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

2026-05-17

AI Summary

This study addresses the lack of convergence guarantees for pricing algorithms in Bayesian Bertrand duopoly with incomplete information, where classical stability conditions—such as monotonicity—fail to hold. The authors analyze the learning dynamics of the regularized Robbins–Monro (RRM) stochastic first-order algorithm within a finite-dimensional symmetric piecewise-linear strategy space. They establish, for the first time, a rigorous theoretical guarantee that RRM converges almost surely to the unique efficient Bayesian Nash equilibrium under private cost structures. By constructing a novel global Lyapunov function, they further prove that this equilibrium is globally asymptotically stable. These findings demonstrate that Euclidean RRM can converge to an efficient equilibrium even when conventional assumptions are violated, providing a theoretical counterexample to prevailing concerns about algorithmic collusion.

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.

Problem

• Bertrand competition
• incomplete information
• stochastic first-order algorithms
• convergence
• Bayes-Nash equilibrium

Innovation

• Stochastic First-Order Algorithms
• Bayesian Bertrand Competition
• Regularized Robbins-Monro
• Lyapunov Function
• Convergence Guarantees