Multi-agent Bayesian Learning with Best Response Dynamics: Convergence and Stability

TL;DR

This paper analyzes how strategic agents learn and adapt their strategies in a game with unknown payoffs using Bayesian updates and best response dynamics, demonstrating convergence and stability under certain conditions.

Contribution

It introduces a model of multi-agent Bayesian learning with best response dynamics, establishing convergence, stability conditions, and the potential for learning the true payoff parameter.

Findings

Beliefs and strategies converge to a fixed point with probability 1.

Conditions for global stability leading to Nash equilibrium are identified.

Local stability of beliefs and strategies is guaranteed under certain conditions.

Abstract

We study learning dynamics induced by strategic agents who repeatedly play a game with an unknown payoff-relevant parameter. In this dynamics, a belief estimate of the parameter is repeatedly updated given players' strategies and realized payoffs using Bayes's rule. Players adjust their strategies by accounting for best response strategies given the belief. We show that, with probability 1, beliefs and strategies converge to a fixed point, where the belief consistently estimates the payoff distribution for the strategy, and the strategy is an equilibrium corresponding to the belief. However, learning may not always identify the unknown parameter because the belief estimate relies on the game outcomes that are endogenously generated by players' strategies. We obtain sufficient and necessary conditions, under which learning leads to a globally stable fixed point that is a complete information Nash equilibrium. We also provide sufficient conditions that guarantee local stability of fixed point beliefs and strategies.