On the Structure of Weakly Acyclic Games

TL;DR

This paper explores the structure of weakly acyclic games, establishing conditions under which they guarantee convergence to pure Nash equilibria, and identifies special cases where multiple equilibria still ensure weak acyclicity.

Contribution

It introduces a novel link between weak acyclicity and the uniqueness of pure Nash equilibria in subgames, and characterizes cases where multiple equilibria imply weak acyclicity.

Findings

Unique pure Nash equilibrium in all subgames implies weak acyclicity.

Multiple equilibria do not always guarantee weak acyclicity.

Certain cases with limited players and strategies ensure sufficiency of multiple equilibria for weak acyclicity.

Abstract

The class of weakly acyclic games, which includes potential games and dominance-solvable games, captures many practical application domains. In a weakly acyclic game, from any starting state, there is a sequence of better-response moves that leads to a pure Nash equilibrium; informally, these are games in which natural distributed dynamics, such as better-response dynamics, cannot enter inescapable oscillations. We establish a novel link between such games and the existence of pure Nash equilibria in subgames. Specifically, we show that the existence of a unique pure Nash equilibrium in every subgame implies the weak acyclicity of a game. In contrast, the possible existence of multiple pure Nash equilibria in every subgame is insufficient for weak acyclicity in general; here, we also systematically identify the special cases (in terms of the number of players and strategies) for which this is sufficient to guarantee weak acyclicity.