Three Brief Proofs of Arrow's Impossibility Theorem

TLDR: Barbera as discussed by the authors replaced the decisive voter with the weaker notion of a pivotal voter, thereby shortening the first step, but complicating the second step, and gave 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).

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).