Convergence of Stochastic First-Order Algorithms in Bertrand Competition Under Incomplete Information

TL;DR

This paper proves that stochastic first-order algorithms converge to equilibrium in Bayesian Bertrand competition, providing rigorous guarantees and challenging claims of collusion in algorithmic pricing.

Contribution

It establishes convergence of Regularized Robbins-Monro algorithms to Bayes-Nash equilibrium in incomplete information settings, with explicit stability analysis.

Findings

RRM algorithms converge almost surely to the unique equilibrium

A global Lyapunov function is constructed for the dynamics

Provides rigorous convergence guarantees in Bayesian pricing games

Abstract

Autonomous pricing agents are widely deployed in online marketplaces, making algorithmic pricing a prominent application of multi-agent learning. Experimental studies often report collusive outcomes, but these findings typically rely on Q-learning in complete-information environments and lack rigorous convergence guarantees. In this paper, we study the stochastic learning dynamics of Regularized Robbins-Monro (RRM) algorithms in a Bayesian Bertrand competition with private costs. We show that this setting violates standard stability conditions, including monotonicity and the Minty variational inequality, rendering classical convergence results for gradient-based learning inapplicable. Despite this, we prove that Euclidean RRM algorithms converge almost surely to the unique, efficient Bayes-Nash equilibrium within a finite-dimensional approximation of the strategy space. By analyzing symmetric piecewise-linear pricing strategies in a duopoly, we explicitly construct a global Lyapunov function for the projected primal dynamics and establish global asymptotic stability of the equilibrium. Our analysis yields rigorous convergence guarantees for stochastic first-order learning algorithms in Bayesian Bertrand competition and provides a principled counterpoint to widespread claims of algorithmic collusion.