Tree Polymatrix Games are PPAD-hard

Abstract

We prove that it is PPAD-hard to compute a Nash equilibrium in a tree polymatrix game with twenty actions per player. This is the first PPAD hardness result for a game with a constant number of actions per player where the interaction graph is acyclic. Along the way we show PPAD-hardness for finding an -fixed point of a 2D-LinearFIXP instance, when is any constant less than ( √ 2 − 1)/2 ≈ 0.2071. This lifts the hardness regime from polynomially small approximations in k-dimensions to constant approximations in two-dimensions, and our constant is substantial when compared to the trivial upper bound of 0.5.

A Survey of Decision Making in Adversarial Games

“…In addition, multi-player polymatrix games [52] can also be equivalently transformed to bilinear games [46]. Generally speaking, the existing literature mainly focuses on the computational complexity and polynomial-time algorithm design for approximating NE of bilinear games [53], bimatrix games [54], polymatrix games [55], and the Colonel Blotto game [56]. Recently, it is shown that NE computation in two-player nonzero-sum games with rank ≥ 2 is PPAD-hard [57], [58].…”

“…And computing a 1/n c s -approximate NE is PPAD-hard even for imitation games for any c > 0 [50], where n s is the number of moves available to the players, and a polynomial-time algorithm was developed for finding an approximate NE in [50]. Also, computing an NE in a tree polymatrix game with twenty actions per player is PPAD-hard [55], and a polynomial-time algorithm for 1/3-approximate NE in bimatrix games was proposed in [54], which is the state-of-the-art in the literature. For the Colonel Blotto game, efficient and simple algorithms have been recently provided in [59]- [61], and meanwhile, various scenarios have been extended for this game, including dynamic Colonel Blotto game [62], generalized Colonel Blotto and generalized lottery Blotto games [63], and multiplayer cases [61], [64].…”