Zero-sum Polymatrix Markov Games: Equilibrium Collapse and Efficient Computation of Nash Equilibria

TL;DR

This paper introduces a new class of zero-sum multi-agent Markov games with pairwise interactions, demonstrating that approximate Nash equilibria can be efficiently computed by leveraging equilibrium set collapse.

Contribution

It defines zero-sum polymatrix Markov games with dynamic interaction graphs and proves that equilibrium set collapse enables efficient equilibrium computation.

Findings

Approximate Nash equilibria can be computed efficiently in the new class.

The equilibrium set collapses from coarse-correlated to Nash equilibria.

Generalizes techniques from zero-sum polymatrix normal-form games.

Abstract

The works of (Daskalakis et al., 2009, 2022; Jin et al., 2022; Deng et al., 2023) indicate that computing Nash equilibria in multi-player Markov games is a computationally hard task. This fact raises the question of whether or not computational intractability can be circumvented if one focuses on specific classes of Markov games. One such example is two-player zero-sum Markov games, in which efficient ways to compute a Nash equilibrium are known. Inspired by zero-sum polymatrix normal-form games (Cai et al., 2016), we define a class of zero-sum multi-agent Markov games in which there are only pairwise interactions described by a graph that changes per state. For this class of Markov games, we show that an $\epsilon$-approximate Nash equilibrium can be found efficiently. To do so, we generalize the techniques of (Cai et al., 2016), by showing that the set of coarse-correlated equilibria collapses to the set of Nash equilibria. Afterwards, it is possible to use any algorithm in the literature that computes approximate coarse-correlated equilibria Markovian policies to get an approximate Nash equilibrium.