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 Q0 and P0. In 1997, Murthy and Parthasarathy proved that if a matrix B belongs to fully copositive (C0f) and Q0, then B also belongs to P0. Building upon these results, our main result states that if the game associated with a fully copositive Q0-matrix B is completely mixed, then B+Dj∈Q for all j from 1 to n, where Dj is a diagonal matrix whose jth diagonal entry is 1 and else 0. Additionally, we prove that if B∈C0f∩Q0 but not a Q-matrix, then GB is completely mixed game if and only if B+Dj∈Q for all j from 1 to n.