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Three brief proofs of Arrow’s Impossibility Theorem

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Arrow’s original proof of his impossibility theorem proceeded in two steps: showing the existence of a decisive voter, and then showing that a decisive voter is a dictator. Barbera replaced the decisive voter with the weaker notion of a pivotal voter, thereby shortening the first step, but complicating the second step. I give three brief proofs, all of which turn on replacing the decisive/pivotal voter with an extremely pivotal voter (a voter who by unilaterally changing his vote can move some alternative from the bottom of the social ranking to the top), thereby simplifying both steps in Arrow’s proof. My first proof is the most straightforward, and the second uses Condorcet preferences (which are transformed into each other by moving the bottom alternative to the top). The third proof proceeds by reinterpreting Step 1 of the first proof as saying that all social decisions are made the same way (neutrality).

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Correspondence to John Geanakoplos.

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Received: 9 July 2001, Revised: 2 September 2004,

JEL Classification Numbers:

D7, D70, D71.

John Geanakoplos: I wish to thank Ken Arrow, Chris Avery, Don Brown, Ben Polak, Herb Scarf, Chris Shannon, Lin Zhou, and especially Eric Maskin for very helpful comments and advice. I was motivated to think of reproving Arrow’s theorem when I undertook to teach it to George Zettler, a mathematician friend. After I presented this paper at MIT, a graduate student there named Luis Ubeda-Rives told me he had worked out the same neutrality argument as I give in my third proof while he was in Spain nine years ago. He said he was anxious to publish on his own and not jointly, so I encourage the reader to consult his forthcoming working paper. The proofs appearing here appeared in my 1996 CFDP working paper. Proofs 2 and 3 originally used May’s notation, which I have dropped on the advice of Chris Avery.

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Geanakoplos, J. Three brief proofs of Arrow’s Impossibility Theorem. Economic Theory 26, 211–215 (2005). https://doi.org/10.1007/s00199-004-0556-7

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  • DOI: https://doi.org/10.1007/s00199-004-0556-7

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