A tight quantitative version of Arrow's impossibility theorem

The well-known Impossibility Theorem of Arrow asserts that any Generalized Social Welfare Function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily nontransitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any > 0, there exists = ( ) such that if a GSWF on three alternatives satisfies the IIA condition and its probability of non-transitive outcome is at most , then the GSWF is at most -far from being a dictatorship or from breaching the Unanimity condition. In 2009, Mossel proved such quantitative version, with ( ) = exp(-C/ 21 ), and generalized it to GSWFs with k alternatives, for all k 3. In this paper we show that the quantitative version holds with ( ) = C

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Nathan Keller
• Keywords: Alternatives, CORR 2010, Education, Quantitative Version, Well-known Impossibility Theorem