  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. The present
paper considers the case when the value is zero. Motivated by the matrix game
result 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. While the concept of
cone-irreducibility for a positive linear transformation is well-known, we
introduce space-irreducibility for a general linear transformation by
reformulating the irreducibility concept of Elsner. Our main result is that for
a Z-transformation with value zero, space-irreducibility is necessary and
sufficient for the completely mixed property. We also extend a recent result of
Parthasarathy et al. on matrix games with value zero to the setting of a
symmetric cone (in a Euclidean Jordan algebra). Additionally, we present a
refined cone/space-irreducibility result for positive transformations on
symmetric cones.
