Three Brief Proofs of Arrow's Impossibility Theorem

6 Pages Posted: 16 Jul 2001

See all articles by John Geanakoplos

John Geanakoplos

Yale University - Cowles Foundation

Date Written: June 2001

Abstract

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 (and shortest) proof proceeds by reinterpreting Step 1 of the first proof as saying that all social decisions are made the same way (neutrality).

Keywords: Arrow Impossibility Theorem, Pivotal, Neutrality

JEL Classification: D7, D70, D71

Suggested Citation

Geanakoplos, John D, Three Brief Proofs of Arrow's Impossibility Theorem (June 2001). Yale Cowles Foundation Discussion Paper No. 1123RRR. Available at SSRN: https://ssrn.com/abstract=275510

John D Geanakoplos (Contact Author)

Yale University - Cowles Foundation ( email )

Box 208281
New Haven, CT 06520-8281
United States
203-432-3397 (Phone)
203-432-6167 (Fax)

HOME PAGE: http://cowles.econ.yale.edu/P/au/d_gean.htm

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