On a Generalization of Markowitz Preference Relation

TL;DR

This paper generalizes Markowitz's portfolio dominance result using a preorder defined by families of functions on topological spaces, establishing the existence of maximal elements under certain compactness conditions.

Contribution

It introduces a broad framework for preference relations on topological spaces, extending classical economic and financial results to more general settings.

Findings

Maximal elements exist under quasi-compact and sequentially compact conditions.

Classical portfolio dominance result is a special case of the general framework.

Applications to various preference and optimization problems are demonstrated.

Abstract

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.