Scholarlink

<p>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 <i>Z</i>-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 <i>Z</i>-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 <i>Z</i>-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.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

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

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2026年4月3日

查看全文
  • M. Seetharama Gowda

摘要

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.