Completely mixed linear games and irreducibility concepts for Z-transformations over self-dual cones
Year: 2026
Type: article
Abstract: In the setting of a self-dual cone in a finite-dimensional real 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.
Author: M. Seetharama Gowda
Institution: University of Maryland, Baltimore County
Language: English
Topic: Game Theory and Applications
Field: Decision Sciences
Domain: Social Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Funder: Indian Statistical Institute
OpenAlex ID: W7148631044