A pedagogical proof of Arrow's Impossibility Theorem

Valentino Dardanoni ()

Social Choice and Welfare, 2001, vol. 18, issue 1, 107-112

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\geq3 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 BarberÁ (1980); Blau (1972); DenicolÔ (1996); Fishburn (1970); Kelly (1988); Mueller (1989); Riker and Ordeshook (1973); Sen (1979, 1986); Suzumura (1988), and Taylor (1995).

Date: 2001-01-10
Note: Received: 5 January 1999/Accepted: 10 December 1999
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Working Paper: A Pedagogical Proof of Arrow's Impossibility Theorem (1999)
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