On the game-theoretic value of a linear transformation relative to a self-dual cone

Abstract

This paper is concerned with a generalization of the concept of value of a (zero-sum) matrix game. Given a finite dimensional real inner product space V with a self-dual cone K, an element e in the interior of K, and a linear transformation L, we define the value of L byv(L):=maxx∈Δ⁡miny∈Δ⁡〈L(x),y〉=miny∈Δ⁡maxx∈Δ⁡〈L(x),y〉, where Δ={x∈K:〈x,e〉=1}. This reduces to the classical value of a square matrix when V=Rn, K=R+n, and e is the vector of ones. In this paper, we extend some classical results of Kaplansky and Raghavan to this general setting. In addition, for a Z-transformation (which is a generalization of a Z-matrix), we relate the value with various properties such as the positive stable property, the S-property, etc. We apply these results to find the values of the Lyapunov transformation LA and the Stein transformation SA on the cone of n×n real symmetric positive semidefinite matrices.