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Judgment aggregators and Boolean algebra homomorphisms

Author

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  • Herzberg, Frederik

    (Center for Mathematical Economics, Bielefeld University)

Abstract

The theory of Boolean algebras can be fruitfully applied to judgment aggregation: Assuming universality, systematicity and a sufficiently rich agenda, there is a correspondence between (i) non-trivial deductively closed judgment aggregators and (ii) Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Furthermore, there is a correspondence between (i) consistent complete judgment aggregators and (ii) 2-valued Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Since the shell of such a homomorphism equals the set of winning coalitions and since (ultra)filters are shells of (2-valued) Boolean algebra homomorphisms, we suggest an explanation for the effectiveness of the (ultra)filter method in social choice theory.

Suggested Citation

  • Herzberg, Frederik, 2010. "Judgment aggregators and Boolean algebra homomorphisms," Center for Mathematical Economics Working Papers 414, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:414
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    File URL: https://pub.uni-bielefeld.de/download/1900280/2319752
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    References listed on IDEAS

    as
    1. Philippe Mongin, 2011. "Judgment aggregation," Working Papers hal-00579346, HAL.
    2. Dietrich, Franz & Mongin, Philippe, 2010. "The premiss-based approach to judgment aggregation," Journal of Economic Theory, Elsevier, vol. 145(2), pages 562-582, March.
    3. Donald J. Brown, 1975. "Aggregation of Preferences," The Quarterly Journal of Economics, Oxford University Press, vol. 89(3), pages 456-469.
    4. 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.
    5. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    6. Franz Dietrich & Christian List, 2008. "Judgment aggregation without full rationality," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 31(1), pages 15-39, June.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Philippe Mongin, 2012. "The doctrinal paradox, the discursive dilemma, and logical aggregation theory," Theory and Decision, Springer, vol. 73(3), pages 315-355, September.
    2. Philippe Mongin & Franz Dietrich, 2011. "An Interpretive Account of Logical Aggregation Theory," Working Papers hal-00625427, HAL.

    More about this item

    Keywords

    Systematicity; Judgment aggregation; Impossibility theorems; Filter; Boolean algebra homomorphism; Ultrafilter;

    JEL classification:

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

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