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Sen’s proofs of the Arrow and Gibbard theorems

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  • Piggins, Ashley

Abstract

The expanded edition of Sen’s classic Collective Choice and Social Welfare contains a new proof of Arrow’s impossibility theorem. The proof relies on two lemmas: Spread of Decisiveness and Contraction of Decisive Sets. The first lemma requires that the social preference relation is quasi-transitive, but the second requires full transitivity. I show how Spread of Decisiveness can be used to prove Gibbard’s oligarchy theorem. Sen himself sketches how this can be done, but my argument is different and simpler. Arrow’s result follows trivially once transitivity is strengthened, omitting the need for Contraction.

Suggested Citation

  • Piggins, Ashley, 2017. "Sen’s proofs of the Arrow and Gibbard theorems," Economics Letters, Elsevier, vol. 161(C), pages 99-101.
    • Handle: RePEc:eee:ecolet:v:161:y:2017:i:c:p:99-101
    DOI: 10.1016/j.econlet.2017.09.038
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    References listed on IDEAS

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    1. Susumu Cato, 2010. "Brief proofs of Arrovian impossibility theorems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 35(2), pages 267-284, July.
    2. Allan Gibbard, 2014. "Intransitive social indifference and the Arrow dilemma," Review of Economic Design, Springer;Society for Economic Design, vol. 18(1), pages 3-10, March.
    3. John Weymark, 2014. "An introduction to Allan Gibbard’s oligarchy theorem paper," Review of Economic Design, Springer;Society for Economic Design, vol. 18(1), pages 1-2, March.
    4. Samuelson, Paul A, 1977. "Reaffirming the Existence of "Reasonable" Bergson-Samuelson Social Welfare Functions," Economica, London School of Economics and Political Science, vol. 44(173), pages 81-88, February.
    5. John Geanakoplos, 2005. "Three brief proofs of Arrow’s Impossibility Theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(1), pages 211-215, July.
    6. Weymark, John A., 2014. "An Introduction To Allan Gibbard’S Harvard Seminar Paper," Economics and Philosophy, Cambridge University Press, vol. 30(3), pages 263-268, November.
    7. Young, H. P., 1974. "An axiomatization of Borda's rule," Journal of Economic Theory, Elsevier, vol. 9(1), pages 43-52, September.
    8. Reny, Philip J., 2001. "Arrow's theorem and the Gibbard-Satterthwaite theorem: a unified approach," Economics Letters, Elsevier, vol. 70(1), pages 99-105, January.
    9. Sen, Amartya, 1995. "Rationality and Social Choice," American Economic Review, American Economic Association, vol. 85(1), pages 1-24, March.
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    Cited by:

    1. Duddy, Conal & Piggins, Ashley, 2020. "A foundation for Pareto optimality," Journal of Mathematical Economics, Elsevier, vol. 88(C), pages 25-30.

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    More about this item

    Keywords

    Preference aggregation; Arrow’s impossibility theorem; Gibbard’s theorem; Transitivity; Decisiveness;
    All these keywords.

    JEL classification:

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

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