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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 (and shortest) 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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    Bibliographic Info

    Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1123R3.

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    Length: 6 pages
    Date of creation: Apr 1996
    Date of revision: Jun 2001
    Publication status: Published in Economic Theory (2005), 26(1): 211-215
    Handle: RePEc:cwl:cwldpp:1123r3

    Note: CFP 1116.
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    Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
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    Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

    Related research

    Keywords: Arrow Impossibility Theorem; pivotal; neutrality;

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    Blog mentions

    As found by, the blog aggregator for Economics research:
    1. Surprises in Mathematics and Theory
      by ? in Gödel's Lost Letter and P=NP on 2009-09-27 21:35:38


    This item is featured on the following reading lists or Wikipedia pages:
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    3. Teorema dell'impossibilità di Arrow in Wikipedia Italian ne '')
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    7. 애로의 불가능성 정리 in Wikipedia Korean ne '')
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