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.
