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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
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    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. Pearce, David G, 1984. "Rationalizable Strategic Behavior and the Problem of Perfection," Econometrica, Econometric Society, vol. 52(4), pages 1029-1050, July.
    3. 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.
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