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)

On the structure of acyclic binary relations

We investigate the structure of acyclic binary relations from different points of view. On the one hand, given a nonempty set we study real-valued bivariate maps that satisfy suitable functional equations, in a way that their associated binary relation is acyclic. On the other hand, we consider acyclic directed graphs as well as their representation by means of incidence matrices. Acyclic binary relations can be extended to the asymmetric part of a linear order, so that, in particular, any directed acyclic graph has a topological sorting.This work has been partially supported by the research projects MTM2012-37894-C02-02, TIN2013-47605-P, ECO2015-65031-R, MTM2015-63608-P (MINECO/FEDER), TIN2016-77356-P and the Research Services of the Public University of Navarre (Spain)

Continuity and continuous multi-utility representations of nontotal preorders: some considerations concerning restrictiveness

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