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Expected utility within a generalized concept of probability — a comprehensive framework for decision making under ambiguity

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Abstract

It had often been complained that the standard framework of decision theory is insufficient. In most applications, neither the maximin paradigm (relying on complete ignorance on the states of nature) nor the classical Bayesian paradigm (assuming perfect probabilistic information on the states of nature) reflect the situation under consideration adequately. Typically one possesses some, but incomplete, knowledge on the stochastic behaviour of the states of nature.

In this paper first steps towards a comprehensive framework for decision making under such complex uncertainty will be provided. Common expected utility theory will be extended to interval probability, a generalized probabilistic setting which has the power to express incomplete stochastic knowledge and to take the extent of ambiguity (non-stochastic uncertainty) into account.

Since two-monotone and totally monotone capacities are special cases of general interval probability, where Choquet integral and interval-valued expectation correspond to one another, the results also show, as a welcome by-product, how to deal efficiently with Choquet Expected Utility and how to perform a neat decision analysis in the case of belief functions.

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References

  1. Augustin, T. (1998): Optimale Tests bei Intervallwahrscheinlichkeit. Vandenhoeck & Ruprecht, Göttingen. (In German, with an English summary on pages 247–249.)

  2. Augustin, T. (2001a): Neyman-Pearson testing under interval probability by globally least favorable pairs — reviewing Huber-Strassen theory and extending it to general interval probability. To appear in: Journal of Statistical Planning and Inference.

  3. Augustin, T. (2001a): On decision making under ambiguous prior and sampling information. In: deCooman, G.; Fine, T.L.; Moral, S., Seidenfeld, T. (eds.): ISIPTA 01: Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications. Cornell University, Ithaca, New York; c.f. also http://decsai.ugr.es/~smc/isipta01/proceedings/index. html

    Google Scholar 

  4. Bazaraa, M.S.; Shetty, CM. (1979): Nonlinear Programming. Theory and Algorithms Wiley. New York.

    Google Scholar 

  5. Berger, J.O. (1984): Statistical Decision Theory and Bayesian Analysis. (2nd edition). Springer. New York.

    Google Scholar 

  6. Chateauneuf, A.; Cohen, M.; Meilijson, I. (1997): New tools to better model behavior under risk and uncertainty: An overview. Finance. 18, 25–46.

    Google Scholar 

  7. de Cooman, G.; Fine, T.L.; Moral, S.; Seidenfeld, T. (eds.): ISIPTA 01: Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications. Cornell University, Ithaca, New York; c.f. also http://decsai.ugr.es/-smc/isipta01/proceedings/index.html

  8. deCooman, G.; Walley, P. (eds.) (2001): The Imprecise Probability Project. http://ippserv.rug.ac.be/

  9. Dempster, A.P. (1967): Upper and lower probabilities induced by a multivalued mapping. The Annals of Mathematical Statistics. 37; 325–339.

    Article  MathSciNet  Google Scholar 

  10. Denneberg, D. (1994): Non-Additive Measure and Integral. Kluwer, Dordrecht.

    MATH  Google Scholar 

  11. Ellsberg, D. (1961): Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics. 75, 643–669.

    Article  Google Scholar 

  12. Fromm, A. (1975): Nichtlineare Optimierungsmodelle. Ausgewahlte Ansatze, Kritih und Anwendungen. Harri Deutsch, Frankfurt/M.

    Google Scholar 

  13. Gilboa, I.; Schmeidler, D. (1989): Maxmin expected utility with nonunique prior. Journal of Mathematical Economics. 18, 141–153.

    Article  MATH  MathSciNet  Google Scholar 

  14. Grabisch, M.; Nguyen, H.T.; Walker, E.A. (1995): Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference. Kluwer, Dordrecht.

    Google Scholar 

  15. Huber, P.J.; Strassen, V. (1973): Minimax tests and the Neyman-Pearson lemma for capacities. Annals of Statistics. 1, 251–263; Correction: 2, 223-224.

