Completely mixed linear games on a self-dual cone

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• 作者：M. Seetharama Gowda ^{gowda@umbc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：91A05 ; 46N10 ; 17C20
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：498
• 期：Complete
• 页码：219-230
• 全文大小：330 K

文摘

Given a finite dimensional real inner product space V with a self-dual cone K, an element e   in K^∘$K^{\circ }$ (the interior of K), and a linear transformation L on V  , the value of the linear game (L,e)$(L,e)$ is defined by where Δ(e)={x∈K:〈x,e〉=1}$\mathrm{\Delta }(e)=\{x\in K:〈x,e〉=1\}$. 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^∘$K^{\circ }$. In particular, we study the structure of the set Ω(L)$\mathrm{\Omega }(L)$ of all e   in K^∘$K^{\circ }$ for which the game (L,e)$(L,e)$ is completely mixed and identify certain classes of transformations for which Ω(L)$\mathrm{\Omega }(L)$ equals K^∘$K^{\circ }$. We also describe necessary and sufficient conditions for a game (L,e)$(L,e)$ to be completely mixed when v(L,e)=0$v(L,e)=0$, thereby generalizing a result of Kaplansky [6].