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

Abstract

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. Results, related to the ranks of the matrices of a bi-matrix game are not “obviously independent” of the kind of restrictions on payoffs invoked in Raghavan (1970) and Oviedo (1996). 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 play a very useful and important role in our analysis.