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.

翻译：纳什均衡（NE）可表述为多线性形式上的一个形式化定理，无需博弈论术语。另一方面，受此形式化启发，我们提出并证明了一个**多线性极小极大定理**，它是冯·诺依曼双线性极小极大定理的推广。与双线性情形类似，该证明基于将底层优化问题与一对原始-对偶线性规划问题（尽管是更复杂的线性规划）相关联。该定理及其证明本身具有独立的研究价值。随后，我们利用该定理将NE中的多线性形式与一个**多线性极小极大松弛**（MMR）相关联，其中原始-对偶解对诱导出一个近似均衡点，该点对任何NE解中**期望收益**的凸组合提供了非平凡上界。实际上，我们证明与玩家相关的任何正概率向量都会诱导出一个相应的**对角缩放**MMR近似均衡及其相关上界。基于多线性极小极大定理的证明，MMR解可在多项式时间内计算得到。另一方面，已知即使在双矩阵博弈中，NE是**PPAD完全**的——这是NP中一个尚未被证明属于P的复杂性类。MMR解的质量及底层线性规划求解的效率有待进一步研究。然而，如另一篇文章所示，对于双矩阵博弈中的大量测试问题，MMR为双方玩家带来的收益不仅优于任何NE收益，其计算时间也优于Lemke-Howsen算法。在大规模问题中，后者甚至无法求解纳什均衡。综上，即使纳什均衡被证明可在多项式时间内计算，求解MMR仍提供了一种有价值的近似方法。