Three Brief Proofs of Arrow's Impossibility Theorem

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Discussion Paper

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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 uses almost no notation, while the second uses May’s notation and is extremely brief. The third proof is perhaps the most interesting, because along the way to proving the existence of an extremely pivotal voter, it shows that the Arrow axioms guarantee issue neutrality, that is, that every choice must be made by exactly the same process.

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