the winning bid distribution be used to recover the true number of active bidders?
• RQ3 Under what conditions can the private value distribution be nonparametrically identified when only winning bids are observed?
• RQ4 Can the framework accommodate unobserved heterogeneity and collusion in bidder participation?
• RQ5 What are the testable conditions under which identification holds in the presence of unobserved competition and bidder uncertainty?

Key findings

• Discontinuities in the winning bid density function arise at the upper bounds of bid distributions due to the strictly increasing nature of the bid quantile function with respect to the number of bidders.
• The locations and sizes of these discontinuities allow for nonparametric identification of the distribution of the unobserved number of bidders.
• The private value distribution can be nonparametrically identified in a second step after the distribution of bidders is recovered.
• Identification remains valid when the number of bidders is latent and only a subset of potential bidders participate with unknown probability.
• The method accommodates additive unobserved heterogeneity drawn from known parametric distributions without requiring full parametric specification of the value distribution.
• In the case of collusion, if winner identities are observed and at least one bidder is always a cartel member, the model still identifies the distribution of bidders and the cartel participation probabilities of other bidders.