IDEAS home Printed from https://ideas.repec.org/a/eee/mateco/v110y2024ics0304406823001295.html
   My bibliography  Save this article

On concave functions over lotteries

Author

Listed:
  • Corrao, Roberto
  • Fudenberg, Drew
  • Levine, David K.

Abstract

This note discusses functions over lotteries that are concave and continuous, but are not necessarily superdifferentiable. Earlier work claims that concave continuous utility for lotteries that satisfy best-outcome independence can be written as the minimum of affine functions. We give a counter-example that cannot be written as the minimum of affine functions, because there is no tangent hyperplane that dominates the functions at the boundary. We then review the fact that concavity and upper semi-continuity are equivalent to a representation as the infimum of affine functions, and show that these assumptions imply continuity for functions on finite-dimensional lotteries. Therefore, in finite-dimensional simplices, concavity and continuity are equivalent to the “infimum” representation. The “minimum” representation is equivalent to the existence of local utilities (supporting affine functions) at every lottery, a property that is equivalent to superdifferentiability.

Suggested Citation

  • Corrao, Roberto & Fudenberg, Drew & Levine, David K., 2024. "On concave functions over lotteries," Journal of Mathematical Economics, Elsevier, vol. 110(C).
    • Handle: RePEc:eee:mateco:v:110:y:2024:i:c:s0304406823001295
    DOI: 10.1016/j.jmateco.2023.102936
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304406823001295
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmateco.2023.102936?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 look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Machina, Mark J., 1984. "Temporal risk and the nature of induced preferences," Journal of Economic Theory, Elsevier, vol. 33(2), pages 199-231, August.
    2. Roberto Corrao & Drew Fudenberg & David K Levine, 2023. "Adversarial forecasters, surprises and randomization," Levine's Working Paper Archive 11694000000000130, David K. Levine.
    3. Chatterjee, Kalyan & Vijay Krishna, R., 2011. "A nonsmooth approach to nonexpected utility theory under risk," Mathematical Social Sciences, Elsevier, vol. 62(3), pages 166-175.
    4. Todd Sarver, 2018. "Dynamic Mixture‐Averse Preferences," Econometrica, Econometric Society, vol. 86(4), pages 1347-1382, July.
    5. Alexander Frankel & Emir Kamenica, 2019. "Quantifying Information and Uncertainty," American Economic Review, American Economic Association, vol. 109(10), pages 3650-3680, October.
    6. Evren, Özgür, 2014. "Scalarization methods and expected multi-utility representations," Journal of Economic Theory, Elsevier, vol. 151(C), pages 30-63.
    7. Shaowei Ke & Qi Zhang, 2020. "Randomization and Ambiguity Aversion," Econometrica, Econometric Society, vol. 88(3), pages 1159-1195, May.
    8. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, December.
    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. Sadowski, Philipp & Sarver, Todd, 2024. "Adaptive preferences: An evolutionary model of non-expected utility and ambiguity aversion," Journal of Economic Theory, Elsevier, vol. 218(C).
    2. Simone Cerreia‐Vioglio & David Dillenberger & Pietro Ortoleva, 2015. "Cautious Expected Utility and the Certainty Effect," Econometrica, Econometric Society, vol. 83, pages 693-728, March.
    3. McCarthy, David & Mikkola, Kalle & Thomas, Teruji, 2021. "Expected utility theory on mixture spaces without the completeness axiom," Journal of Mathematical Economics, Elsevier, vol. 97(C).
    4. Chambers, Christopher P. & Liu, Ce & Rehbeck, John, 2020. "Costly information acquisition," Journal of Economic Theory, Elsevier, vol. 186(C).
    5. Leandro Gorno, 2018. "The structure of incomplete preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(1), pages 159-185, July.
    6. Dillenberger, David & Raymond, Collin, 2019. "On the consensus effect," Journal of Economic Theory, Elsevier, vol. 183(C), pages 384-416.
    7. McCarthy, David & Mikkola, Kalle & Thomas, Teruji, 2017. "Representation of strongly independent preorders by sets of scalar-valued functions," MPRA Paper 79284, University Library of Munich, Germany.
    8. McCarthy, David & Mikkola, Kalle & Thomas, Teruji, 2020. "Utilitarianism with and without expected utility," Journal of Mathematical Economics, Elsevier, vol. 87(C), pages 77-113.
    9. Yosuke Hashidate, 2018. "Preferences for Randomization and Anticipated Utility," CIRJE F-Series CIRJE-F-1083, CIRJE, Faculty of Economics, University of Tokyo.
    10. Campi, Luciano & Zabaljauregui, Diego, 2020. "Optimal market making under partial information with general intensities," LSE Research Online Documents on Economics 104612, London School of Economics and Political Science, LSE Library.
    11. Kaido, Hiroaki, 2017. "Asymptotically Efficient Estimation Of Weighted Average Derivatives With An Interval Censored Variable," Econometric Theory, Cambridge University Press, vol. 33(5), pages 1218-1241, October.
    12. Andrea Attar & Thomas Mariotti & François Salanié, 2021. "Entry-Proofness and Discriminatory Pricing under Adverse Selection," American Economic Review, American Economic Association, vol. 111(8), pages 2623-2659, August.
    13. Nocetti, Diego C., 2013. "The LeChatelier principle for changes in risk," Journal of Mathematical Economics, Elsevier, vol. 49(6), pages 460-466.
    14. Askoura, Youcef & Billot, Antoine, 2021. "Social decision for a measure society," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    15. Xiaohong Chen & Andres Santos, 2018. "Overidentification in Regular Models," Econometrica, Econometric Society, vol. 86(5), pages 1771-1817, September.
    16. He, Wei & Sun, Yeneng, 2013. "Stationary Markov Perfect Equilibria in Discounted Stochastic Games," MPRA Paper 51274, University Library of Munich, Germany.
    17. Duggan, John, 2011. "General conditions for the existence of maximal elements via the uncovered set," Journal of Mathematical Economics, Elsevier, vol. 47(6), pages 755-759.
    18. Goeree, Jacob K. & Holt, Charles A. & Palfrey, Thomas R., 2002. "Quantal Response Equilibrium and Overbidding in Private-Value Auctions," Journal of Economic Theory, Elsevier, vol. 104(1), pages 247-272, May.
    19. Eduardo Perez & Delphine Prady, 2012. "Complicating to Persuade?," Working Papers hal-03583827, HAL.
    20. Romain Blanchard & Laurence Carassus & Miklós Rásonyi, 2018. "No-arbitrage and optimal investment with possibly non-concave utilities: a measure theoretical approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(2), pages 241-281, October.

    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:eee:mateco:v:110:y:2024:i:c:s0304406823001295. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jmateco .

    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.