Dynamics in Near-Potential Games

TL;DR

This paper investigates how the dynamic properties of potential games extend to near-potential games, providing a systematic framework for analyzing convergence in adaptive learning dynamics based on the game's proximity to potential games.

Contribution

It formalizes the concept of near-potential games and analyzes the convergence of various adaptive dynamics, showing they inherit properties from potential games.

Findings

Sequences of pure strategies converge to approximate equilibria

Stationary distributions relate to the game's distance from a potential game

Empirical frequencies converge to neighborhoods of equilibria

Abstract

Except for special classes of games, there is no systematic framework for analyzing the dynamical properties of multi-agent strategic interactions. Potential games are one such special but restrictive class of games that allow for tractable dynamic analysis. Intuitively, games that are "close" to a potential game should share similar properties. In this paper, we formalize and develop this idea by quantifying to what extent the dynamic features of potential games extend to "near-potential" games. We study convergence of three commonly studied classes of adaptive dynamics: discrete-time better/best response, logit response, and discrete-time fictitious play dynamics. For better/best response dynamics, we focus on the evolution of the sequence of pure strategy profiles and show that this sequence converges to a (pure) approximate equilibrium set, whose size is a function of the "distance" from a close potential game. We then study logit response dynamics and provide a characterization of the stationary distribution of this update rule in terms of the distance of the game from a close potential game and the corresponding potential function. We further show that the stochastically stable strategy profiles are pure approximate equilibria. Finally, we turn attention to fictitious play, and establish that the sequence of empirical frequencies of player actions converges to a neighborhood of (mixed) equilibria of the game, where the size of the neighborhood increases with distance of the game to a potential game. Thus, our results suggest that games that are close to a potential game inherit the dynamical properties of potential games. Since a close potential game to a given game can be found by solving a convex optimization problem, our approach also provides a systematic framework for studying convergence behavior of adaptive learning dynamics in arbitrary finite strategic form games.