Reciprocity and consistency of fuzzy preference relations

Preference relations are the most common representation structures of information used in decision making problems because they are useful tool in modelling decision processes, above all when we want to aggregate experts’ preferences into group preferences. Therefore, to establish rationality properties to be verified by preference relations is very important in the design of good decision making models. There are three fundamental and hierarchical levels of rationality assumptions when dealing with preference relations: the first one requires indifference between any alternative and itself, the second one assumes the property of reciprocity in the pairwise comparison between any two alternatives, and the third one is associated with the transitivity in the pairwise comparison among any three alternatives. Furthermore, it would also be desirable to maintain the rationality assumptions on the individual preferences in the aggregation process, so that the collective preferences verify the same ones. However, as this is not always the case, establishing conditions that guarantee the preservation of these rationality properties throughout the aggregation process becomes very important. In this article we address this problem and present a review of the main results that we have obtained about reciprocity and consistency properties of fuzzy preference relations. In particular, we present a characterization of fuzzy consistency based on the additive transitivity property which facilitates the verification of consistency in the case of fuzzy preference relations. Using this new characterization we give a method to construct consistent fuzzy preference relations from n−1 given preference values. We also discuss some questions concerning the compatibility between the three levels of rationality, as well as the conflict that appears between the additive consistency property and the scale used to provide fuzzy preferences. Finally, we provide aggregation operators that provide reciprocal and consistent collective preference relations when the individual preference relations are reciprocal and consistent.