Continuity and Continuous Multi-utility Representations of Nontotal Preorders: Some Considerations Concerning Restrictiveness

Abstract

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.

Characterization of Useful Topologies in Mathematical Utility Theory by Countable Chain Conditions

“…Then σ(X, C(X, t, R)) is also useful. Indeed, every total preorder that is continuous with respect to σ(X, C(X, t, R)) is also continuous with respect to t according to Bosi and Zuanon, Theorem 2.23, ( 9) ⇒ (1) of [21]. Then there exists a utility representation u : X → R that is continuous with respect to t, hence u is continuous with respect to σ(X, C(X, t, R)).…”

Topologies for the Continuous Representability of All Continuous Total Preorders

“…Bosi and Herden [14,Theorem 3.1] showed that a topology is useful if and only if the topology generated by every complete separable system is second countable. In particular, this latter result can be regarded as the simplest axiomatization of useful topologies, even if it, nevertheless, invokes the concept of a complete separable system (see Bosi and Zuanon [16,Definition 2.20]).…”

Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders

“…Therefore, we have that E xy is a complete decreasing separable system on (X, t) such that x ∈ E , y ∈ E for every E ∈ E xy . Then the thesis follows from Bosi and Zuanon Remark 2.21, 2 in [19], since to every complete decreasing separable system we can associate a continuous increasing function in a very natural way.…”