Open questions in utility theory

Throughout this paper, our main idea is to explore different classical questions arising in Utility Theory, with a particular attention to those that lean on numerical representations of preference orderings. We intend to present a survey of open questions in that discipline, also showing the state-of-art of the corresponding literature.This work is partially supported by the research projects ECO2015-65031-R, MTM2015-63608-P (MINECO/ AEI-FEDER, UE), and TIN2016-77356-P (MINECO/ AEI-FEDER, UE)

Continuous representability of interval orders: The topological compatibility setting

In this paper, we go further on the problem of the continuous numerical representability of interval orders defined on topological spaces. A new condition of compatibility between the given topology and the indifference associated to the main trace of an interval order is introduced. Provided that this condition is fulfilled, a semiorder has a continuous interval order representation through a pair of continuous real-valued functions. Other necessary and sufficient conditions for the continuous representability of interval orders are also discussed, and, in particular, a characterization is achieved for the particular case of interval orders defined on a topological space of finite support

Conditions for the Upper Semicontinuous Representability of Preferences with Nontransitive Indifference

We present different conditions for the existence of a pair of upper semicontinuous functions representing an interval order on a topological space without imposing any restrictive assumptions neither on the topological space nor on the representing functions. The particular case of second countable topological spaces, which is particularly interesting and frequent in economics, is carefully considered. Some final considerations concerning semiorders finish the paper

Semiorders and continuous Scott–Suppes representations. Debreu’s Open Gap Lemma with a threshold

The problem of finding a utility function for a semiorder has been studied since 1956, when the notion of semiorder was introduced by Luce. But few results on continuity and no result like Debreu’s Open Gap Lemma, but for semiorders, was found. In the present paper, we characterize semiorders that accept a continuous representation (in the sense of Scott–Suppes). Two weaker theorems are also proved, which provide a programmable approach to Open Gap Lemma, yield a Debreu’s Lemma for semiorders, and enable us to remove the open-closed and closed-open gaps of a set of reals while keeping the threshold.Asier Estevan acknowledges financial support from the Ministry of Science and Innovation of Spain under grants PID2020-119703RB-I00 and PID2021-127799NB-I00 as well as from the UPNA, Spain under grant JIUPNA19-2022

A selection of maximal elements under non-transitive indifferences

In this work we are concerned with maximality issues under intransitivity of the indifference. Our approach relies on the analysis of "undominated maximals" (cf., Peris and Subiza, J Math Psychology 2002). Provided that an agent's binary relation is acyclic, this is a selection of its maximal elements that can always be done when the set of alternatives is finite. In the case of semiorders, proceeding in this way is the same as using Luce's selected maximals. We put forward a sufficient condition for the existence of undominated maximals for interval orders without any cardinality restriction. Its application to certain type of continuous semiorders is very intuitive and accommodates the well-known "sugar example" by Luce.Maximal element; Selection of maximals; Acyclicity; Interval order; Semiorder

Biased quantitative measurement of interval ordered homothetic preferences

We represent interval ordered homothetic preferences with a quantitative homothetic utility function and a multiplicative bias. When preferences are weakly ordered (i.e. when indifference is transitive), such a bias equals 1. When indifference is intransitive, the biasing factor is a positive function smaller than 1 and measures a threshold of indifference. We show that the bias is constant if and only if preferences are semiordered, and we identify conditions ensuring a linear utility function. We illustrate our approach with indifference sets on a two dimensional commodity space.Weak order, semiorder, interval order, intransitive indifference, independence, homothetic, representation, linear utility

New trends on the numerical representability of semiordered structures

[EN] We introduce a survey, including the historical back-ground, on different techniques that have recently been issued in the search for a characterization of the representability of semiordered structures, in the sense of Scott and Suppes, by means of a real-valued function and a strictly positive threshold of discrimination.This work has been supported by the research projects MTM2007-62499, ECO2008-01297, MTM2009-12872-C02-02 and MTM2010-17844 (Spain)Abrísqueta, F.; Campión, M.; Catalán, R.; De Miguel, J.; Estevan, A.; Induráin, E.; Zudaire, M.... (2012). New trends on the numerical representability of semiordered structures. Mathware & Soft Computing Magazine. 19(1):25-37. http://hdl.handle.net/10251/57632S253719

A new approach on distributed systems: orderings and representability

In the present paper we propose a new approach on `distributed systems': the processes are represented through total orders and the communications are characterized by means of biorders. The resulting distributed systems capture situations met in various fields (such as computer science, economics and decision theory). We investigate questions associated to the numerical representability of order structures, relating concepts of economics and computing to each other. The concept of `quasi-finite partial orders' is introduced as a finite family of chains with a communication between them. The representability of this kind of structure is studied, achieving a construction method for a finite (continuous) Richter-Peleg multi-utility representation