THE SET THEORETIC AMBIT OF ARROW'S THEOREM

Abstract

Set theoretic formulation of Arrow's theorem, viewed in light of a taxonomy of transitive relations, serves to unmask the theorem's understated generality. Under the impress of the independence of irrelevant alternatives, the antipode of ceteris paribus reasoning, a purported compiler function either breaches some other rationality premise or produces the effet Condorcet. Types of cycles, each the seeming handiwork of a virtual voter disdaining transitivity, are rigorously defined. Arrow's theorem erects a dilemma between cyclic indecision and dictatorship. Maneuvers responsive thereto are explicable in set theoretic terms. None of these gambits rival in simplicity the unassisted escape of strict linear orderings, which, by virtue of the Arrow-Sen reflexivity premise, are not captured by the theorem. Yet these are the relations among whose n-tuples the effet Condorcet is most frequent. A generalization and stronger theorem encompasses these and all other linear orderings and total tierings. Revisions to the Arrow-Sen definitions of 'choice set' and 'rationalization' similarly enable one to generalize Sen's demonstration that some rational choice function always exists. Similarly may one generalize Debreu's theorems establishing conditions under which a binary relation may be represented by a continuous real-valued order homomorphism. Arrow has recounted that an early interest in mathematical logic, piqued by the work of Russell and a course from Tarksi, formed the seedbed for his celebrated impossibility theorem on the 'rational' aggregation of certain binary relations. 1 Arrow's departure from the nomenclature of set theory to some extent clouds the set theoretic compass of the theorem. This point has itself been eclipsed in a rich discussion of related impossibility results and of consequences for economic decisionmaking. The following treatment first casts the theorem and the occurrence of cycles in precise set theoretical terms. It then traces in the same terms responses to the dilemma that the theorem poses. The fruit is the discovery that the theorem is at once uneconomical and unnecessarily restricted in a respect whose cure furnishes a stronger result and generalization. The same is also true of Sen's companion results on the inferability of certain choice functions. These generalizations may seem within economics to enlarge scope only by including relations infrequently exhibited by consumers. Their general theoretical significance lies in the sweep of conclusions across all manner of transitive relations that position each member of a set with respect to each other.