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Impossibility results for infinite-electorate abstract aggregation rules

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Author Info

  • Frederik Herzberg

    (Institute of Mathematical Economics, Bielefeld University)

  • Daniel Eckert

    (Institute of Public Economics, Graz University)

Abstract

It is well known that the literature on judgment aggregation inherits the impossibility results from the aggregation of preferences that it generalises. This is due to the fact that the typical judgment aggregation problem induces an ultrafilter on the the set of individuals, as was shown in a model theoretic framework by Herzberg and Eckert (2009), generalising the Kirman-Sondermann correspondence and extending the methodology of Lauwers and Van Liedekerke (1995). In the finite case, dictatorship then immediately follows from the principality of an ultrafilter on a finite set. This is not the case for an infinite set of individuals, where there exist free ultrafilters, as Fishburn already stressed in 1970. The main problem associated with free ultrafilters in the literature on aggregation problems is however, the arbitrariness of their selection combined with the limited anonymity they guarantee (which already led Kirman and Sondermann (1972) to speak about invisible dictators). Following another line of Lauwers and Van Liedekerke's (1995) seminal paper, this note explores another source of impossibility results for free ultrafilters: The domain of an ultraproduct over a free ultrafilter extends the individual factor domains, such that the preservation of the truth value of some sentences by the aggregate model --- if this is as usual to be restricted to the original domain --- may again require the exclusion of free ultrafilters, leading to dictatorship once again.

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File URL: http://www.imw.uni-bielefeld.de/papers/files/imw-wp-427.pdf
File Function: First version, 2010
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Bibliographic Info

Paper provided by Bielefeld University, Center for Mathematical Economics in its series Working Papers with number 427.

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Length: 10 pages
Date of creation: Feb 2010
Date of revision:
Handle: RePEc:bie:wpaper:427

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Related research

Keywords: Arrow-type preference aggregation; judgment aggregation; model theory; first-order predicate logic; filter; ultrafilter; reduced product; ultraproduct; existential quantifier;

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Citations

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Cited by:
  1. Mongin, Philippe & Dietrich, Franz, 2011. "An interpretive account of logical aggregation theory," Les Cahiers de Recherche 941, HEC Paris.
  2. Frederik Herzberg, 2013. "Aggregation of Monotonic Bernoullian Archimedean preferences: Arrovian impossibility results," Working Papers 488, Bielefeld University, Center for Mathematical Economics.
  3. Herzberg, Frederik, 2013. "Arrovian aggregation of MBA preferences: An impossibility result," Annual Conference 2013 (Duesseldorf): Competition Policy and Regulation in a Global Economic Order 79957, Verein für Socialpolitik / German Economic Association.
  4. Mongin, Philippe, 2012. "The doctrinal paradox, the discursive dilemma, and logical aggregation theory," MPRA Paper 37752, University Library of Munich, Germany.

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