The interplay between transitivity and completeness: Generalized NaP-preferences

ElsevierVolume 108, June 2022, 102667Journal of Mathematical PsychologyAbstract

We analyze the interaction between two classical tenets of economic rationality: transitivity and completeness. For a single binary relation, the combination of these properties yields a total preorder. For two binary relations, their interplay entails a NaP-preference: this is a nested pair ≿N,≿P of relations satisfying mixed forms of transitivity and completeness. It is known that a NaP-preference ≿N,≿P is characterized by a family of total preorders having ≿N and ≿P as their intersection and union, respectively. This paper examines extensions of this approach to more than two relations. First we define generalized NaP-preferences (GNaP-preferences): these are diamond-shaped quadruples of binary relations on a set such that transitivity and completeness are suitably spread throughout all four components. Then we characterize a GNaP-preference by a family of total preorders witnessing all relations by different combinations of union and intersection. We also show that a suitable amalgamation of NaP-preferences generates a GNaP-preference, and vice versa each GNaP-preference arises by this procedure. Finally, we introduce modal preference structures: these are lattices of binary relations such that the satisfaction of transitivity and completeness is guided by the action of the universal and the existential quantifiers over a family of total preorders. Applications of these novel preference structures to economics and psychology emerge naturally.

Keywords

Transitivity

Completeness

Transitive coherence

Mixed completeness

NaP-preference

GNaP-preference

Modal preference structure

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