Completely mixed linear games and irreducibility concepts for Z-transformations over self-dual cones.

Abstract: 

In the setting of a self-dual cone in a finite-dimensional real inner product space (in particular, over a symmetric cone in an Euclidean Jordan algebra), we consider zero-sum linear games.  Motivated by dynamical systems, we concentrate on $Z$-transformations (which are generalizations of $Z$-matrices). It is known that a $Z$-transformation with positive (game) value is completely mixed (thus yielding uniqueness of optimal strategies). The present talk deals with the case of value 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.

Posted on Sep 23, 2024 in Optimization and Equilibrium, Seminars