Completely mixed linear games on a self-dual cone

Given a finite dimensional real inner product space V with a self-dual cone K, an element e in K∘ (the interior of K), and a linear transformation L on V, the value of the linear game (L,e) is defined byv(L,e):=maxx∈Δ(e)⁡miny∈Δ(e)⁡〈L(x),y〉=miny∈Δ(e)⁡maxx∈Δ(e)⁡〈L(x),y〉, where Δ(e)={x∈K:〈x,e〉=1}. In [5], various properties of a linear game and its value were studied and some classical results of Kaplansky [6] and Raghavan [8] were extended to this general setting. In the present paper, we study how the value and properties change as e varies in K∘. In particular, we study the structure of the set Ω(L) of all e in K∘ for which the game (L,e) is completely mixed and identify certain classes of transformations for which Ω(L) equals K∘. We also describe necessary and sufficient conditions for a game (L,e) to be completely mixed when v(L,e)=0, thereby generalizing a result of Kaplansky [6].