Statistics Papers

Document Type

Journal Article

Date of this Version

6-2012

Publication Source

Social Choice and Welfare

Volume

39

Issue

1

Start Page

127

Last Page

140

DOI

10.1007/s00355-011-0547-0

Abstract

Arrow’s theorem implies that a social welfare function satisfying Transitivity, the Weak Pareto Principle (Unanimity), and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are also allowed, a dictatorial social welfare function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions, since non-strict preferences of the dictator are not necessarily followed. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow’s theorem, in the case of non-strict preferences, does not provide a complete characterization of all social welfare functions satisfying Transitivity, the Weak Pareto Principle, and IIA.

The main results of this article provide such a characterization for Arrow’s theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow’s and Wilson’s result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Weak Pareto Principle). Additionally, we derive formulae for the number of functions satisfying these conditions.

Copyright/Permission Statement

The final publication is available at Springer via http://dx.doi.org/10.1007/s00355-011-0547-0.

Comments

At the time of publication, author Elchanan Mossel was affiliated with University of California, Berkeley. Currently, he is a faculty member at the Statistics Department at the University of Pennsylvania.

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Date Posted: 27 November 2017

This document has been peer reviewed.