On Copositive Matrices and Completely Mixed Games

In 1945, Kaplansky [4] introduced the concept of the games being completely mixed and presented a necessary and sufficient condition for a game associated with a skew-symmetric matrix to be completely mixed. Recently, we have provided an additional condition for such games. It is known that skew symmetric matrices are \(Q_0\) and \(P_0\) . In 1997, Murthy and Parthasarathy proved that if a matrix B belongs to fully copositive ( \(C_0^f\) ) and \(Q_0\) , then B also belongs to \(P_0\) . Building upon these results, our main result states that if the game associated with a fully copositive \(Q_0\) -matrix B is completely mixed, then \(B + D_j \in Q\) for all j from 1 to n, where \(D_j\) is a diagonal matrix whose \(j^{th}\) diagonal entry is 1 and else 0. Additionally, we prove that if \(B \in C^f_0 \cap Q_0\) but not a Q-matrix, then \(G_B\) is completely mixed game if and only if \(B + D_j \in Q\) for all j from 1 to n.