Utility Decoupling for Distributed Nash Equilibrium Seeking in Weakly Acyclic Games

This article addresses the problem of distributed Nash equilibrium seeking over networks for games with finite action sets. Gradient-like and consensus-based methods commonly used for continuous action spaces fail to work for this case. To this end, we propose a utility decoupling method to reformulate the original game into an augmented game, which preserves the Nash equilibrium and weakly acyclic property, yet enjoys a utility coupling network the same as the communication network. In this way, a variety of full-information game-theoretic learning dynamics for the augmented game turns into partial-information Nash equilibrium seeking dynamics for the original game. We proceed to apply the developed utility decoupling method to formulate three types of distributed Nash equilibrium seeking dynamics, including distributed best-response dynamics, distributed fictitious play, and distributed regret matching for weakly acyclic games. In the last, a typical color assignment game is utilized to empirically illustrate the validity and effectiveness of our approach.