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Judgment aggregators and Boolean algebra homomorphisms

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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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    References listed on IDEAS

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    1. Anand, Paul & Pattanaik, Prasanta & Puppe, Clemens (ed.), 2009. "The Handbook of Rational and Social Choice," OUP Catalogue, Oxford University Press, number 9780199290420, Decembrie.
    2. Mongin, Philippe, 2011. "Judgment aggregation," HEC Research Papers Series 942, HEC Paris.
    3. 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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    6. Donald J. Brown, 1975. "Aggregation of Preferences," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 89(3), pages 456-469.
    7. 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.
    8. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    9. Bengt Hansson, 1976. "The existence of group preference functions," Public Choice, Springer, vol. 28(1), pages 89-98, December.
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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. Mongin, Philippe & Dietrich, Franz, 2011. "An interpretive account of logical aggregation theory," HEC Research Papers Series 941, HEC Paris.

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    Keywords

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