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Dual theory of choice under multivariate risks

Author

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  • Alfred Galichon

    (ECON - Département d'économie (Sciences Po) - Sciences Po - Sciences Po - CNRS - Centre National de la Recherche Scientifique)

  • Marc Henry

    (Départment de sciences économiques - UdeM - Université de Montréal)

Abstract

We propose a multivariate extension of Yaari's dual theory of choice under risk. We show that a decision maker with a preference relation on multidimensional prospects that preserves ¯rst order stochastic dominance and satis¯es comonotonic in-dependence behaves as if evaluating prospects with a weighted sum of quantiles. Both the notions of quantiles and of comonotonicity are extended to the multivariate framework using optimal transportation maps. Finally risk averse decision makers are characterized, and we show how to efficiently compute the functionals they use to evaluate prospects.

Suggested Citation

  • Alfred Galichon & Marc Henry, 2012. "Dual theory of choice under multivariate risks," Post-Print hal-01024582, HAL.
    • Handle: RePEc:hal:journl:hal-01024582
    DOI: 10.1016/j.jet.2011.06.002
    Note: View the original document on HAL open archive server: https://sciencespo.hal.science/hal-01024582
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    References listed on IDEAS

    as
    1. Puccetti, Giovanni & Scarsini, Marco, 2010. "Multivariate comonotonicity," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 291-304, January.
    2. Carlier, G. & Dana, R.-A. & Galichon, A., 2012. "Pareto efficiency for the concave order and multivariate comonotonicity," Journal of Economic Theory, Elsevier, vol. 147(1), pages 207-229.
    3. Yaari, Menahem E, 1987. "The Dual Theory of Choice under Risk," Econometrica, Econometric Society, vol. 55(1), pages 95-115, January.
    4. Thibault Gajdos & John Weymark, 2005. "Multidimensional generalized Gini indices," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(3), pages 471-496, October.
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    7. Hong, Chew Soo & Karni, Edi & Safra, Zvi, 1987. "Risk aversion in the theory of expected utility with rank dependent probabilities," Journal of Economic Theory, Elsevier, vol. 42(2), pages 370-381, August.
    8. repec:hal:spmain:info:hdl:2441/5rkqqmvrn4tl22s9mc0p00hch is not listed on IDEAS
    9. François Bourguignon & Satya R. Chakravarty, 2019. "The Measurement of Multidimensional Poverty," Themes in Economics, in: Satya R. Chakravarty (ed.), Poverty, Social Exclusion and Stochastic Dominance, pages 83-107, Springer.
    10. Weymark, John A., 1981. "Generalized gini inequality indices," Mathematical Social Sciences, Elsevier, vol. 1(4), pages 409-430, August.
    11. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    12. Rothschild, Michael & Stiglitz, Joseph E., 1970. "Increasing risk: I. A definition," Journal of Economic Theory, Elsevier, vol. 2(3), pages 225-243, September.
    13. Atkinson, Anthony B., 1970. "On the measurement of inequality," Journal of Economic Theory, Elsevier, vol. 2(3), pages 244-263, September.
    14. Serge-Christophe Kolm, 1977. "Multidimensional Egalitarianisms," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 91(1), pages 1-13.
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    Cited by:

    1. repec:hal:spmain:info:hdl:2441/7o52iohb7k6srk09mj4in40o4 is not listed on IDEAS
    2. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.
    3. repec:spo:wpmain:info:hdl:2441/7o52iohb7k6srk09mj4in40o4 is not listed on IDEAS
    4. Fan, Yanqin & Henry, Marc, 2023. "Vector copulas," Journal of Econometrics, Elsevier, vol. 234(1), pages 128-150.
    5. Alfred Galichon & Damien Bosc, 2010. "Extreme dependence for multivariate data," Working Papers hal-03588294, HAL.
    6. Alfred Galichon & Damien Bosc, 2010. "Extreme dependence for multivariate data," SciencePo Working papers hal-03588294, HAL.
    7. Damien Bosc & Alfred Galichon, 2014. "Extreme dependence for multivariate data," SciencePo Working papers hal-03470461, HAL.
    8. Alfred Galichon & Damien Bosc, 2010. "Extreme dependence for multivariate data," SciencePo Working papers Main hal-03588294, HAL.
    9. Damien Bosc & Alfred Galichon, 2014. "Extreme dependence for multivariate data," SciencePo Working papers Main hal-03470461, HAL.
    10. Yanqin Fan & Marc Henry & Brendan Pass & Jorge A. Rivero, 2022. "Lorenz map, inequality ordering and curves based on multidimensional rearrangements," Papers 2203.09000, arXiv.org, revised Apr 2024.
    11. repec:hal:spmain:info:hdl:2441/8pttci1na9qmqnud8j8lvbamu is not listed on IDEAS
    12. Yanqin Fan & Marc Henry, 2020. "Vector copulas," Papers 2009.06558, arXiv.org, revised Apr 2021.
    13. repec:spo:wpmain:info:hdl:2441/8pttci1na9qmqnud8j8lvbamu is not listed on IDEAS
    14. Damien Bosc & Alfred Galichon, 2014. "Extreme dependence for multivariate data," Post-Print hal-03470461, HAL.
    15. Dilip B. Madan, 2016. "Conic Portfolio Theory," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-42, May.
    16. Victor Chernozhukov & Alfred Galichon & Marc Hallin & Marc Henry, 2014. "Monge-Kantorovich Depth, Quantiles, Ranks, and Signs," Papers 1412.8434, arXiv.org, revised Sep 2015.

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    More about this item

    Keywords

    Risk; Rank dependent utility theory; Multivariate comonotonicity; Optimal transportation; Multi-attribute inequality; Gini evaluation functions;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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