Naturality for ranking from pairwise comparisons

Abstract

Pairwise comparison is the most important procedure of the Analytic Hierarchy Process. In the result of any aggregation of pairwise comparisons, we expect that the top alternative beats all others at every pairwise comparison. There, however, is no procedure which satisfies the requirement when cycles of preferences occur. So, I modify it to a condition that the better alternative must win to the worse alternative directly or indirectly in pair comparison. In this paper, the condition, called naturality, is represented by a covariant functor in the framework of category theory. To satisfy the naturality, ranking from pairwise comparisons becomes constructing a category and a covariant functor, and I provide a simple procedure to achieve it. Moreover, rating procedure to decide weights of the alternatives, and a link diagram to visualize the naturality are introduced.