Continuous Utility Representation Theorems in Arbitrary Concrete Categories

Abstract

In this paper the continuous utility representation problem will be discussed in arbitrary concrete categories. In particular, generalizations of the utility representation theorems of Eilenberg, Debreu and Estévez and Hervés will be presented that also hold if the codomain of a utility function is an arbitrary totally ordered set and not just the real line. In addition, we shall prove and apply a general result on the characterization of structures that have the property that every continuous total preorder has a continuous utility representation. Finally, generalizations of the utility representation theorems of Debreu and Eilenberg will be discussed that are valid if we consider arbitrary binary relations and allow a utility function to have values in an arbitrary totally ordered set.

Continuous multi-utility representations of preorders

“…Since has a continuous linear extension there exists a family C of comonotonic respectively, similarly ordered continuous and isotonic real-valued functions f on (X, , t) such that for every pair (v, w) ∈≺ there exists some function f vw ∈ C such that f vw (v) < f vw (w) (cf. Schmeidler, 1986 respectively, Skala, 1998 respectively, Bosi andHerden, 2008). The reader may recall that comonotonic respectively, similarly ordered means that for all functions f ∈ C and all functions g ∈ C and all pairs (s, t) …”