Optimization Seminar: Dr. M. Seetharama Gowda

In the classical (zero-sum) matrix game setting, Kaplansky's well known result provides a necessary and sufficient condition for a matrix game with value zero to be completely mixed. About 50 years later, Kaplansky gave another such result for skew-symmetric matrices of odd order in terms of the so-called Pfaffians. Considering Z-matrices (which are square real matrices with nonpositive off-diagonal entries), Raghavan showed that if the corresponding matrix game has positive value, then it is completely mixed.

In the works of Gowda and Ravindran, the results of Kaplansky and Raghavan were generalized to the setting of a linear transformation over a self-dual cone in a finite dimensional real inner product space; in particular, it was shown that for a Z-transformation (which is a generalization of a Z-matrix), positive value implies the completely mixed property. Motivated by the result (in the classical setting) that a Z-matrix with value zero is completely mixed if and only if it is irreducible, in the present talk, we focus on Z-transformations with value zero and describe the completely mixed property under certain (old and new) irreducibility conditions.