The Set Theoretic Ambit Of Arrow's Theorem

Author: Guenin L.M.

Source: Synthese, Volume 126, Number 3, March 2001 , pp. 443-472(30)

Publisher: Springer

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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.

Language: English

Document Type: Regular paper

Affiliations: 1: Harvard University, Boston, MA U.S.A. E-mail: guenin@hms.harvard.edu

Publication date: 2001-03-01