Identification and Estimation of Level-k Auctions

I develop a structural econometric framework for first-price auctions by generalizing the assumption of Bayesian Nash Equilibrium within the context of a level-k behavioral model, which nests equilibrium by allowing bidders to hold heterogeneous beliefs about opponents’ bidding strategies. While behavioral heterogeneity causes identification to fail under benchmark equilibrium conditions, independence and exclusion restrictions recover identification of the joint distribution over valuations and bidder-types in heterogeneous populations. Establishing consistent maximum likelihood sieve estimation with an upper semicontinuous population loglikelihood function leads to a natural semi-nonparametric maximum likelihood estimator based on Legendre polynomials. The level-k model introduces a mixture structure to the estimation problem, requiring a generalized expectation maximization algorithm. Presenting evidence from a pilot study of vintage computer auctions, I find a high level of bidder sophistication in the field. To further apply the econometric framework, I characterize expected revenues in first price auctions with level-k bidders, establishing a partial identification result for expected revenues in unidentified models. An empirical analysis of USFS timber auctions finds that a misspecified equilibrium optimal reserve price could reduce expected revenues up to 30% relative to an unbinding reserve price. ∗Department of Economics; University of California, San Diego; 9500 Gilman Dr., Mailcode 0534; La Jolla, CA 92093-0534. I owe special thanks to Vincent Crawford and to my adviser Allan Timmermann for their guidance and support. I thank Sellam Ismael for providing me with his data on vintage computer auctions and Philip Haile for providing data on USFS timber auctions. The paper has benefitted from comments and helpful feedback from Nageeb Ali, Dirk Bergemann, Gray Calhoun, Graham Elliott, Michael Ewens, Hiroaki Kaido, Ivana Komunjer, Jacob LaRiviere, David Miller, Andres Santos, Matt Shum, Nicholas Shunda, Joel Sobel, Suyong Song, Bryan Tomlin, Liang Choon Wang, Quang Vuong, Hal White and seminar participants at UCSD and Stanford’s SITE conference. Comments, feedback, and suggestions are welcome at [email protected].