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Impossibility results for infinite-electorate abstract aggregation rules

Author

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  • Herzberg, Frederik

    (Center for Mathematical Economics, Bielefeld University)

  • Eckert, Daniel

    (Center for Mathematical Economics, Bielefeld 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.

Suggested Citation

  • Herzberg, Frederik & Eckert, Daniel, 2017. "Impossibility results for infinite-electorate abstract aggregation rules," Center for Mathematical Economics Working Papers 427, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:427
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    File URL: https://pub.uni-bielefeld.de/download/2909316/2909318
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    Citations

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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.
    3. Frederik S. Herzberg, 2013. "The (im)possibility of collective risk measurement: Arrovian aggregation of variational preferences," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(1), pages 69-92, May.
    4. Herzberg, Frederik, 2014. "Aggregation of Monotonic Bernoullian Archimedean preferences: Arrovian impossibility results," Center for Mathematical Economics Working Papers 488, Center for Mathematical Economics, Bielefeld University.
    5. Frederik Herzberg, 2015. "Aggregating infinitely many probability measures," Theory and Decision, Springer, vol. 78(2), pages 319-337, February.
    6. Herzberg, Frederik, 2013. "Arrovian aggregation of MBA preferences: An impossibility result," VfS Annual Conference 2013 (Duesseldorf): Competition Policy and Regulation in a Global Economic Order 79957, Verein für Socialpolitik / German Economic Association.

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