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Complete characterization of functions satisfying the conditions of Arrow’s theorem

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  • Elchanan Mossel
  • Omer Tamuz

Abstract

Arrow's theorem implies that a social choice function satisfying Transitivity, the Pareto Principle (Unanimity) and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are allowed, a dictatorial social choice function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow's theorem, in the case of non-strict preferences, does not provide a complete characterization of all social choice functions satisfying Transitivity, the Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow's theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow's and Wilson's result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Pareto Principle). Additionally, we derive formulas for the number of functions satisfying these conditions.
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  • Elchanan Mossel & Omer Tamuz, 2012. "Complete characterization of functions satisfying the conditions of Arrow’s theorem," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(1), pages 127-140, June.
    • Handle: RePEc:spr:sochwe:v:39:y:2012:i:1:p:127-140
    DOI: 10.1007/s00355-011-0547-0
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    1. repec:bla:econom:v:38:y:1971:i:152:p:413-20 is not listed on IDEAS
    2. James M. Buchanan, 1954. "Social Choice, Democracy, and Free Markets," Journal of Political Economy, University of Chicago Press, vol. 62(2), pages 114-114.
    3. Kenneth J. Arrow, 1950. "A Difficulty in the Concept of Social Welfare," Journal of Political Economy, University of Chicago Press, vol. 58(4), pages 328-328.
    4. James M. Buchanan, 1954. "Individual Choice in Voting and the Market," Journal of Political Economy, University of Chicago Press, vol. 62(4), pages 334-334.
    5. Wilson, Robert, 1972. "Social choice theory without the Pareto Principle," Journal of Economic Theory, Elsevier, vol. 5(3), pages 478-486, December.
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    Cited by:

    1. Juan Candeal, 2013. "Invariance axioms for preferences: applications to social choice theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(3), pages 453-471, September.
    2. Maurice Salles, 2014. "‘Social choice and welfare’ at 30: its role in the development of social choice theory and welfare economics," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(1), pages 1-16, January.
    3. Nicolas Gabriel Andjiga & Issofa Moyouwou & Monge Kleber Kamdem Ouambo, 2017. "Avoiding Majority Dissatisfaction on a Series of Majority Decisions," Group Decision and Negotiation, Springer, vol. 26(3), pages 453-471, May.
    4. Salvador Barberà & Dolors Berga & Bernardo Moreno & Antonio Nicolò, 2021. "Pairwise Justifiable Changes in Collective Choices," Working Papers 1256, Barcelona School of Economics.

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