Completely mixed bimatrix games

Kaplansky (Ann. Math.46(3) (1945) 474–479) introduced the notion of completely mixed games. Fifty years later in 1995, he wrote another beautiful paper where he gave a set of necessary and sufficient conditions for a skew symmetric matrix game to be completely mixed. In this work, we attempt to answer when bimatrix games will be completely mixed. In particular, we give a set of necessary and sufficient condition for a bimatrix game (A, B) to be completely mixed when A and B are skew symmetric matrices of order 3. We give several examples to show the sharpness of our result. We also formulate a conjecture when A and B are skew symmetric matrices of order 5.