General preferences and utility functions. An approach based on the dual Kantorovich problem

Abstract

In this paper, with help of a special dual Kantorov� ich mass transfer problem new existence conditions are obtained for utility functions of general (nontotal and nontransitive) preference relations. Further, by a preference on a set X of alternatives we mean an arbitrary binary relation � (totality or transi� tivity in general is not required). If xy (in such a case, we will write also yx), then the alternative x is more preferable than the alternative y. We connect withtwo binary relations: the strict preference � (xy means that xy holds but yx fails) and the equal worth relation ~ (x ~ y means that both xy and yx hold true). Alternatives x and y are incomparable if none of them is more preferable than other, i.e. none of relations xy and yx is satisfied. A preference is called total if any two alternatives are comparable. A preference with properties of reflexivity (xx ∀x ∈ X) and transitivity (∀x, y, z ∈ X: xy, yz ⇒ xz) is called a preorder (or quasiordering in other termi� nology).