Multi-Player Zero-Sum Markov Games with Networked Separable Interactions

TL;DR

This paper introduces a new class of multi-player zero-sum Markov games with networked separable interactions, analyzing their equilibrium properties, computational hardness, and proposing learning algorithms with convergence guarantees.

Contribution

The paper defines zero-sum NMGs with networked separable payoffs, characterizes equilibrium sets, proves hardness results, and develops algorithms with convergence guarantees for these complex games.

Findings

Set of Markov CCE collapses to Markov NE in zero-sum NMGs.

Finding approximate stationary CCE is PPAD-hard unless network is star-shaped.

Proposed fictitious-play-type dynamics converge to Markov NE in star networks.

Abstract

We study a new class of Markov games, \emph(multi-player) zero-sum Markov Games} with \emph{Networked separable interactions} (zero-sum NMGs), to model the local interaction structure in non-cooperative multi-agent sequential decision-making. We define a zero-sum NMG as a model where {the payoffs of the auxiliary games associated with each state are zero-sum and} have some separable (i.e., polymatrix) structure across the neighbors over some interaction network. We first identify the necessary and sufficient conditions under which an MG can be presented as a zero-sum NMG, and show that the set of Markov coarse correlated equilibrium (CCE) collapses to the set of Markov Nash equilibrium (NE) in these games, in that the product of per-state marginalization of the former for all players yields the latter. Furthermore, we show that finding approximate Markov \emph{stationary} CCE in infinite-horizon discounted zero-sum NMGs is \texttt{PPAD}-hard, unless the underlying network has a ``star topology''. Then, we propose fictitious-play-type dynamics, the classical learning dynamics in normal-form games, for zero-sum NMGs, and establish convergence guarantees to Markov stationary NE under a star-shaped network structure. Finally, in light of the hardness result, we focus on computing a Markov \emph{non-stationary} NE and provide finite-iteration guarantees for a series of value-iteration-based algorithms. We also provide numerical experiments to corroborate our theoretical results.