Richter–Peleg multi-utility representations of preorders

Abstract

The 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.\ud We give several applications that include the analysis of countable Richter-Peleg multi-utility representations

Partial Representations of Orderings

“…While a multi-utility representation exists for every not necessarily total preorder on X (see Evren and Ok 16 ), its application is in some sense limited, since if we start from a binary relation on set X and it admits the representation above, then it must be necessarily a preorder (i.e., a reflexive and transitive binary relation). Nevertheless, it is interesting to search for a continuous multi-utility representation of a preorder when the set X is endowed with a topology τ (cf., for instance, Evren and Ok 16 , Bosi and Herden 5 and Alcantud et al 1 ). The existence of a finite multi-utility representation was studied by Ok 29 and Kaminski 20 , who refers to representation by means of multi-objective functions.…”

On continuous multi-utility representations of semi-closed and closed preorders

“…A well-known and in some sense best approach (cf. Evren and Ok [6], Bosi and Herden [2] and Alcantud, Bosi and Zuanon [1]) of representing a preorder (reflexive and transitive relation) on a topological space (X, t) is to find a family F of continuous real-valued functions f on (X, t) such that "x y ⇔ f (x) ≤ f (y)" for all f ∈ F. This equivalence, clearly, implies that every function f ∈ F is increasing. Such a representation of is called a continuous multi-utility representation of .…”

“…In this case, however, the converse implications, in general, fail to be true. In order to verify that the implication "C0 ⇒ C1" does not hold for not necessarily total (complete) preorders let (X, t) := ({1, 2, 3, 4}, P({1, 3, 4}) ∪ {{1, 2, 3, 4}}) and := ∆ X ∪ { (1,3)}. Then one easily verifies that satisfies condition C0 but has the property that neither d({1}) = {1} nor i({3}) = {3} are closed subsets of (X, t).…”

Multiobjective Optimization, Scalarization, and Maximal Elements of Preorders

“…, ) : → R is a (Richter-Peleg) multiutility representation of the preorder ≾ u (≾ u , respectively) allow us to present various results concerning the existence of solutions to the multiobjective optimization problem, also in the classical case when the design space is a compact topological space. We recall that the concept of a (finite) multiutility representation of a preorder was introduced and studied by Ok [12] and Evren and Ok [13], while Richter-Peleg multiutility representations were introduced by Minguzzi [14] and then studied by Alcantud et al [15].…”