    Article  MATH  MathSciNet  Google Scholar 

  16. Jaffray, J.Y. (1989): Linear utility theory and belief functions. Operations Research Letters. 8, 107–122.

    Article  MATH  MathSciNet  Google Scholar 

  17. Kofler, E. (1989): Prognosen und Stabilitat bei unvollstandiger Information. Campus, Frankfurt/Main.

    Google Scholar 

  18. Kofler, E.; Menges, G. (1976): Entscheidungen bei unvollstandiger Information. Springer, Berlin (Lecture Notes in Economics and Mathematical Systems, 136).

    Google Scholar 

  19. Papamarcou, A.; Fine, T.L. (1991): Unstable collectives and envelopes of probability measures. Annals of Probability. 19, 893–906.

    Article  MATH  MathSciNet  Google Scholar 

  20. Schmeidler, D. (1989): Subjective probability and expected utility without additivity. Econometrica. 57, 571–587.

    Article  MATH  MathSciNet  Google Scholar 

  21. Shafer, G. (1976): A Mathematical Theory of Evidence. Princeton University Press. Princeton.

    MATH  Google Scholar 

  22. Shapley, S. (1971): Cores of convex games. International Journal of Game Theory. 1, 12–26.

    MathSciNet  Google Scholar 

  23. Smithson, M. (1999): Human judgement research on imprecise probabilities — a reference list distributed at ISIPTA99. Division of Psychology, The Australian National University.

  24. Vidakovic, B. (2000): Г-minimax: A paradigm for conservative robust Bayesians. In: Insua, D.R., and Ruggeri, F. (eds.) Robust Bayesian Analysis. Springer.

  25. Walley, P. (1991): Statistical Reasoning with Imprecise Probabilities. Chapman & Hall, London.

    MATH  Google Scholar 

  26. Walley, P.; Fine, T.L. (1982): Towards a frequentist theory of upper and lower probability. The Annals of Statistics. 10, 741–761.

    Article  MATH  MathSciNet  Google Scholar 

  27. Weichselberger, K. (1995): Axiomatic foundations of the theory of interval-probability. In: Mammitzsch, V., Schneeweiβ, H. (eds.) Symposia Gaussiana Conference B. de Gruyter, Berlin; 47–64.

  28. Weichselberger, K. (1996): Interval-probability on finite sample-spaces. In: Rieder, H. (ed.) Robust Statistics, Data Analysis and Computer Intensive Methods. Springer, New York; 391–409.

    Google Scholar 

  29. Weichselberger, K. (2000): The theory of interval-probability as a unifying concept for uncertainty. International Journal of Approximate Reasoning. 24, 149–170.

    Article  MATH  MathSciNet  Google Scholar 

  30. Weichselberger, K. (2001): Elementare Grundbegriffe einer allgemeineren Wahrscheinlichkeitsrechnung I. Intervallwahrscheinlichkeit als umfassendes Konzept. Physika, Heidelberg.

    MATH  Google Scholar 

  31. Weichselberger, K.; Augustin, T. (1998): Analysing Ellsberg’s Paradox by means of interval-probability. In: Galata, R.; Küchenhof, H. (eds.): Econometrics in Theory and Practice. (Festschrift for Hans Schneeweifi.) Physika. Heidelberg; 291–304.

    Google Scholar 

  32. Yager, R.R.; Fedrizzi, M.; Kacprzyk, J. (eds.) (1994): Advances in the Dempster-Shafer Theory of Evidence. Wiley, New York.

    MATH  Google Scholar 

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  1. Department of Statistics, University of Munich, Akademiestraβe 1, 80799, München, Germany

    Thomas Augustin

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Augustin, T. Expected utility within a generalized concept of probability — a comprehensive framework for decision making under ambiguity. Statistical Papers 43, 5–22 (2002). https://doi.org/10.1007/s00362-001-0083-6

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  • DOI: https://doi.org/10.1007/s00362-001-0083-6

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