## Optimization Seminar: Dr. M. Seetharama Gowda

#### UMBC

Thursday, September 21, 2023 · 10:30 AM - 12 PM

**Title:**

*C**ompletely mixed games*

**Speaker:**Dr. M. Seetharama Gowda

**Abstract:**

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.