Juan Carlos Candeal

The non-existence of a utility function and the structure of non-representable preference relations

In this paper we investigate the structure of non-representable preference relations. While there is a vast literature on different kinds of preference relations that can be represented by a real-valued utility function, very little is known or understood about preference relations that cannot be represented by a real-valued utility function. There has been no systematic analysis of the non-representation problem. In this paper we give a complete description of non-representable preference relations which are total preorders or chains. We introduce and study the properties of four classes of non-representable chains: long chains, planar chains, Aronszajn-like chains and Souslin chains. In the main theorem of the paper we prove that a chain is non-representable if and only it is a long chain, a planar chain, an Aronszajn-like chain or a Souslin chain.

Numerical Representations of Interval Orders

In the framework of the analysis of orderings whose associated indifference relation is not necessarily transitive, we study the structure of an interval order and its representability through a pair of real-valued functions. We obtain a list of characterizations of the existence of a representation, showing that the three main techniques that have been used in the literature to achieve numerical representations of interval orders are indeed equivalent.

Utility and entropy

Summary. In this paper we study an astonishing similarity between the utility representation problem in economics and the entropy representation problem in thermodynamics.

Numerical representability of semiorders

In the framework of the analysis of orderings whose associated indifference relation is not necessarily transitive, we study the structure of a semiorder, and its representability through a real-valued function and a threshold. Inspired in a recent characterization of the representability of interval orders, we obtain a full characterization of the existence of numerical representations for semiorders. This is an extension to the general case of the classical Scott-Suppes theorem concerning the representability of semiorders defined on finite sets.

Some results on representation and extension of preferences

Abstract In this note we study properties about the existence of continuous utility representations and extensions of continuous preorders. We provide a unified treatment in a more general setting of results that Monteiro (1987) and Yi (1993) gave for path connected topological spaces.

INTERVAL-VALUED REPRESENTABILITY OF QUALITATIVE DATA: THE CONTINUOUS CASE

In the framework of the representability of ordinal qualitative data by means of interval-valued correspondences, we study interval orders defined on a nonempty set X. We analyse the continuous case, that corresponds to a set endowed with a topology that furnishes an idea of continuity, so that it becomes natural to ask for the existence of quantifications based on interval-valued mappings from the set of data into the real numbers under preservation of order and topology. In the present paper we solve a continuous representability problem for interval orders. We furnish a characterization of the representability of an interval order through a pair of continuous real-valued functions so that each element in X has associated in a continuous manner a characteristic interval or equivalently a symmetric triangular fuzzy number.

Continuous representability of homothetic preferences by means of homogeneous utility functions

Abstract We provide a full characterization of those complete preorders defined on a real cone that admit a representation by means of a utility function which is continuous and homogeneous of degree one. Our approach is based on the solution of the functional equation of homotheticity .

Representability of binary relations through fuzzy numbers

We analyse the representability of different classes of binary relations on a set by means of suitable fuzzy numbers. In particular, we show that symmetric triangular fuzzy numbers can be considered as the best codomain to represent interval orders. We also pay attention to the representability of other classes of acyclic binary relations.

Homothetic and weakly homothetic preferences

Abstract This paper describes properties of homothetic and weakly homothetic preferences defined on a cone of Rn. We obtain a general result concerning the structure of weakly homothetic preferences. As an application of it, we answer a recent conjecture posed by Dow and Werlang. Further results are also proved, including a full classification of weakly homothetic preferences defined on the real line (n = 1), and a characterization of the generalized Cobb-Douglas preorders on R n ++ .

Some issues related to the topological aggregation of preferences

This paper deals with the topological approach to social choice theory initiated by Chichilnisky. We study several issues concerning the existence and uniqueness of Chichilnisky rules defined on preference spaces. We show that on topological vector spaces the only additive, anonymous, and unanimous aggregation n-rule is the convex mean. We study the case of infinite agents and show that an infinite Chichilnisky rule might be considered as the limit of rules for finitely many agents. Finally, we show that under some restrictions on the preference space, the existence of a Chichilnisky rule for every finite case implies the existence of a weak Chichilnisky rule for the infinite case.