IDEAS home Printed from https://ideas.repec.org/a/spr/etbull/v1y2013i1d10.1007_s40505-013-0004-6.html
   My bibliography  Save this article

The (im)possibility of collective risk measurement: Arrovian aggregation of variational preferences

Author

Listed:
  • Frederik S. Herzberg

    (Institut für Mathematische Wirtschaftsforschung, Universität Bielefeld
    Munich Center for Mathematical Philosophy, Ludwig-Maximilians-Universität München)

Abstract

This paper studies collective decision making when individual preferences can be represented by convex risk measures. It addresses the question whether there exist non-dictatorial aggregation functions of convex risk measures satisfying the following Arrow-type rationality axioms: a weak form of universality, systematicity (a strong variant of independence), and the Pareto principle. Herein, convex risk measures are identified with variational preferences on account of the Maccheroni et al. (Econometrica 74(6):1447–1498, 2006) axiomatisation of variational preference relations and Föllmer’s and Schied’s (Finance Stoch 6(4):429–447, 2002; Stochastic Finance. An Introduction in Discrete Time. de Gruyter Studies in Mathematics, vol. 27, 2nd edn. de Gruyter, Berlin, 2004) representation theorem for concave monetary utility functionals. The cases of both finite and infinite electorates are considered. For finite electorates, we prove a variational analogue of Arrow’s impossibility theorem. For infinite electorates, the possibility of rational aggregation depends on an equicontinuity condition for the variational preference profiles; we prove a variational analogue of Fishburn’s possibility theorem and point to potential analogues of Campbell’s impossibility theorem. The proof methodology is based on a model-theoretic approach to aggregation theory inspired by Lauwers and Van Liedekerke (J Math Econ 24(3):217–237, 1995). Diverse applications of these results are conceivable, in particular (a) the constitutional design of panels of risk managers and (b) microfoundations for (macroeconomic) multiplier preferences.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:etbull:v:1:y:2013:i:1:d:10.1007_s40505-013-0004-6
    DOI: 10.1007/s40505-013-0004-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s40505-013-0004-6
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s40505-013-0004-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Michel Le Breton & John A. Weymark, 2002. "Social choice with analytic preferences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(3), pages 637-657.
    2. Chateauneuf, Alain & Eichberger, Jurgen & Grant, Simon, 2007. "Choice under uncertainty with the best and worst in mind: Neo-additive capacities," Journal of Economic Theory, Elsevier, vol. 137(1), pages 538-567, November.
    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. List, Christian & Pettit, Philip, 2002. "Aggregating Sets of Judgments: An Impossibility Result," Economics and Philosophy, Cambridge University Press, vol. 18(1), pages 89-110, April.
    5. Mongin, Philippe, 1998. "The paradox of the Bayesian experts and state-dependent utility theory," Journal of Mathematical Economics, Elsevier, vol. 29(3), pages 331-361, April.
    6. Gajdos, T. & Tallon, J.-M. & Vergnaud, J.-C., 2008. "Representation and aggregation of preferences under uncertainty," Journal of Economic Theory, Elsevier, vol. 141(1), pages 68-99, July.
    7. Alain Chateauneuf & Jürgen Eichberger & Simon Grant, 2003. "A simple axiomatization and constructive representation proof for choquet expected utility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 22(4), pages 907-915, November.
    8. Franz Dietrich & Christian List, 2007. "Arrow’s theorem in judgment aggregation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 29(1), pages 19-33, July.
    9. Neuefeind, Wilhelm & Trockel, Walter, 1995. "Continuous Linear Representability of Binary Relations," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 6(2), pages 351-356, July.
    10. 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.
    11. Luciano Castro & Alain Chateauneuf, 2011. "Ambiguity aversion and trade," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(2), pages 243-273, October.
    12. Campbell, Donald E., 1990. "Intergenerational social choice without the Pareto principle," Journal of Economic Theory, Elsevier, vol. 50(2), pages 414-423, April.
