IDEAS home Printed from https://ideas.repec.org/a/spr/stmapp/v29y2020i2d10.1007_s10260-019-00486-6.html
   My bibliography  Save this article

What finite-additivity can add to decision theory

Author

Listed:
  • Mark J. Schervish

    (Carnegie Mellon University)

  • Teddy Seidenfeld

    (Carnegie Mellon University)

  • Rafael B. Stern

    (Federal University of São Carlos)

  • Joseph B. Kadane

    (Carnegie Mellon University)

Abstract

We examine general decision problems with loss functions that are bounded below. We allow the loss function to assume the value $$\infty $$∞. No other assumptions are made about the action space, the types of data available, the types of non-randomized decision rules allowed, or the parameter space. By allowing prior distributions and the randomizations in randomized rules to be finitely-additive, we prove very general complete class and minimax theorems. Specifically, under the sole assumption that the loss function is bounded below, we show that every decision problem has a minimal complete class and all admissible rules are Bayes rules. We also show that every decision problem has a minimax rule and a least-favorable distribution and that every minimax rule is Bayes with respect to the least-favorable distribution. Some special care is required to deal properly with infinite-valued risk functions and integrals taking infinite values.

Suggested Citation

  • Mark J. Schervish & Teddy Seidenfeld & Rafael B. Stern & Joseph B. Kadane, 2020. "What finite-additivity can add to decision theory," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(2), pages 237-263, June.
    • Handle: RePEc:spr:stmapp:v:29:y:2020:i:2:d:10.1007_s10260-019-00486-6
    DOI: 10.1007/s10260-019-00486-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10260-019-00486-6
    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/s10260-019-00486-6?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. Mark J. Schervish & Teddy Seidenfeld & Joseph B. Kadane, 2009. "Proper Scoring Rules, Dominated Forecasts, and Coherence," Decision Analysis, INFORMS, vol. 6(4), pages 202-221, December.
    2. P. Battigalli & S. Cerreia‐Vioglio & F. Maccheroni & M. Marinacci, 2016. "A Note on Comparative Ambiguity Aversion and Justifiability," Econometrica, Econometric Society, vol. 84, pages 1903-1916, September.
    3. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-1050, July.
    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. Calford, Evan M., 2020. "Uncertainty aversion in game theory: Experimental evidence," Journal of Economic Behavior & Organization, Elsevier, vol. 176(C), pages 720-734.
    2. Kuzmics, Christoph, 2017. "Abraham Wald's complete class theorem and Knightian uncertainty," Games and Economic Behavior, Elsevier, vol. 104(C), pages 666-673.
    3. Guarino, Pierfrancesco & Ziegler, Gabriel, 2022. "Optimism and pessimism in strategic interactions under ignorance," Games and Economic Behavior, Elsevier, vol. 136(C), pages 559-585.
    4. Calford, Evan M., 2021. "Mixed strategies and preference for randomization in games with ambiguity averse agents," Journal of Economic Theory, Elsevier, vol. 197(C).
    5. Burkhard Schipper & Hee Yeul Woo, 2012. "Political Awareness and Microtargeting of Voters in Electoral Competition," Working Papers 124, University of California, Davis, Department of Economics.
    6. Andrés Perea & Elias Tsakas, 2019. "Limited focus in dynamic games," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(2), pages 571-607, June.
    7. Battigalli, Pierpaolo & Bonanno, Giacomo, 1997. "The Logic of Belief Persistence," Economics and Philosophy, Cambridge University Press, vol. 13(1), pages 39-59, April.
    8. Renou, Ludovic & Schlag, Karl H., 2010. "Minimax regret and strategic uncertainty," Journal of Economic Theory, Elsevier, vol. 145(1), pages 264-286, January.
    9. Schipper, Burkhard C., 2021. "Discovery and equilibrium in games with unawareness," Journal of Economic Theory, Elsevier, vol. 198(C).
    10. Martin Meier & Burkhard Schipper, 2014. "Bayesian games with unawareness and unawareness perfection," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(2), pages 219-249, June.
    11. Radzvilas, Mantas, 2016. "Hypothetical Bargaining and the Equilibrium Selection Problem in Non-Cooperative Games," MPRA Paper 70248, University Library of Munich, Germany.
    12. Stephen Morris & Satoru Takahashi & Olivier Tercieux, 2012. "Robust Rationalizability Under Almost Common Certainty Of Payoffs," The Japanese Economic Review, Japanese Economic Association, vol. 63(1), pages 57-67, March.
    13. Jorge M. Streb & Gustavo Torrens, 2011. "Meaningful talk," CEMA Working Papers: Serie Documentos de Trabajo. 443, Universidad del CEMA, revised May 2017.
    14. Vincent J. Vannetelbosch & P. Jean-Jacques Herings, 2000. "The equivalence of the Dekel-Fudenberg iterative procedure and weakly perfect rationalizability," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 15(3), pages 677-687.
    15. Ambrus, Attila, 2006. "Coalitional Rationalizability," Scholarly Articles 3200266, Harvard University Department of Economics.
    16. Pierpaolo Battigalli, 2006. "Rationalization In Signaling Games: Theory And Applications," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 67-93.
    17. Jackson, Matthew O. & Sonnenschein, Hugo F., 2003. "The Linking of Collective Decisions and Efficiency," Working Papers 1159, California Institute of Technology, Division of the Humanities and Social Sciences.
    18. Srihari Govindan & Robert Wilson, 2009. "On Forward Induction," Econometrica, Econometric Society, vol. 77(1), pages 1-28, January.
    19. Roger Guesnerie & Pedro Jara-Moroni, 2007. "Expectational coordination in a class of economic models: Strategic substitutabilities versus strategic complementarities," PSE Working Papers halshs-00587837, HAL.
    20. Andrés Carvajal, 2003. "Testable Restrictions of Nash Equilibrium in Games with Continuous Domains," Borradores de Economia 229, Banco de la Republica de Colombia.

    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:stmapp:v:29:y:2020:i:2:d:10.1007_s10260-019-00486-6. 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: 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. RePEc uses bibliographic data supplied by the respective publishers.