Continuous representation of a preference relation on a connected topological space (original) (raw)

Mathematical Topics on Representations of Ordered Structures and Utility Theory

Studies in Systems, Decision and Control, 2020

A continuous multi-utility fully represents a not necessarily total preorder on a topological space by means of a family of continuous increasing functions. While it is very attractive for obvious reasons, and therefore it has been applied in different contexts, such as expected utility for example, it is nevertheless very restrictive. In this paper we first present some general characterizations of the existence of a continuous order-preserving function, and respectively a continuous multi-utility representation, for a preorder on a topological space. We then illustrate the restrictiveness associated to the existence of a continuous multi-utility representation, by referring both to appropriate continuity conditions which must be satisfied by a preorder admitting this kind of representation, and to the Hausdorff property of the quotient order topology corresponding to the equivalence relation induced by the preorder. We prove a very restrictive result, which may concisely described as follows: the continuous multi-utility representability of all closed (or equivalently weakly continuous) preorders on a topological space is equivalent to the requirement according to which the quotient topology with respect to the equivalence corresponding to the coincidence of all continuous functions is discrete.

Continuous representability of homothetic preferences by means of homogeneous utility functions

Journal of Mathematical Economics, 2000

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder. 

On the Preference Relations with Negatively Transitive Asymmetric Part. I

Comptes rendus de l'Académie bulgare des sciences: sciences mathématiques et naturelles

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 ι ∈ I with their inverse images p −1 (ι) 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 A 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.

Representing preferences with nontransitive indifference by a single real-valued function

Journal of Mathematical Economics, 1995

Let + be an interval order on a topological space (X, r), and let x < L y if and only if [y<z~x+z], and x+2' y if and only if [z <x * z + y]. Then <t and <t" are complete preorders. In the particular case when 4 is a semiorder, let x <t!. y if and only if x.<* _ ~1 and x<"* y. Then 4% is a complete preorder, too. We present sufficient conditions for the existence of continuous utility functions representing < t , -X 2 * and <'1 , by using the notion of strong separability of a preference relation, which was introduced by Chateauneuf (Journal of Mathemarical Economics, 1987, 16, 139-146). Finally, we discuss the existence of a pair of continuous functions u, u representing a strongly separable interval order + on a measurable topological space (X, T, CL, A).

Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof

Journal of Mathematical Economics, 2002

In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces.

A note on the representation of preferences

Mathematical Social Sciences, 1995

We consider a class of relations which includes irreflexive preference relations and interdependent preferences. For this class, we obtain necessary and sufficient conditions for representation of the relation by two numerical functions in the sense of a ~ x if and only if u(a) < vex).

From preference to utility: a problem of descriptive set theory

Notre Dame Journal of Formal Logic, 1985

Some years ago J. H. Silver proved that a co-analytic equivalence relation on a Polish space has either countably many or continuum many equivalence classes. Later L. Harrington greatly simplified the complicated original proof. The present paper is a sort of footnote to Harrington's lectures on these matters. It will be shown that information developed in his proof settles a problem of (hyper-)theoretical mathematical economics first investigated by Wesley [13] and Mauldin [8]. Namely, it will be shown that any family of closed preference orders that is parametrized in a Borel fashion can be represented by a family of continuous utility functions parametrized in an absolutely measurable fashion. Though the author is greatly indebted to Mauldin's work [8], the treatment of the problem here will be self-contained. Background and motivation for problems of this kind can be found in [6], Section 2.1. Terminology and notation pertaining to descriptive set theory will be as in [9]. 2 Definitions Throughout let ψ be a topological space. A preference order on 'ψ is any transitive, connected binary relation <*. Associated are the strict preference and indifference relations given by: x <* y <-> x <* y & ~y <* x X =* y +-> x <* y & y <* χ m Note that Ξ* i s an equivalence relation, and that <* induces a linear order on its equivalence classes, [x]* will denote the equivalence class of x. <* will be *Research in part supported by USA National Science Foundation Grant MCS 8003254.

Continuity of utility functions representing fuzzy preferences

Social Choice and Welfare, 2011

In a previous paper, we established necessary and sufficient conditions for a given binary fuzzy relation to be representable by a utility function. In this article, we construct a crisp order topology associated to a given weakly complete fuzzy pre-order and introduce the notion of “continuous fuzzy pre-order.” We show that this new condition and the conditions introduced in the previous paper are together necessary and sufficient for a numerical representation of a given weakly complete fuzzy pre-order by a continuous utility function.