Extending Utility Functions on Arbitrary Sets

TL;DR

This paper investigates how to extend functions defined on subsets of arbitrary sets to the entire set in a way that preserves strict monotonicity with respect to a preorder, introducing conditions and methods for such extensions.

Contribution

The paper characterizes when a function can be extended monotonically without continuity constraints and proposes a class of such extensions based on utility representations.

Findings

Extension is possible if and only if the function is gap-safe increasing.

A class of extensions using utility representations is proposed and analyzed.

Simplifications occur when the subset is a Pareto set.

Abstract

We consider the problem of extending a function $f^{}_P$ defined on a subset $P$ of an arbitrary set $X$ to $X$ strictly monotonically with respect to a preorder $\succcurlyeq$ defined on $X$, without imposing continuity constraints. We show that whenever $\succcurlyeq$ has a utility representation, $f^{}_P$ is extendable if and only if it is gap-safe increasing. A class of extensions involving an arbitrary utility representation of $\succcurlyeq$ is proposed and investigated. Connections to related topological results are discussed. The condition of extendability and the form of the extension are simplified when $P$ is a Pareto set.