Multi-agent Bayesian Learning with Adaptive Strategies: Convergence and Stability

TL;DR

This paper analyzes how strategic agents learn and adapt their strategies over time in a game with unknown payoffs, proving convergence and stability of beliefs and strategies under certain conditions.

Contribution

It introduces a Bayesian learning framework for multi-agent games, establishing convergence, stability conditions, and criteria for learning the true payoff parameter.

Findings

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

Conditions for local and global stability of fixed points are provided.

Complete information Nash equilibrium convergence is not always guaranteed.

Abstract

We study learning dynamics induced by strategic agents who repeatedly play a game with an unknown payoff-relevant parameter. In each step, an information system estimates a belief distribution of the parameter based on the players' strategies and realized payoffs using Bayes' rule. Players adjust their strategies by accounting for an equilibrium strategy or a best response strategy based on the updated belief. We prove that beliefs and strategies converge to a fixed point with probability 1. We also provide conditions that guarantee local and global stability of fixed points. Any fixed point belief consistently estimates the payoff distribution given the fixed point strategy profile. However, convergence to a complete information Nash equilibrium is not always guaranteed. We provide a sufficient and necessary condition under which fixed point belief recovers the unknown parameter. We also provide a sufficient condition for convergence to complete information equilibrium even when parameter learning is incomplete.