Operators and Laws for Combining Preference Relations

The paper is a theoretical study of a generalization of the lexicographic rule for combining ordering relations. We define the concept of priority operator: a priority operator maps a family of relations to a single relation which represents their lexicographic combination according to a certain priority on the family of relations. We present four kinds of results. We show that the lexicographic rule is the only way of combining preference relations which satisfies natural conditions (similar to those proposed by Arrow). We show in what circumstances the lexicographic rule propagates various conditions on preference relations, thus extending Grosof’s results. We give necessary and sufficient conditions on the priority relation to determine various relationships between combinations of preferences. We give an algebraic treatment of this form of generalized prioritization. Two operators, called but and on the other hand, are sufficient to express any prioritization. We present a complete equational axiomatization of these two operators. These results can be applied in the theory of social choice (a branch of economics), in non-monotonic reasoning (a branch of artificial intelligence), and more generally wherever relations have to be combined.