Computing Constrained Approximate Equilibria in Polymatrix Games

Abstract

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.Lemma 3. In G, the total payoff for every variable player is at most 0, and the total payoff for every clause player c j is at most 1. Moreover, if c j gets payoff 1, then c j and the variable players connected to c j play pure strategies.Lemma 4. If φ is a "Yes" instance, there is an NE for G with social welfare m.We now prove that if there is a strategy profile of G with social welfare m then φ is a "Yes" instance. Clearly, if this statement holds for any strategy profile, it also holds for all ǫ-NE for any ǫ.

Tree Polymatrix Games are PPAD-hard

“…On the positive side, Cai and Daskalakis [2], proved that NE can be efficiently found in polymatrix games where every 2-player game is zero-sum. Ortiz and Irfan [13] and Deligkas, Fearnley, and Savani [7] produced QPTASs for polymatrix games of bounded treewidth (in addition to the FPTAS of [13] for tree polymatrix games mentioned above). For general polymatrix games, the only positive result to date is a polynomial-time algorithm to compute a ( 1 2 + δ)-NE [8].…”

Approximating the Existential Theory of the Reals

“…Constrained Nash equilibria attracted the attention of many authors, who proved NP-completeness for two-player games [23,14,6] and ETR-completeness for three-player games [6,7,8,9,22] for constrained exact Nash equilibria. Constrained approximate equilibria have been studied, but so far only lower bounds have been derived [2,25,11,19,18]. It has been observed that sampling methods can give QPTASs for finding constrained approximate Nash equilibria for certain constraints in two player games [19].…”

“…Subsequently, it was used to produce algorithms for finding approximate equilibria in normal form games with many players [3], sparse bimatrix games [4], tree polymatrix [5], and Lipschitz games [20]. It has also been used to find constrained approximate equilibria in polymatrix games with bounded treewidth [18].…”

An Empirical Study on Computing Equilibria in Polymatrix Games

“…In our results, we check 201 different points for α in each iteration. For a justification of the reasonableness of this choice, see Appendix A in the full version of this paper [21].…”

“…The underlying reason is that the number of players in the polymatrix game (i.e., number of types) affects the running time much more that the number of actions (i.e., number of items/troops). Also see Appendix B in the full version of this paper[21].…”