Learning in nonatomic games, Part I: Finite action spaces and population games

TL;DR

This paper analyzes the long-term behavior of various learning dynamics in finite-action nonatomic games, including fictitious play, best-reply, and dual averaging, covering potential, monotone, and evolutionarily stable games.

Contribution

It provides a comprehensive analysis of the asymptotic behavior of multiple learning dynamics in finite-action nonatomic games, extending understanding beyond prior specific cases.

Findings

Dynamics converge to equilibrium in potential games.

Time-averages of play exhibit stability in monotone games.

Evolutionarily stable states attract the long-run behavior.

Abstract

We examine the long-run behavior of a wide range of dynamics for learning in nonatomic games, in both discrete and continuous time. The class of dynamics under consideration includes fictitious play and its regularized variants, the best-reply dynamics (again, possibly regularized), as well as the dynamics of dual averaging / "follow the regularized leader" (which themselves include as special cases the replicator dynamics and Friedman's projection dynamics). Our analysis concerns both the actual trajectory of play and its time-average, and we cover potential and monotone games, as well as games with an evolutionarily stable state (global or otherwise). We focus exclusively on games with finite action spaces; nonatomic games with continuous action spaces are treated in detail in Part II of this paper.