On a Generalization of Markowitz Preference Relation

Given two families of continuous functions $u$ and $v$ on a topological space $X$, we define a preorder $R=R(u,v)$ on $X$ by the condition that any member of $u$ is an $R$-increasing and any member of $v$ is an $R$-decreasing function. It turns out that if the topological space $X$ is quasi-compact and sequentially compact, then any element of $X$ is $R$-dominated by an $R$-maximal element of $X$. In particular, since the $(n-1)$-dimensional simplex is a compact subset of the real $n$-dimensional vector space, then considering its members as portfolios consisting of $n$ financial assets, we obtain the classical 1952 result of Harry Markowitz that any portfolio is dominated by an efficient portfolio. Moreover, several other examples of possible application of this general setup are presented.