Numerical Solutions of Asymmetric, First-Price, Independent Private Values Auctions Wayne-Roy Gayle ∗ and Jean Francois Richard †‡ February 25, 2008 Abstract We propose a powerful, fully automated, and numerically robust algorithm to com- pute (inverse) equilibrium bid functions for asymmetric, Independent Private Values, First-Price auctions. The algorithm relies upon a built-in algebra of local Taylor-series expansions in order to compute highly accurate solutions to the set of differential equations characterizing first order conditions. It offers an extensive user friendly menu whereby one can assign commonly-used distributions to bidders and can also create arbitrary (non-inclusive) coalitions. In addition to (inverse) bid functions, the algorithm also computes a full range of auxiliary statistics of interest (expected rev- enues, probabilities of winning, probability of retention under reserve pricing and, on request, optimal reserve price). The algorithm also includes a built-in numerical pro- cedure designed to automatically produce local Taylor-series expansions for any user- supplied distribution, whether analytical or tabulated (empirical, parametric, semi- or non-parametric). It provides a tool of unparalleled flexibility for the numerical investi- gation of theoretical conjectures of interest and/or for easy implementation within any numerical empirical inference procedure relying upon inverse bid functions. ∗ Corresponding author. Economics Department, University of Virginia, 2015 Ivy Rd, Room 312, Char- lottesville, VA 22903. Phone:434-243-4336, FAX: 434-982-2904, e-mail: wg4b@virginia.edu. † Economics Department, University of Pittsburgh, 4901 W. W. Posvar Hall, 230 South Bouquet Street, Pittsburgh, PA 15260. ‡ The authors are grateful to Andreas Blume, George Gayle, Robert Marshall and Leslie Marx for discus- sions and insightful comments. They thank the editor-in-chief and an anonymous referee for constructive comments which resulted in improvement of the paper. They also thank participants at seminars at Bates White, Carnegie Mellon University and SUNY-Stony Brook. The work was supported by grant SES-0136408 from the National Science Foundation. All errors are our own. 1