A pedagogical proof of Arrow's Impossibility Theorem |
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Authors: | Valentino Dardanoni |
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Institution: | (1) Università di Palermo, Istituto di Scienze Finanziarie, Facoltà di Economia, Viale delle Scienze, 90132 Palermo, Italy (e-mail: vdardano@unipa.it), IT |
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Abstract: | In this note I consider a simple proof of Arrow's Impossibility Theorem (Arrow 1963). I start with the case of three individuals
who have preferences on three alternatives. In this special case there are 133=2197 possible combinations of the three individuals' rational preferences. However, by considering the subset of linear preferences, and employing the full strength of the IIA axiom, I reduce the number of cases necessary to completely describe
the SWF to a small number, allowing an elementary proof suitable for most undergraduate students.
This special case conveys the nature of Arrow's result. It is well known that the restriction to three options is not really
limiting (any larger set of alternatives can be broken down into triplets, and any inconsistency within a triplet implies
an inconsistency on the larger set). However, the general case of n≥3 individuals can be easily considered in this framework, by building on the proof of the simpler case. I hope that a motivated
student, having mastered the simple case of three individuals, will find this extension approachable and rewarding.
This approach can be compared with the traditional simple proofs of Barberà (1980); Blau (1972); Denicolò (1996); Fishburn
(1970); Kelly (1988); Mueller (1989); Riker and Ordeshook (1973); Sen (1979, 1986); Suzumura (1988), and Taylor (1995).
Received: 5 January 1999/Accepted: 10 December 1999 |
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