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Judgment aggregators and Boolean algebra homomorphisms

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  • Herzberg, Frederik

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. From the (ultra)filter property of the set of winning coalitions, one obtains two general impossibility theorems for judgment aggregation on finite electorates, even without assuming the Pareto principle.

Suggested Citation

  • Herzberg, Frederik, 2010. "Judgment aggregators and Boolean algebra homomorphisms," Journal of Mathematical Economics, Elsevier, vol. 46(1), pages 132-140, January.
  • Handle: RePEc:eee:mateco:v:46:y:2010:i:1:p:132-140
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    References listed on IDEAS

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    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.
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    Cited by:

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

    More about this item

    Keywords

    Judgment aggregation Systematicity Impossibility theorems Filter Ultrafilter Boolean algebra homomorphism;

    JEL classification:

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

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