Gradient Dynamics in First-Price Auctions: Iterative Strategy Elimination via Cubic Potentials

Abstract

We show that in discretised first-price auctions with complete information, if the buyers learn to bid with online gradient ascent, in time-average the outcome is (almost) the efficient outcome of the second-price auction. Our proof rests on two novel innovations in the analysis of online gradient ascent in normal-form games, which may be useful in a wider range of applications. First, we develop a potential-function-based argument for the analysis of gradient ascent in normal-form games, allowing us to deduce that certain strategies will not be played in time-average. We provide sufficient conditions which ensure this argument can be applied iteratively, resulting in a procedure reminiscent of iterative elimination of dominated strategies. Second, we develop a novel class of cubic "candidate potential functions", classifying a family of quadratic strategy modifications on the probability simplex against which online gradient ascent incurs no regret.

43 pages, accepted to EC'26. This version incorporates reviewer feedback from the conference review process