The Bimatrix Nash Equilibrium (NE) for $m \times n$ real matrices $R$ and $C$, denoted as the {\it Row} and {\it Column} players, is characterized as follows: Let $\Delta =S_m \times S_n$, where $S_k$ denotes the unit simplex in $\mathbb{R}^k$. For a given point $p=(x,y) \in \Delta$, define $R[p]=x^TRy$ and $C[p]=x^TCy$. Consequently, there exists a subset $\Delta_* \subset \Delta$ such that for any $p_*=(x_*,y_*) \in \Delta_*$, $\max_{p \in \Delta, y=y_*}R[p]=R[p_*]$ and $\max_{p \in \Delta, x=x_* } C[p]=C[p_*]$. The computational complexity of bimatrix NE falls within the class of {\it PPAD-complete}. Although the von Neumann Minimax Theorem is a special case of bimatrix NE, we introduce a novel extension termed {\it Trilinear Minimax Relaxation} (TMR) with the following implications: Let $\lambda^*=\min_{\alpha \in S_{2}} \max_{p \in \Delta} (\alpha_1 R[p]+ \alpha_2C[p])$ and $\lambda_*=\max_{p \in \Delta} \min_{\alpha \in S_{2}} (\alpha_1 R[p]+ \alpha_2C[p])$. $\lambda^* \geq \lambda_*$. $\lambda^*$ is computable as a linear programming in $O(mn)$ time, ensuring $\max_{p_* \in \Delta_*}\min \{R[p_*], C[p_*]\} \leq \lambda^*$, meaning that in any Nash Equilibrium it is not possible to have both players' payoffs to exceed $\lambda^*$. $\lambda^*=\lambda_*$ if and only if there exists $p^* \in \Delta$ such that $\lambda^*= \min\{R[p^*], C[p^*]\}$. Such a $p^*$ serves as an approximate Nash Equilibrium. We analyze the cases where such $p^*$ exists and is computable. Even when $\lambda^* > \lambda_*$, we derive approximate Nash Equilibria. In summary, the aforementioned properties of TMR and its efficient computational aspects underscore its significance and relevance for Nash Equilibrium, irrespective of the computational complexity associated with bimatrix Nash Equilibrium. Finally, we extend TMR to scenarios involving three or more players.

翻译：双矩阵纳什均衡（NE）针对实矩阵$R$和$C$（$m \times n$维），分别对应行玩家与列玩家，其定义如下：设$\Delta =S_m \times S_n$，其中$S_k$表示$\mathbb{R}^k$中的单位单纯形。对于给定点$p=(x,y) \in \Delta$，定义$R[p]=x^TRy$与$C[p]=x^TCy$。由此存在子集$\Delta_* \subset \Delta$，使得对任意$p_*=(x_*,y_*) \in \Delta_*$，有$\max_{p \in \Delta, y=y_*}R[p]=R[p_*]$且$\max_{p \in \Delta, x=x_* } C[p]=C[p_*]$。双矩阵NE的计算复杂度属于PPAD完全类。尽管冯·诺依曼极小极大定理是双矩阵NE的特例，我们提出一种新型扩展——三线性极小极大松弛（TMR），其性质如下：设$\lambda^*=\min_{\alpha \in S_{2}} \max_{p \in \Delta} (\alpha_1 R[p]+ \alpha_2C[p])$且$\lambda_*=\max_{p \in \Delta} \min_{\alpha \in S_{2}} (\alpha_1 R[p]+ \alpha_2C[p])$。则有$\lambda^* \geq \lambda_*$。$\lambda^*$可通过$O(mn)$时间的线性规划计算，确保$\max_{p_* \in \Delta_*}\min \{R[p_*], C[p_*]\} \leq \lambda^*$，即任何纳什均衡中双方收益不可能同时超过$\lambda^*$。当且仅当存在$p^* \in \Delta$使得$\lambda^*= \min\{R[p^*], C[p^*]\}$时，有$\lambda^*=\lambda_*$。此类$p^*$可视为近似纳什均衡。我们分析了此类$p^*$存在且可计算的情形。即使$\lambda^* > \lambda_*$，我们仍可推导出近似纳什均衡。综上，TMR的上述性质及其高效计算特性凸显了其在纳什均衡中的重要意义与相关性，无论双矩阵纳什均衡的计算复杂度如何。最后，我们将TMR扩展至三个或更多玩家的情形。