Numerical Solutions of Asymmetric, First-Price, Independent Private Values Auctions

Abstract

We propose a powerful, fully automated, and numerically robust algorithm to compute (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 revenues, probabilities of winning, probability of retention under reserve pricing and, on request, optimal reserve price). The algorithm also includes a built-in numerical procedure 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 investigation of theoretical conjectures of interest and/or for easy implementation within any numerical empirical inference procedure relying upon inverse bid functions.