On the Preference Relations with Negatively Transitive Asymmetric Part. I

TL;DR

This paper characterizes preference relations with negatively transitive asymmetric parts using a ballooning construction over linearly ordered sets, and proves the existence of continuous utility functions under certain topological conditions.

Contribution

It introduces a novel ballooning framework to represent negatively transitive preferences and establishes conditions for continuous utility representation.

Findings

Preference relations with negatively transitive asymmetric parts can be represented via ballooning.

A continuous generalized utility function exists under natural topological conditions.

The difference between linearity and negative transitivity is characterized by balloons.

Abstract

Given a linearly ordered set I, every surjective map p: A --> I endows the set A with a structure of set of preferences by "replacing" the elements of I with their inverse images via p considered as "balloons" (sets endowed with an equivalence relation), lifting the linear order on A, and "agglutinating" this structure with the balloons. Every ballooning A of a structure of linearly ordered set I is a set of preferences whose preference relation (not necessarily complete) is negatively transitive and every such structure on a given set A can be obtained by ballooning of certain structure of a linearly ordered set I, intrinsically encoded in A. In other words, the difference between linearity and negative transitivity is constituted of balloons. As a consequence of this characterization, under certain natural topological conditions on the set of preferences A furnished with its interval topology, the existence of a continuous generalized utility function on A is proved.