A matrix game is considered completely mixed if all the optimal pairs of strategies in the game are completely mixed. In this paper, we establish that a matrix game A, with a value of zero, is completely mixed if and only if the value of the game associated with \(A +D_i \) is positive for all i, where \(D_i\) represents a diagonal matrix where ith diagonal entry is 1 and else 0. Additionally, we address Kaplansky’s question from 1945 regarding whether an odd-ordered symmetric game can be completely mixed, and provide characterizations for odd-ordered skew-symmetric matrices to be completely mixed. Moreover, we demonstrate that if A is an almost skew-symmetric matrix and the game associated with A has value positive, then \(A +D_i \in Q\) for all i, where \(D_i\) is a diagonal matrix whose ith diagonal entry is 1 and else 0. Skew-symmetric matrices and almost skew-symmetric matrices with value positive fall under the class of \(P_0\) and \(Q_0\), making them amenable to processing through Lemke’s algorithm.

如果博弈中所有最优策略对都完全混合，则矩阵博弈被认为是完全混合的。在本文中，我们建立了一个值为 0 的矩阵博弈A是完全混合的，当且仅当与\(A +D_i \)相关的博弈值对于所有i均为正，其中\(D_i\ )表示一个对角矩阵，其中第i 个对角元素为 1，否则为 0。此外，我们还解决了 Kaplansky 从 1945 年提出的关于奇数阶对称博弈是否可以完全混合的问题，并提供了奇数阶斜对称矩阵的表征：完全混合。此外，我们证明，如果A是一个几乎斜对称矩阵，并且与A相关的博弈具有正值，则对于所有i来说\(A +D_i \in Q\)，其中\(D_i\)是一个对角矩阵，其第 i个对角线条目为 1，否则为 0。值为正的斜对称矩阵和几乎斜对称矩阵属于\(P_0\)和\(Q_0\)类，使它们适合通过 Lemke 算法进行处理。