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Judgment aggregators and Boolean algebra homomorphisms

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  • Frederik Herzberg

    ()
    (Institute of 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. 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 the Pareto principle.

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File URL: http://www.imw.uni-bielefeld.de/papers/files/imw-wp-414.pdf
File Function: First version, 2009
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Bibliographic Info

Paper provided by Bielefeld University, Center for Mathematical Economics in its series Working Papers with number 414.

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Length: 12 pages
Date of creation: Feb 2009
Date of revision:
Handle: RePEc:bie:wpaper:414

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Keywords: judgment aggregation; systematicity; impossibility theorems; filter; ultrafilter; Boolean algebra; homomorphism;

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  1. Franz Dietrich & Christian List, 2008. "Judgment aggregation without full rationality," Social Choice and Welfare, Springer, vol. 31(1), pages 15-39, June.
  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.
  3. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
  4. Brown, Donald J, 1975. "Aggregation of Preferences," The Quarterly Journal of Economics, MIT Press, vol. 89(3), pages 456-69, August.
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Cited by:
  1. Mongin, Philippe, 2012. "The doctrinal paradox, the discursive dilemma, and logical aggregation theory," MPRA Paper 37752, University Library of Munich, Germany.
  2. Mongin, Philippe & Dietrich, Franz, 2011. "An interpretive account of logical aggregation theory," Les Cahiers de Recherche 941, HEC Paris.

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