EXISTENCE OF MAXIMAL ELEMENTS OF SEMICONTINUOUS PREORDERS

We discuss the existence of an upper semicontinuous multi-utility representation of a preorder on a topological space. We then prove that every weakly upper semicontinuous preorder $\precsim$ is extended by an upper semicontinuous preorder and use this fact in order to show that every weakly upper semicontinuous preorder on a compact topological space admits a maximal element

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

Multiobjective Optimization, Scalarization, and Maximal Elements of Preorders

We characterize the existence of (weak) Pareto optimal solutions to the classical multiobjective optimization problem by referring to the naturally associated preorders and their finite (Richter-Peleg) multiutility representation. The case of a compact design space is appropriately considered by using results concerning the existence of maximal elements of preorders. The possibility of reformulating the multiobjective optimization problem for determining the weak Pareto optimal solutions by means of a scalarization procedure is finally characterized

Existence of Order-Preserving Functions for Nontotal Fuzzy Preference Relations under Decisiveness

Looking at decisiveness as crucial, we discuss the existence of an order-preserving function for the nontotal crisp preference relation naturally associated to a nontotal fuzzy preference relation. We further present conditions for the existence of an upper semicontinuous order-preserving function for a fuzzy binary relation on a crisp topological space

Mathematical utility theory and the representability of demand by continuous homogeneous functions

The resort to utility-theoretical issues will permit us to propose a constructive procedure for deriving a homogeneous of degree one continuous function that gives raise to a primitive demand function under suitably mild conditions. This constitutes the first self-contained and elementary proof of a necessary and sufficient condition for an integrability problem to have a solution by continuous (subjective utility) functions

Isotonies on ordered cones throught the concept of a decreasing scale

Using techniques based on decreasing scales, necessary and sufficient conditions are presented for the existence of a continuous and homogeneous of degree one real-valued function representing a (not necessarily complete) preorder defined on a cone of a real vector space. Applications to measure theory and expected utility are given as consequences

Richter–Peleg multi-utility representations of preorders

3siThe existence of a Richter–Peleg multi-utility representation of a preorder by means of upper semicontinuous or continuous functions is discussed in connection with the existence of a Richter–Peleg utility representation. We give several applications that include the analysis of countable Richter–Peleg multi-utility representations.partially_openembargoed_20170311Alcantud, José Carlos R.; Bosi, Gianni; Zuanon, MagalìAlcantud, José Carlos R.; Bosi, Gianni; Zuanon, Magal

Continuous representability of homothetic preorders by means of sublinear order-preserving functions

2nonenoneBOSI G.; ZUANON M.Bosi, Gianni; Zuanon, M

Maximal elements of quasi upper semicontinuous preorders on compact spaces

We introduce the concept of quasi upper semicontinuity of a not necessarily total preorder on a topological space and we prove that there exists a maximal element for a preorder on a compact topological space provided that it is quasi upper semicontinuous. In this way, we generalize many classical and well known results in the literature. We compare the concept of quasi upper semicontinuity with the other semicontinuity concepts to arrive at the conclusion that our definition can be viewed as the most appropriate and natural when dealing with maximal elements of preorders on compact spaces

Upper semicontinuous representations of interval orders

Given an interval order on a topological space, we characterize its representability by means of a pair of upper semicontinuous real-valued functions. This characterization is only based on separability and continuity conditions related to both the interval order and one of its two traces. As a corollary, we obtain the classical Rader's theorem concerning the existence of an upper semicontinuous representation for an upper semicontinuous total preorder on a second countable topological space