Transitive Matrices, Strict Preference Order and Ordinal Evaluation Operators

Abstract

Let X ={x ₁, x ₂, ..., x _n } be a set of alternatives and a _ij a positive number expressing how much the alternative x _i is preferred to the alternative x _j . Under suitable hypothesis of no indifference and transitivity over the pairwise comparison matrix A=(a _ij ), the actual qualitative ranking on the set X is achievable. Then a coherent priority vector is a vector giving a weighted ranking agreeing with the actual ranking and an ordinal evaluation operator is a functional F that, acting on the row vectors $\underline{a}_{i}$ , translates A in a coherent priority vector. In this paper we focus our attention on the matrix A, looking for conditions ensuring the existence of coherent priority vectors. Then, given a type of matrices, we look for ordinal evaluation operators, including OWA operators, associated to it.