Completely mixed linear games corresponding to Z-transformations over self-dual cones

Abstract

In the setting of a self-dual cone in a finite dimensional inner product space, we consider (zero-sum) linear games. In our previous work, we showed that a Z-transformation with positive value is completely mixed. In the present paper, we consider the case when the value is zero. 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, we formulate our general results based on the concepts of cone-irreducibility and space-irreducibility. In the setting of a symmetric cone (in a Euclidean Jordan algebra), we show that the space-irreducibility condition is necessary for a Z-transformation with value zero to be completely mixed and that it is sufficient when the Z-transformation is the difference of a Lyapunov-like transformation and a positive transformation. Additionally, we show that cone-irreducibility and space-irreducibility are equivalent for a positive transformation on a symmetric cone.