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The model-theoretic approach to aggregation: Impossibility results for finite and infinite electorates

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  • Herzberg, Frederik
  • Eckert, Daniel

Abstract

It is well known that the literature on judgement aggregation inherits the impossibility results from the aggregation of preferences that it generalises. This is due to the fact that the typical judgement aggregation problem induces an ultrafilter on the set of individuals. We propose a model-theoretic framework for the analysis of judgement aggregation and show that the conditions typically imposed on aggregators induce an ultrafilter on the set of individuals, thus establishing a generalised version of the Kirman–Sondermann correspondence. 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. Following Lauwers and Van Liedekerke’s (1995) seminal paper, we investigate 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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  • Herzberg, Frederik & Eckert, Daniel, 2012. "The model-theoretic approach to aggregation: Impossibility results for finite and infinite electorates," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 41-47.
  • Handle: RePEc:eee:matsoc:v:64:y:2012:i:1:p:41-47
    DOI: 10.1016/j.mathsocsci.2011.08.004
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    References listed on IDEAS

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    1. Fishburn, Peter C., 1970. "Arrow's impossibility theorem: Concise proof and infinite voters," Journal of Economic Theory, Elsevier, vol. 2(1), pages 103-106, March.
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    5. List, Christian & Polak, Ben, 2010. "Introduction to judgment aggregation," Journal of Economic Theory, Elsevier, vol. 145(2), pages 441-466, March.
    6. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
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    8. Rubinstein, Ariel, 1984. "The Single Profile Analogues to Multi Profile Theorems: Mathematical Logic's Approach," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(3), pages 719-730, October.
    9. Dietrich, F.K. & List, C., 2009. "Propositionwise judgment aggregation," Research Memorandum 020, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    10. Rubinstein, Ariel & Fishburn, Peter C., 1986. "Algebraic aggregation theory," Journal of Economic Theory, Elsevier, vol. 38(1), pages 63-77, February.
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    Cited by:

    1. Wesley H. Holliday & Eric Pacuit, 2020. "Arrow’s decisive coalitions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 54(2), pages 463-505, March.
    2. 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.
    3. 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.
    4. 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.
    5. Schoch, Daniel, 2015. "Game Form Representation for Judgement and Arrovian Aggregation," MPRA Paper 64311, University Library of Munich, Germany.
    6. Frederik Herzberg, 2015. "Aggregating infinitely many probability measures," Theory and Decision, Springer, vol. 78(2), pages 319-337, February.

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