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A pedagogical proof of Arrow's Impossibility Theorem


  • Valentino Dardanoni

    () (UniversitÁ di Palermo, Istituto di Scienze Finanziarie, FacoltÁ di Economia, Viale delle Scienze, 90132 Palermo, Italy)


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).

Suggested Citation

  • Valentino Dardanoni, 2001. "A pedagogical proof of Arrow's Impossibility Theorem," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(1), pages 107-112.
  • Handle: RePEc:spr:sochwe:v:18:y:2001:i:1:p:107-112
    Note: Received: 5 January 1999/Accepted: 10 December 1999

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    References listed on IDEAS

    1. Barbera, Salvador, 1980. "Pivotal voters : A new proof of arrow's theorem," Economics Letters, Elsevier, vol. 6(1), pages 13-16.
    2. Blau, Julian H, 1972. "A Direct Proof of Arrow's Theorem," Econometrica, Econometric Society, vol. 40(1), pages 61-67, January.
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    Cited by:

    1. Susumu Cato, 2013. "Alternative proofs of Arrow’s general possibility theorem," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(2), pages 131-137, November.
    2. Susumu Cato, 2010. "Brief proofs of Arrovian impossibility theorems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 35(2), pages 267-284, July.
    3. Mark Fey, 2014. "A straightforward proof of Arrow's theorem," Economics Bulletin, AccessEcon, vol. 34(3), pages 1792-1797.

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