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Judgment aggregation functions and ultraproducts

Author

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  • Herzberg, Frederik

    (Center for Mathematical Economics, Bielefeld University)

Abstract

The relationship between propositional model theory and social decision making via premise-based procedures is explored. A one-to-one correspondence between ultrafilters on the population set and weakly universal, unanimity-respecting, systematic judgment aggregation functions is established. The proof constructs an ultraproduct of profiles, viewed as propositional structures, with respect to the ultrafilter of decisive coalitions. This representation theorem can be used to prove other properties of such judgment aggregation functions, in particular sovereignty and monotonicity, as well as an impossibility theorem for judgment aggregation in finite populations. As a corollary, Lauwers and Van Liedekerke's (1995) representation theorem for preference aggregation functions is derived.

Suggested Citation

  • Herzberg, Frederik, 2011. "Judgment aggregation functions and ultraproducts," Center for Mathematical Economics Working Papers 405, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:405
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    References listed on IDEAS

    as
    1. Daniel Eckert & Bernard Monjardet, 2009. "Guilbaud's Theorem : An early contribution to judgment aggregation," Post-Print halshs-00404185, HAL.
    2. 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.
    3. Gã„Rdenfors, Peter, 2006. "A Representation Theorem For Voting With Logical Consequences," Economics and Philosophy, Cambridge University Press, vol. 22(2), pages 181-190, July.
    4. Dietrich, Franz & List, Christian, 2010. "Majority voting on restricted domains," Journal of Economic Theory, Elsevier, vol. 145(2), pages 512-543, March.
    5. Mongin, Philippe, 2011. "Judgment aggregation," HEC Research Papers Series 942, HEC Paris.
    6. Armstrong, Thomas E., 1980. "Arrow's theorem with restricted coalition algebras," Journal of Mathematical Economics, Elsevier, vol. 7(1), pages 55-75, March.
    7. 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.
    8. Christian Klamler & Daniel Eckert, 2009. "A simple ultrafilter proof for an impossibility theorem in judgment aggregation," Economics Bulletin, AccessEcon, vol. 29(1), pages 319-327.
    9. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    10. Bengt Hansson, 1976. "The existence of group preference functions," Public Choice, Springer, vol. 28(1), pages 89-98, December.
    11. 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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    Cited by:

    1. Daniel Eckert & Bernard Monjardet, 2010. "Guilbaud's 1952 theorem on the logical problem of aggregation," Post-Print halshs-00504959, HAL.
    2. Christian Klamler & Daniel Eckert, 2009. "A simple ultrafilter proof for an impossibility theorem in judgment aggregation," Economics Bulletin, AccessEcon, vol. 29(1), pages 319-327.

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    More about this item

    Keywords

    Judgment aggregation function; Ultrafilter; Ultraproduct;
    All these keywords.

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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