Approximating Nash Equilibrium Via Multilinear Minimax

TL;DR

This paper introduces a multilinear minimax theorem and an associated relaxation method to approximate Nash equilibria efficiently, providing better payoffs and computational advantages over traditional algorithms in large games.

Contribution

The paper generalizes the minimax theorem to multilinear forms and develops a polynomial-time computable approximation method for Nash equilibria.

Findings

MMR solutions outperform NE payoffs in test problems.

MMR can be computed in polynomial time.

In large games, MMR is more efficient and often more accurate than existing NE algorithms.

Abstract

Nash equilibrium} (NE) can be stated as a formal theorem on a multilinear form, free of game theory terminology. On the other hand, inspired by this formalism, we state and prove a {\it multilinear minimax theorem}, a generalization of von Neumann's bilinear minimax theorem. As in the bilinear case, the proof is based on relating the underlying optimizations to a primal-dual pair of linear programming problems, albeit more complicated LPs. The theorem together with its proof is of independent interest. Next, we use the theorem to associate to a multilinear form in NE a {\it multilinear minimax relaxation} (MMR), where the primal-dual pair of solutions induce an approximate equilibrium point that provides a nontrivial upper bound on a convex combination of {\it expected payoffs} in any NE solution. In fact we show any positive probability vector associated to the players induces a corresponding {\it diagonally-scaled} MMR approximate equilibrium with its associated upper bound. By virtue of the proof of the multilinear minimax theorem, MMR solution can be computed in polynomial-time. On the other hand, it is known that even in bimatrix games NE is {\it PPAD-complete}, a complexity class in NP not known to be in P. The quality of MMR solution and the efficiency of solving the underlying LPs are the subject of further investigation. However, as shown in a separate article, for a large set of test problems in bimatrix games, not only the MMR payoffs for both players are better than any NE payoffs, so is the computing time of MMR in contrast with that of Lemke-Howsen algorithm. In large size problems the latter algorithm even fails to produce a Nash equilibrium. In summary, solving MMR provides a worthy approximation even if Nash equilibrium is shown to be computable in polynomial-time.