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From Arrow to cycles, instability, and chaos by untying alternatives

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  • Thomas Schwartz

    (Department of Political Science, UCLA, Los Angeles, CA 90095-1472, USA)

Abstract

From remarkably general assumptions, Arrow's Theorem concludes that a social intransitivity must afflict some profile of transitive individual preferences. It need not be a cycle, but all others have ties. If we add a modest tie-limit, we get a chaotic cycle, one comprising all alternatives, and a tight one to boot: a short path connects any two alternatives. For this we need naught but (1) linear preference orderings devoid of infinite ascent, (2) profiles that unanimously order a set of all but two alternatives, and with a slightly fortified tie-limit, (3) profiles that deviate ever so little from singlepeakedness. With a weaker tie-limit but not (2) or (3), we still get a chaotic cycle, not necessarily tight. With an even weaker one, we still get a dominant cycle, not necessarily chaotic (every member beats every outside alternative), and with it global instability (every alternative beaten). That tie-limit is necessary for a cycle of any sort, and for global instability too (which does not require a cycle unless alternatives are finite in number). Earlier Arrovian cycle theorems are quite limited by comparison with these.

Suggested Citation

  • Thomas Schwartz, 2001. "From Arrow to cycles, instability, and chaos by untying alternatives," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(1), pages 1-22.
  • Handle: RePEc:spr:sochwe:v:18:y:2001:i:1:p:1-22
    Note: Received: 31 July 1999/Accepted: 15 October 1999
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    Cited by:

    1. Josep Freixas & Sascha Kurz, 2019. "Bounds for the Nakamura number," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 52(4), pages 607-634, April.
    2. Thomas Schwartz, 2011. "Social choice and individual values in the electronic republic," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 37(4), pages 621-632, October.
    3. John Duggan, 2007. "A systematic approach to the construction of non-empty choice sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(3), pages 491-506, April.

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