# Computing Constrained Approximate Equilibria in Polymatrix Games

## TL;DR

This paper investigates the computational complexity of finding constrained approximate Nash equilibria in polymatrix games, establishing NP-hardness results for various constraints and providing a deterministic QPTAS for certain graph structures.

## Contribution

It proves NP-hardness for deciding constrained approximate equilibria in polymatrix games and offers a deterministic QPTAS for games with bounded treewidth and limited actions.

## Key findings

- NP-hardness for 9 natural constraints in polymatrix games
- Contrast with bimatrix games requiring non-constant actions
- A deterministic QPTAS for bounded treewidth interaction graphs

## Abstract

This paper is about computing constrained approximate Nash equilibria in polymatrix games, which are succinctly represented many-player games defined by an interaction graph between the players. In a recent breakthrough, Rubinstein showed that there exists a small constant $\epsilon$, such that it is PPAD-complete to find an (unconstrained) $\epsilon$-Nash equilibrium of a polymatrix game. In the first part of the paper, we show that is NP-hard to decide if a polymatrix game has a constrained approximate equilibrium for 9 natural constraints and any non-trivial approximation guarantee. These results hold even for planar bipartite polymatrix games with degree 3 and at most 7 strategies per player, and all non-trivial approximation guarantees. These results stand in contrast to similar results for bimatrix games, which obviously need a non-constant number of actions, and which rely on stronger complexity-theoretic conjectures such as the exponential time hypothesis. In the second part, we provide a deterministic QPTAS for interaction graphs with bounded treewidth and with logarithmically many actions per player that can compute constrained approximate equilibria for a wide family of constraints that cover many of the constraints dealt with in the first part.