Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity

Consistency of preferences is related to rationality, which is associated with the transitivity property. Many properties suggested to model transitivity of preferences are inappropriate for reciprocal preference relations. In this paper, a functional equation is put forward to model the “cardinal consistency in the strength of preferences” of reciprocal preference relations. We show that under the assumptions of continuity and monotonicity properties, the set of representable uninorm operators is characterized as the solution to this functional equation. Cardinal consistency with the conjunctive representable cross ratio uninorm is equivalent to Tanino's multiplicative transitivity property. Because any two representable uninorms are order isomorphic, we conclude that multiplicative transitivity is the most appropriate property for modeling cardinal consistency of reciprocal preference relations. Results toward the characterization of this uninorm consistency property based on a restricted set of $(n-1)$ preference values, which can be used in practical cases to construct perfect consistent preference relations, are also presented.