Finite-time convergence to an $\epsilon$-efficient Nash equilibrium in potential games

TL;DR

This paper establishes the first finite-time convergence bounds for log-linear learning to an $psilon$-efficient Nash equilibrium in general potential games, improving previous asymptotic and subclass-specific results.

Contribution

It proves finite-time convergence in general potential games with polynomial bounds in $1/psilon$, and shows robustness under limited feedback and perturbations.

Findings

Finite-time convergence bounds depend polynomially on $1/psilon$.

Convergence is robust to small perturbations and noise.

A variant of log-linear learning with less feedback also converges efficiently.

Abstract

This paper investigates the convergence time of log-linear learning to an $\epsilon$-efficient Nash equilibrium in potential games, where an efficient Nash equilibrium is defined as the maximizer of the potential function. Previous literature provides asymptotic convergence rates to efficient Nash equilibria, and existing finite-time rates are limited to potential games with further assumptions such as the interchangeability of players. We prove the first finite-time convergence to an $\epsilon$-efficient Nash equilibrium in general potential games. Our bounds depend polynomially on $1/\epsilon$, an improvement over previous bounds for subclasses of potential games that are exponential in $1/\epsilon$. We then strengthen our convergence result in two directions: first, we show that a variant of log-linear learning requiring a constant factor less feedback on the utility per round enjoys a similar convergence time; second, we demonstrate the robustness of our convergence guarantee if log-linear learning is subject to small perturbations such as alterations in the learning rule or noise-corrupted utilities.