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A note on Murakami’s theorems and incomplete social choice without the Pareto principle

Author

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  • Wesley H. Holliday

    (University of California)

  • Mikayla Kelley

    (Stanford University)

Abstract

In Arrovian social choice theory assuming the independence of irrelevant alternatives, Murakami (Logic and social choice, Dover Publications, New York, 1968) proved two theorems about complete and transitive collective choice rules satisfying strict non-imposition (citizens’ sovereignty), one being a dichotomy theorem about Paretian or anti-Paretian rules and the other a dictator-or-inverse-dictator impossibility theorem without the Pareto principle. It has been claimed in the later literature that a theorem of Malawski and Zhou (Soc Choice Welf 11(2):103–107, 1994) is a generalization of Murakami’s dichotomy theorem and that Wilson’s impossibility theorem (J Econ Theory 5(3):478–486, 1972) is stronger than Murakami’s impossibility theorem, both by virtue of replacing Murakami’s assumption of strict non-imposition with the assumptions of non-imposition and non-nullness. In this note, we first point out that these claims are incorrect: non-imposition and non-nullness are together equivalent to strict non-imposition for all transitive collective choice rules. We then generalize Murakami’s dichotomy and impossibility theorems to the setting of incomplete social preference. We prove that if one drops completeness from Murakami’s assumptions, his remaining assumptions imply (i) that a collective choice rule is either Paretian, anti-Paretian, or dis-Paretian (unanimous individual preference implies noncomparability) and (ii) that adding proposed constraints on noncomparability, such as the regularity axiom of Eliaz and Ok (Games Econ Behav 56:61–86, 2006), restores Murakami’s dictator-or-inverse-dictator result.

Suggested Citation

  • Wesley H. Holliday & Mikayla Kelley, 2020. "A note on Murakami’s theorems and incomplete social choice without the Pareto principle," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 55(2), pages 243-253, August.
  • Handle: RePEc:spr:sochwe:v:55:y:2020:i:2:d:10.1007_s00355-020-01238-2
    DOI: 10.1007/s00355-020-01238-2
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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. Miller, Michael K., 2009. "Social choice theory without Pareto: The pivotal voter approach," Mathematical Social Sciences, Elsevier, vol. 58(2), pages 251-255, September.
    3. Andreu Mas-Colell & Hugo Sonnenschein, 1972. "General Possibility Theorems for Group Decisions," Review of Economic Studies, Oxford University Press, vol. 39(2), pages 185-192.
    4. Sen, Amartya, 1970. "Interpersonal Aggregation and Partial Comparability," Econometrica, Econometric Society, vol. 38(3), pages 393-409, May.
    5. Fishburn, Peter C., 1970. "Arrow's impossibility theorem: Concise proof and infinite voters," Journal of Economic Theory, Elsevier, vol. 2(1), pages 103-106, March.
    6. Campbell, Donald E., 1990. "Intergenerational social choice without the Pareto principle," Journal of Economic Theory, Elsevier, vol. 50(2), pages 414-423, April.
    7. Border, Kim C., 1983. "Social welfare functions for economic environments with and without the pareto principle," Journal of Economic Theory, Elsevier, vol. 29(2), pages 205-216, April.
    8. David Kelsey, 1984. "Acyclic Choice without the Pareto Principle," Review of Economic Studies, Oxford University Press, vol. 51(4), pages 693-699.
    9. Susumu Cato, 2016. "Weak independence and the Pareto principle," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(2), pages 295-314, August.
    10. Susumu Cato, 2013. "Quasi-decisiveness, quasi-ultrafilter, and social quasi-orderings," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(1), pages 169-202, June.
    11. Amartya Sen, 1969. "Quasi-Transitivity, Rational Choice and Collective Decisions," Review of Economic Studies, Oxford University Press, vol. 36(3), pages 381-393.
    12. Ceyhun Coban & M. Sanver, 2014. "Social choice without the Pareto principle under weak independence," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 43(4), pages 953-961, December.
    13. Susumu Cato, 2012. "Social choice without the Pareto principle: a comprehensive analysis," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(4), pages 869-889, October.
    14. Cato, Susumu, 2018. "Incomplete decision-making and Arrow’s impossibility theorem," Mathematical Social Sciences, Elsevier, vol. 94(C), pages 58-64.
    15. Susumu Cato, 2020. "Incompleteness, regularity, and collective preference," Metroeconomica, Wiley Blackwell, vol. 71(2), pages 333-344, May.
    16. Fountain, John & Suzumura, Kotaro, 1982. "Collective Choice Rules without the Pareto Principle," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 23(2), pages 299-308, June.
    17. Wilson, Robert, 1972. "Social choice theory without the Pareto Principle," Journal of Economic Theory, Elsevier, vol. 5(3), pages 478-486, December.
    18. Bengt Hansson, 1976. "The existence of group preference functions," Public Choice, Springer, vol. 28(1), pages 89-98, December.
    19. Guha, Ashok, 1972. "Neutrality, Monotonicity, and the Right of Veto," Econometrica, Econometric Society, vol. 40(5), pages 821-826, September.
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    Cited by:

    1. Wesley H. Holliday & Mikayla Kelley, 2021. "Escaping Arrow's Theorem: The Advantage-Standard Model," Papers 2108.01134, arXiv.org.

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