Existence of Richter-Peleg Representation for General Preferences

TL;DR

This paper characterizes preferences that can be represented by Richter-Peleg functions without requiring completeness or transitivity, using conditions like strong acyclicity and separability, with implications for decision theory.

Contribution

It provides a general characterization of preferences admitting Richter-Peleg representation under minimal assumptions, introducing conditions like strong acyclicity and separability.

Findings

Countable sets: Richter-Peleg representation iff strongly acyclic.

Preorders: Richter-Peleg representation iff separable.

Implications for decision theory and scalar maximization.

Abstract

This paper provides a general characterization of preferences that admit a Richter-Peleg representation without imposing completeness or transitivity. We establish that a binary relation on a nonempty set admits a Richter-Peleg representation if and only if it is "strongly acyclic" and its transitive closure is "separable". Strong acyclicity rules out problematic cycles among indifference classes, while separability limits the structural complexity of the relation. Our main result has two significant corollaries. First, when the set of alternatives is countable, a binary relation admits a Richter-Peleg representation if and only if it is strongly acyclic. Second, a preorder admits a Richter-Peleg representation if and only if it is separable. These findings have important implications for decision theory, particularly for obtaining maximal elements through scalar maximization in the presence of indecisiveness.