IDEAS home Printed from https://ideas.repec.org/p/bie/wpaper/414.html
   My bibliography  Save this paper

Judgment aggregators and Boolean algebra homomorphisms

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://pub.uni-bielefeld.de/download/1900280/2319752
    File Function: First Version, 2009
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Philippe Mongin, 2011. "Judgment aggregation," Working Papers hal-00579346, HAL.
    2. 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.
    3. 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.
    4. Bengt Hansson, 1976. "The existence of group preference functions," Public Choice, Springer, vol. 28(1), pages 89-98, December.
    5. Donald J. Brown, 1975. "Aggregation of Preferences," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 89(3), pages 456-469.
    6. 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.
    7. Anand, Paul & Pattanaik, Prasanta & Puppe, Clemens (ed.), 2009. "The Handbook of Rational and Social Choice," OUP Catalogue, Oxford University Press, number 9780199290420, Decembrie.
    8. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. Philippe Mongin & Franz Dietrich, 2011. "An Interpretive Account of Logical Aggregation Theory," Working Papers hal-00579343, HAL.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Herzberg, Frederik S., 2008. "Judgement aggregation functions and ultraproducts," MPRA Paper 10546, University Library of Munich, Germany, revised 10 Sep 2008.
    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. Bossert, Walter & Cato, Susumu, 2020. "Acyclicity, anonymity, and prefilters," Journal of Mathematical Economics, Elsevier, vol. 87(C), pages 134-141.
    4. Mongin, Philippe & Dietrich, Franz, 2011. "An interpretive account of logical aggregation theory," HEC Research Papers Series 941, HEC Paris.
    5. Bossert, Walter & Cato, Susumu, 2021. "Superset-robust collective choice rules," Mathematical Social Sciences, Elsevier, vol. 109(C), pages 126-136.
    6. Philippe Mongin, 2012. "The doctrinal paradox, the discursive dilemma, and logical aggregation theory," Theory and Decision, Springer, vol. 73(3), pages 315-355, September.
    7. Pierre Batteau, 1978. "Stability of Aggregation Procedures, Ultrafilters and Simple Games," Discussion Papers 318, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    8. Susumu Cato, 2013. "Quasi-decisiveness, quasi-ultrafilter, and social quasi-orderings," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(1), pages 169-202, June.
    9. Cato, Susumu, 2017. "Unanimity, anonymity, and infinite population," Journal of Mathematical Economics, Elsevier, vol. 71(C), pages 28-35.
    10. 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.
    11. Susumu Cato, 2013. "Social choice, the strong Pareto principle, and conditional decisiveness," Theory and Decision, Springer, vol. 75(4), pages 563-579, October.
    12. Susumu Cato, 2022. "Stable preference aggregation with infinite population," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 59(2), pages 287-304, August.
    13. Susumu Cato, 2019. "The possibility of Paretian anonymous decision-making with an infinite population," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(4), pages 587-601, December.
    14. Susumu Cato, 2018. "Collective rationality and decisiveness coherence," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 50(2), pages 305-328, February.
    15. Herzberg, Frederik, 2010. "A representative individual from Arrovian aggregation of parametric individual utilities," Journal of Mathematical Economics, Elsevier, vol. 46(6), pages 1115-1124, November.
    16. Cato, Susumu, 2021. "Preference aggregation and atoms in measures," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    17. Susumu Cato, 2020. "Quasi-stationary social welfare functions," Theory and Decision, Springer, vol. 89(1), pages 85-106, July.
    18. Daniel Eckert & Bernard Monjardet, 2009. "Guilbaud's Theorem : An early contribution to judgment aggregation," Post-Print halshs-00404185, HAL.
    19. Wesley H. Holliday & Mikayla Kelley, 2020. "A note on Murakami’s theorems and incomplete social choice without the Pareto principle," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 55(2), pages 243-253, August.
    20. Zoi Terzopoulou & Ulle Endriss, 2019. "Strategyproof judgment aggregation under partial information," Post-Print hal-04809526, HAL.

    More about this item

    Keywords

    Systematicity; Judgment aggregation; Impossibility theorems; Filter; Boolean algebra homomorphism; Ultrafilter;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bie:wpaper:414. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Bettina Weingarten (email available below). General contact details of provider: https://edirc.repec.org/data/imbiede.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.