    13. Maccheroni, Fabio & Marinacci, Massimo & Rustichini, Aldo, 2006. "Dynamic variational preferences," Journal of Economic Theory, Elsevier, vol. 128(1), pages 4-44, May.
    14. Itzhak Gilboa & Dov Samet & David Schmeidler, 2004. "Utilitarian Aggregation of Beliefs and Tastes," Journal of Political Economy, University of Chicago Press, vol. 112(4), pages 932-938, August.
    15. 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.
    16. Alain Chateauneuf & Luciano De Castro, 2011. "Ambiguity Aversion and Absence of Trade," Discussion Papers 1535, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    17. Herzberg, Frederik & Lauwers, Luc & van Liedekerke, Luc & Fianu, Emmanuel Senyo, 2010. "Addendum to L. Lauwers and L. Van Liedekerke, "Ultraproducts and aggregation" [J. Math. Econ. 24 (3) (1995) 217-237]," Journal of Mathematical Economics, Elsevier, vol. 46(2), pages 277-278, March.
    18. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and Dynamic Convex Risk Measures," SFB 649 Discussion Papers SFB649DP2005-006, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    19. Dietrich, Franz & List, Christian, 2010. "Majority voting on restricted domains," Journal of Economic Theory, Elsevier, vol. 145(2), pages 512-543, March.
    20. Gilboa, Itzhak & Schmeidler, David, 1989. "Maxmin expected utility with non-unique prior," Journal of Mathematical Economics, Elsevier, vol. 18(2), pages 141-153, April.
    21. Dokow, Elad & Holzman, Ron, 2010. "Aggregation of binary evaluations with abstentions," Journal of Economic Theory, Elsevier, vol. 145(2), pages 544-561, March.
    22. Kirman, Alan P. & Sondermann, Dieter, 1972. "Arrow's theorem, many agents, and invisible dictators," Journal of Economic Theory, Elsevier, vol. 5(2), pages 267-277, October.
    23. John C. Harsanyi, 1955. "Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility," Journal of Political Economy, University of Chicago Press, vol. 63, pages 309-309.
    24. Riedel, Frank, 2004. "Dynamic coherent risk measures," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 185-200, August.
    25. Fabio Maccheroni & Massimo Marinacci & Aldo Rustichini, 2006. "Ambiguity Aversion, Robustness, and the Variational Representation of Preferences," Econometrica, Econometric Society, vol. 74(6), pages 1447-1498, November.
    26. Ning Yu, 2012. "A one-shot proof of Arrow’s impossibility theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 50(2), pages 523-525, June.
    27. Nehring, Klaus & Puppe, Clemens, 2007. "The structure of strategy-proof social choice -- Part I: General characterization and possibility results on median spaces," Journal of Economic Theory, Elsevier, vol. 135(1), pages 269-305, July.
    28. Simone Cerreia-Vioglio & Paolo Ghirardato & Fabio Maccheroni & Massimo Marinacci & Marciano Siniscalchi, 2011. "Rational preferences under ambiguity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(2), pages 341-375, October.
    29. H. Moulin, 1980. "On strategy-proofness and single peakedness," Public Choice, Springer, vol. 35(4), pages 437-455, January.
    30. 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.
    31. 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.
    32. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    33. Hylland, Aanund & Zeckhauser, Richard J, 1979. "The Impossibility of Bayesian Group Decision Making with Separate Aggregation of Beliefs and Values," Econometrica, Econometric Society, vol. 47(6), pages 1321-1336, November.
    34. Kai Detlefsen & Giacomo Scandolo, 2005. "Conditional and dynamic convex risk measures," Finance and Stochastics, Springer, vol. 9(4), pages 539-561, October.
    35. Robert Nau, 2011. "Risk, ambiguity, and state-preference theory," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 48(2), pages 437-467, October.
    36. Dokow, Elad & Holzman, Ron, 2010. "Aggregation of binary evaluations," Journal of Economic Theory, Elsevier, vol. 145(2), pages 495-511, March.
    37. Ehud Kalai & Eitan Muller & Mark Satterthwaite, 1979. "Social welfare functions when preferences are convex, strictly monotonic, and continuous," Public Choice, Springer, vol. 34(1), pages 87-97, March.
    38. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    Full references (including those not matched with items on IDEAS)

