Nash Convergence of Mean-Based Learning Algorithms in First Price Auctions

Abstract

Understanding the convergence properties of learning dynamics in repeated auctions is a timely and important question in the area of learning in auctions, with numerous applications in, e.g., online advertising markets. This work focuses on repeated first price auctions where bidders with fixed values for the item learn to bid using mean-based algorithms - a large class of online learning algorithms that include popular no-regret algorithms such as Multiplicative Weights Update and Follow the Perturbed Leader. We completely characterize the learning dynamics of mean-based algorithms, in terms of convergence to a Nash equilibrium of the auction, in two senses: (1) time-average: the fraction of rounds where bidders play a Nash equilibrium approaches 1 in the limit; (2) last-iterate: the mixed strategy profile of bidders approaches a Nash equilibrium in the limit. Specifically, the results depend on the number of bidders with the highest value: Our discovery opens up new possibilities in the study of convergence dynamics of learning algorithms.