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Arrow's Theorem, countably many agents, and more visible invisible dictators

Author

Listed:
  • H. Reiju Mihara

    (Kagawa University)

Abstract

For infinite societies, Fishburn (1970), Kirman and Sondermann (1972), and Armstrong (1980) gave a nonconstructive proof of the existence of a social welfare function satisfying Arrow's conditions (Unanimity, Independence, and Nondictatorship). This paper improves on their results by (i) giving a concrete example of such a function, and (ii) showing how to compute, from a description of a profile on a pair of alternatives, which alternative is socially preferred under the function. The introduction of a certain "oracle" resolves Mihara's impossibility result (1997) about computability of social welfare functions.

Suggested Citation

  • H. Reiju Mihara, 1997. "Arrow's Theorem, countably many agents, and more visible invisible dictators," Public Economics 9705001, University Library of Munich, Germany, revised 01 Jun 2004.
    • Handle: RePEc:wpa:wuwppe:9705001
    Note: Journal of Mathematical Economics (1999) 32: 267-287
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    File URL: https://econwpa.ub.uni-muenchen.de/econ-wp/pe/papers/9705/9705001.pdf
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    References listed on IDEAS

    as
    1. Spear, Stephen E, 1989. "Learning Rational Expectations under Computability Constraints," Econometrica, Econometric Society, vol. 57(4), pages 889-910, July.
    2. 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.
    3. H. Reiju Mihara, 2001. "Existence of a coalitionally strategyproof social choice function: A constructive proof," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(3), pages 543-553.
    4. Luca Anderlini & Leonardo Felli, 1994. "Incomplete Written Contracts: Undescribable States of Nature," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 109(4), pages 1085-1124.
    5. Kelly, Jerry S., 1988. "Social choice and computational complexity," Journal of Mathematical Economics, Elsevier, vol. 17(1), pages 1-8, February.
    6. H. Reiju Mihara, 1997. "Arrow's Theorem and Turing computability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(2), pages 257-276.
    7. Kirman, Alan P. & Sondermann, Dieter, 1972. "Arrow's theorem, many agents, and invisible dictators," Journal of Economic Theory, Elsevier, vol. 5(2), pages 267-277, October.
    8. Lewis, Alain A., 1988. "An infinite version of arrow's theorem in the effective setting," Mathematical Social Sciences, Elsevier, vol. 16(1), pages 41-48, August.
    9. Barton L. Lipman, 1995. "Information Processing and Bounded Rationality: A Survey," Canadian Journal of Economics, Canadian Economics Association, vol. 28(1), pages 42-67, February.
    10. Elisha A. Pazner & Eugene Wesley, 1978. "Cheatproofness Properties of the Plurality Rule in Large Societies," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 45(1), pages 85-91.
    11. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    12. Ehud Kalai, 1987. "Bounded Rationality and Strategic Complexity in Repeated Games," Discussion Papers 783, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    13. Armstrong, Thomas E., 1980. "Arrow's theorem with restricted coalition algebras," Journal of Mathematical Economics, Elsevier, vol. 7(1), pages 55-75, March.
    14. Armstrong, Thomas E., 1985. "Precisely dictatorial social welfare functions : Erratum and Addendum to `arrows theorem with restricted coalition algebras'," Journal of Mathematical Economics, Elsevier, vol. 14(1), pages 57-59, February.
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    Citations

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    Cited by:

    1. Kumabe, Masahiro & Mihara, H. Reiju, 2011. "Preference aggregation theory without acyclicity: The core without majority dissatisfaction," Games and Economic Behavior, Elsevier, vol. 72(1), pages 187-201, May.
    2. Kumabe, Masahiro & Mihara, H. Reiju, 2008. "Computability of simple games: A characterization and application to the core," Journal of Mathematical Economics, Elsevier, vol. 44(3-4), pages 348-366, February.
    3. Mihara, H. Reiju, 2004. "Nonanonymity and sensitivity of computable simple games," Mathematical Social Sciences, Elsevier, vol. 48(3), pages 329-341, November.
    4. Kumabe, Masahiro & Mihara, H. Reiju, 2011. "Computability of simple games: A complete investigation of the sixty-four possibilities," Journal of Mathematical Economics, Elsevier, vol. 47(2), pages 150-158, March.
    5. Andrei Gomberg & César Martinelli & Ricard Torres, 2005. "Anonymity in large societies," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 25(1), pages 187-205, October.
    6. Norbert Brunner & H. Reiju Mihara, 1999. "Arrow's theorem, Weglorz' models and the axiom of choice," Public Economics 9902001, University Library of Munich, Germany, revised 01 Jun 2004.
    7. Potgieter, Petrus H. & Rosinger, Elemér E., 2007. "Is Economics Entering its Post-Witchcraft Era?," MPRA Paper 3340, University Library of Munich, Germany.
    8. Mabrouk, Mohamed, 2006. "Allais-anonymity as an alternative to the discounted-sum criterion in the calculus of optimal growth I: Consensual optimality," MPRA Paper 10512, University Library of Munich, Germany.
    9. Ricard Torres, 2002. "Smallness of Invisible Dictators," Working Papers 0213, Centro de Investigacion Economica, ITAM, revised Sep 2003.

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    More about this item

    Keywords

    Arrow impossibility theorem; Turing computability; recursion theory; oracle algorithms; free ultrafilters;
    All these keywords.

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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    1. Ultrafilter in Wikipedia English
    2. Arrow's impossibility theorem in Wikipedia English

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