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

Thomas Schwartz
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Thomas Schwartz: Department of Political Science, UCLA, Los Angeles, CA 90095-1472, USA

Social Choice and Welfare, 2001, vol. 18, issue 1, 22 pages

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.

Date: 2001-01-10
Note: Received: 31 July 1999/Accepted: 15 October 1999
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