Stochastic Stability of Perturbed Learning Automata in Positive-Utility Games

TL;DR

This paper analyzes the long-term stability of perturbed learning automata in positive-utility games by characterizing invariant measures of the induced Markov chain, offering a new approach beyond traditional Lyapunov-based methods.

Contribution

It introduces a novel framework for stochastic stability analysis that does not rely on potential functions, applicable to a broader class of positive-utility games.

Findings

Provides a method to compute invariant measures in positive-utility games

Demonstrates the approach with an example in coordination games

Offers insights into the global convergence behavior of learning automata

Abstract

This paper considers a class of reinforcement-based learning (namely, perturbed learning automata) and provides a stochastic-stability analysis in repeatedly-played, positive-utility, finite strategic-form games. Prior work in this class of learning dynamics primarily analyzes asymptotic convergence through stochastic approximations, where convergence can be associated with the limit points of an ordinary-differential equation (ODE). However, analyzing global convergence through an ODE-approximation requires the existence of a Lyapunov or a potential function, which naturally restricts the analysis to a fine class of games. To overcome these limitations, this paper introduces an alternative framework for analyzing asymptotic convergence that is based upon an explicit characterization of the invariant probability measure of the induced Markov chain. We further provide a methodology for computing the invariant probability measure in positive-utility games, together with an illustration in the context of coordination games.