Completely Mixed Bi-matrix Games Without Restrictions on Payoffs at Equilibrium

We consider bi-matrix games of the type discussed in Raghavan (1970) and Oviedo (1996). A strategy profile is said to be completely mixed if it assigns positive probabilities to all pure strategies for both players and a bi-matrix game is said to be completely mixed if all equilibria of the game are completely mixed. Our first theorem in this note extends necessary conditions for completely mixed bi-matrix games that comprise theorem 1 in Raghavan (1970) and prove that theorem 1 in Raghavan (1970) holds without assuming zero-valued equilibrium pay-offs. Of particular significance is the result that the pay-off matrices of all completely mixed bi-matrix games are square matrices, with the rank of the matrices being at least one less than the dimension of the matrices. Results concerning ranks of matrices in bi-matrix games are in general not independent of equilibrium payoffs. An immediate corollary of our first theorem is that for completely mixed two-person zero-sum (TPZS) games with value zero, the pure-strategy pay-off matrices are square matrices with the rank of the matrices being one less than the dimension of the matrices. We apply this corollary to obtain a complete characterization of all completely mixed TPZS games that have value zero. This characterization is different from the characterization available in Kaplansky (1945). The Complementary Slackness Condition for bi-matrix games plays a very useful and important role in our analysis.