Computing Constrained Approximate Equilibria in Polymatrix Games

This paper is about computing constrained approximate Nash equilibria in polymatrix games, which are succinctly represented manyplayer games defined by an interaction graph between the players. In a recent breakthrough, Rubinstein showed that there exists a small constant ǫ, such that it is PPAD-complete to find an (unconstrained) ǫ-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.