Convex Markov Games and Beyond: New Proof of Existence, Characterization and Learning Algorithms for Nash Equilibria

TL;DR

This paper extends the theory of convex Markov games to a broader class called GUMGs, providing new existence proofs, equilibrium characterizations, and learning algorithms, including policy gradient methods with complexity guarantees.

Contribution

It introduces GUMGs, proves NE existence via fixed points, characterizes NE as stationary points, and develops policy gradient algorithms with complexity bounds.

Findings

Nash equilibria coincide with fixed points of pseudo-gradient dynamics

Existence of Markov perfect equilibria in GUMGs

Sample complexity bounds for policy gradient algorithms

Abstract

Convex Markov Games (cMGs) were recently introduced as a broad class of multi-agent learning problems that generalize Markov games to settings where strategic agents optimize general utilities beyond additive rewards. While cMGs expand the modeling frontier, their theoretical foundations, particularly the structure of Nash equilibria (NE) and guarantees for learning algorithms, are not yet well understood. In this work, we address these gaps for an extension of cMGs, which we term General Utility Markov Games (GUMGs), capturing new applications requiring coupling between agents' occupancy measures. We prove that in GUMGs, Nash equilibria coincide with the fixed points of projected pseudo-gradient dynamics (i.e., first-order stationary points), enabled by a novel agent-wise gradient domination property. This insight also yields a simple proof of NE existence using Brouwer's fixed-point theorem. We further show the existence of Markov perfect equilibria. Building on this characterization, we establish a policy gradient theorem for GUMGs and design a model-free policy gradient algorithm. For potential GUMGs, we establish iteration complexity guarantees for computing approximate-NE under exact gradients and provide sample complexity bounds in both the generative model and on-policy settings. Our results extend beyond prior work restricted to zero-sum cMGs, providing the first theoretical analysis of common-interest cMGs.