Constrained Pure Nash Equilibria in Polymatrix Games

TL;DR

This paper investigates the computational complexity of finding constrained pure Nash equilibria in a specific class of polymatrix games modeled on weighted directed graphs, revealing complexity results even in simple cases.

Contribution

It characterizes the complexity of checking the existence of constrained pure Nash equilibria in polymatrix games on weighted directed graphs, identifying tractable cases and NP/coNP-complete problems.

Findings

NP-completeness for unweighted DAGs

coNP-completeness results

tractable cases identified

Abstract

We study the problem of checking for the existence of constrained pure Nash equilibria in a subclass of polymatrix games defined on weighted directed graphs. The payoff of a player is defined as the sum of nonnegative rational weights on incoming edges from players who picked the same strategy augmented by a fixed integer bonus for picking a given strategy. These games capture the idea of coordination within a local neighbourhood in the absence of globally common strategies. We study the decision problem of checking whether a given set of strategy choices for a subset of the players is consistent with some pure Nash equilibrium or, alternatively, with all pure Nash equilibria. We identify the most natural tractable cases and show NP or coNP-completness of these problems already for unweighted DAGs.