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Three Brief Proofs of Arrow's Impossibility Theorem

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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).

Suggested Citation

  • John Geanakoplos, 1996. "Three Brief Proofs of Arrow's Impossibility Theorem," Cowles Foundation Discussion Papers 1123R3, Cowles Foundation for Research in Economics, Yale University, revised Jun 2001.
  • Handle: RePEc:cwl:cwldpp:1123r3
    Note: CFP 1116.
    as

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    File URL: http://cowles.yale.edu/sites/default/files/files/pub/d11/d1123-r3.pdf
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    Blog mentions

    As found by EconAcademics.org, the blog aggregator for Economics research:
    1. Surprises in Mathematics and Theory
      by ? in Gödel's Lost Letter and P=NP on 2009-09-28 02:35:38

    More about this item

    Keywords

    Arrow Impossibility Theorem; pivotal; neutrality;

    JEL classification:

    • D7 - Microeconomics - - Analysis of Collective Decision-Making
    • D70 - Microeconomics - - Analysis of Collective Decision-Making - - - General
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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    1. Arrow'un imkânsızlık teoremi in Wikipedia Turkish ne '')
    2. Wikipedia:Articles for deletion/David Linden in Wikipedia English ne '')
    3. Teorema dell'impossibilità di Arrow in Wikipedia Italian ne '')
    4. Arrows teorem in Wikipedia Norwegian ne '')
    5. Paradoja de Arrow in Wikipedia Spanish ne '')
    6. Wikipedia:Articles for deletion/Log/2008 July 3 in Wikipedia English ne '')
    7. 애로의 불가능성 정리 in Wikipedia Korean ne '')
    8. قضیه عدم امکان ارو in Wikipedia Persian ne '')

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