    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, 2014. "Aggregation of Monotonic Bernoullian Archimedean preferences: Arrovian impossibility results," Center for Mathematical Economics Working Papers 488, Center for Mathematical Economics, Bielefeld University.
    2. 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.
    3. Zuber, Stéphane, 2016. "Harsanyi’s theorem without the sure-thing principle: On the consistent aggregation of Monotonic Bernoullian and Archimedean preferences," Journal of Mathematical Economics, Elsevier, vol. 63(C), pages 78-83.
    4. Frederik Herzberg, 2015. "Aggregating infinitely many probability measures," Theory and Decision, Springer, vol. 78(2), pages 319-337, February.
    5. Philippe Mongin, 2012. "The doctrinal paradox, the discursive dilemma, and logical aggregation theory," Theory and Decision, Springer, vol. 73(3), pages 315-355, September.
    6. Nascimento, Leandro, 2012. "The ex-ante aggregation of opinions under uncertainty," Theoretical Economics, Econometric Society, vol. 7(3), September.
    7. List, Christian & Polak, Ben, 2010. "Introduction to judgment aggregation," Journal of Economic Theory, Elsevier, vol. 145(2), pages 441-466, March.
    8. Xiangyu Qu, 2017. "Separate aggregation of beliefs and values under ambiguity," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 63(2), pages 503-519, February.
    9. Acciaio, Beatrice & Föllmer, Hans & Penner, Irina, 2012. "Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles," LSE Research Online Documents on Economics 50118, London School of Economics and Political Science, LSE Library.
    10. Bellini, Fabio & Laeven, Roger J.A. & Rosazza Gianin, Emanuela, 2021. "Dynamic robust Orlicz premia and Haezendonck–Goovaerts risk measures," European Journal of Operational Research, Elsevier, vol. 291(2), pages 438-446.
    11. Crès, Hervé & Gilboa, Itzhak & Vieille, Nicolas, 2011. "Aggregation of multiple prior opinions," Journal of Economic Theory, Elsevier, vol. 146(6), pages 2563-2582.
    12. Bozbay, İrem & Dietrich, Franz & Peters, Hans, 2014. "Judgment aggregation in search for the truth," Games and Economic Behavior, Elsevier, vol. 87(C), pages 571-590.
    13. José Faro, 2013. "Cobb-Douglas preferences under uncertainty," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(2), pages 273-285, October.
    14. Yehuda Izhakian & Zur Izhakian, 2015. "Decision making in phantom spaces," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 58(1), pages 59-98, January.
    15. Beatrice Acciaio & Hans Föllmer & Irina Penner, 2012. "Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles," Finance and Stochastics, Springer, vol. 16(4), pages 669-709, October.
    16. Ji, Ronglin & Shi, Xuejun & Wang, Shijie & Zhou, Jinming, 2019. "Dynamic risk measures for processes via backward stochastic differential equations," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 43-50.
    17. 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.
    18. Fleurbaey, Marc & Zuber, Stéphane, 2017. "Fair management of social risk," Journal of Economic Theory, Elsevier, vol. 169(C), pages 666-706.
    19. Philippe Mongin & Marcus Pivato, 2020. "Social preference under twofold uncertainty," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 70(3), pages 633-663, October.
    20. Nehring, Klaus & Pivato, Marcus & Puppe, Clemens, 2014. "The Condorcet set: Majority voting over interconnected propositions," Journal of Economic Theory, Elsevier, vol. 151(C), pages 268-303.

    More about this item

    Keywords

    Arrow-type preference aggregation; Abstract aggregation theory; Variational preferences; Convex risk measures; Ultraproduct;
    All these keywords.

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

    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:spr:etbull:v:1:y:2013:i:1:d:10.1007_s40505-013-0004-6. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: http://www.springer.com .

    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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.