IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2309.05816.html
   My bibliography  Save this paper

A duality between utility transforms and probability distortions

Author

Listed:
  • Christopher P. Chambers
  • Peng Liu
  • Ruodu Wang

Abstract

In this paper, we establish a mathematical duality between utility transforms and probability distortions. These transforms play a central role in decision under risk by forming the foundation for the classic theories of expected utility, dual utility, and rank-dependent utility. Our main results establish that probability distortions are characterized by commutation with utility transforms, and utility transforms are characterized by commutation with probability distortions. These results require no additional conditions, and hence each class can be axiomatized with only one property. Moreover, under monotonicity, rank-dependent utility transforms can be characterized by set commutation with either utility transforms or probability distortions.

Suggested Citation

  • Christopher P. Chambers & Peng Liu & Ruodu Wang, 2023. "A duality between utility transforms and probability distortions," Papers 2309.05816, arXiv.org, revised Mar 2024.
    • Handle: RePEc:arx:papers:2309.05816
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2309.05816
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    2. Quiggin, John, 1982. "A theory of anticipated utility," Journal of Economic Behavior & Organization, Elsevier, vol. 3(4), pages 323-343, December.
    3. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
    4. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    5. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    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. Muqiao Huang & Ruodu Wang, 2024. "Coherent risk measures and uniform integrability," Papers 2404.03783, arXiv.org.

    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. Ruodu Wang & Zuo Quan Xu & Xun Yu Zhou, 2019. "Dual utilities on risk aggregation under dependence uncertainty," Finance and Stochastics, Springer, vol. 23(4), pages 1025-1048, October.
    2. Haiyan Liu & Bin Wang & Ruodu Wang & Sheng Chao Zhuang, 2023. "Distorted optimal transport," Papers 2308.11238, arXiv.org.
    3. Shi, Yun & Cui, Xiangyu & Zhou, Xunyu, 2020. "Beta and Coskewness Pricing: Perspective from Probability Weighting," SocArXiv 5rqhv, Center for Open Science.
    4. Itzhak Gilboa & Andrew Postlewaite & Larry Samuelson & David Schmeidler, 2019. "What are axiomatizations good for?," Theory and Decision, Springer, vol. 86(3), pages 339-359, May.
    5. Lovric, M. & Kaymak, U. & Spronk, J., 2008. "A Conceptual Model of Investor Behavior," ERIM Report Series Research in Management ERS-2008-030-F&A, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    6. Simone Cerreia‐Vioglio & David Dillenberger & Pietro Ortoleva, 2015. "Cautious Expected Utility and the Certainty Effect," Econometrica, Econometric Society, vol. 83, pages 693-728, March.
    7. Massimiliano Amarante & Mario Ghossoub & Edmund Phelps, 2012. "Contracting for Innovation under Knightian Uncertainty," Cahiers de recherche 18-2012, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    8. Mohammed Abdellaoui & Olivier L’Haridon & Horst Zank, 2010. "Separating curvature and elevation: A parametric probability weighting function," Journal of Risk and Uncertainty, Springer, vol. 41(1), pages 39-65, August.
    9. Groneck, Max & Ludwig, Alexander & Zimper, Alexander, 2016. "A life-cycle model with ambiguous survival beliefs," Journal of Economic Theory, Elsevier, vol. 162(C), pages 137-180.
    10. Mohammed Abdellaoui & Horst Zank, 2023. "Source and rank-dependent utility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 75(4), pages 949-981, May.
    11. Ulrich Schmidt & Horst Zank, 2008. "Risk Aversion in Cumulative Prospect Theory," Management Science, INFORMS, vol. 54(1), pages 208-216, January.
    12. Tsang, Ming, 2020. "Estimating uncertainty aversion using the source method in stylized tasks with varying degrees of uncertainty," Journal of Behavioral and Experimental Economics (formerly The Journal of Socio-Economics), Elsevier, vol. 84(C).
    13. Yehuda Izhakian, 2012. "Ambiguity Measurement," Working Papers 12-01, New York University, Leonard N. Stern School of Business, Department of Economics.
    14. R. Luce & A. Marley, 2005. "Ranked Additive Utility Representations of Gambles: Old and New Axiomatizations," Journal of Risk and Uncertainty, Springer, vol. 30(1), pages 21-62, January.
    15. Ivan Moscati, 2022. "Behavioral and heuristic models are as-if models too — and that’s ok," BAFFI CAREFIN Working Papers 22177, BAFFI CAREFIN, Centre for Applied Research on International Markets Banking Finance and Regulation, Universita' Bocconi, Milano, Italy.
    16. Epper, Thomas & Fehr-Duda, Helga, 2017. "A Tale of Two Tails: On the Coexistence of Overweighting and Underweighting of Rare Extreme Events," Economics Working Paper Series 1705, University of St. Gallen, School of Economics and Political Science.
    17. Anna Conte & John D. Hey, 2018. "Assessing multiple prior models of behaviour under ambiguity," World Scientific Book Chapters, in: Experiments in Economics Decision Making and Markets, chapter 7, pages 169-188, World Scientific Publishing Co. Pte. Ltd..
    18. Gul, Faruk & Pesendorfer, Wolfgang, 2015. "Hurwicz expected utility and subjective sources," Journal of Economic Theory, Elsevier, vol. 159(PA), pages 465-488.
    19. Fox, Craig R. & Weber, Martin, 2002. "Ambiguity Aversion, Comparative Ignorance, and Decision Context," Organizational Behavior and Human Decision Processes, Elsevier, vol. 88(1), pages 476-498, May.
    20. Robin Cubitt & Gijs van de Kuilen & Sujoy Mukerji, 2020. "Discriminating Between Models of Ambiguity Attitude: a Qualitative Test," Journal of the European Economic Association, European Economic Association, vol. 18(2), pages 708-749.

    More about this item

    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:arx:papers:2309.05816. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.