Decision Making in the Environment of Heterogeneous Uncertainty

  • Phan H. Giang
Part of the Studies in Computational Intelligence book series (SCI, volume 502)


The environment of heterogeneous uncertainty is characterized by the presence of variables in multiple uncertainty formalisms. This paper provides an overview of decision models under several uncertainty frameworks including probability theory, Dempster-Shafer belief function theory and possibility theory. It explores the challenges in pulling them together for decision making. We show that the information of sequence of variable resolution, which was often neglected, actually plays a key role in decision making under heterogeneous uncertainty. A novel approach, based on the well-known folding-back principle, to find the certainty equivalent of acts under heterogeneous uncertainty is proposed.


Decision making Possibility theory Dempster-Shafer belief function Ignorance 


  1. 1.
    Allais, M.: Le comportement de l’homme rationel devant le risque: Critique des postulats et axioms de l’ecole americaine. Econometrica 21, 503–546 (1953)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arrow, K.J., Hurwicz, L.: An optimality criterion for decision making under ignorance. In: Arrow, K.J., Hurwicz, L. (eds.) Studies in Resource Allocation Processes. Cambridge University Press, Cambridge (1977)Google Scholar
  3. 3.
    Birnbaum, A.: On the foundation of statistical inference. J. Am. Stat. Assoc. 57(298), 269–306 (1962)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cox, D.R., Hinkley, D.V.: Theoretical Statistics. Chapman and Hall, London (1974)CrossRefMATHGoogle Scholar
  5. 5.
    Dempster, A.: Upper and lower probability induced by multivalued mapping. Ann. Probab. 38, 325–339 (1967)MathSciNetMATHGoogle Scholar
  6. 6.
    Dempster, A.: A generalization of Bayesian inference. J. Roy. Stat. Soc. B 30, 205–247 (1968) with discussionGoogle Scholar
  7. 7.
    Dubois, D., Godo, L., Prade, H., Zapico, A.: Making decision in a qualitative setting: from decision under uncertaity to case-based decision. In: Cohn, A., Schubert, L., Shapiro, S. (eds.) Sixth International Conference on Principles of Knowledge Representation and Reasoning (KR98), pp. 594–605. Morgan Kaufmann, California (CA) (6 1998)Google Scholar
  8. 8.
    Dubois, D., Nguyen, T.H., Prade, H.: Possibility theory, probability and fuzzy sets. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets, pp. 344–438. Kluwer Academic, Boston (2000)CrossRefGoogle Scholar
  9. 9.
    Ellsberg, D.: Risk, ambiguity and the Savage’s axioms. Q. J. Econ. 75(4), 643–669 (1961)CrossRefGoogle Scholar
  10. 10.
    Giang, P.H.: A new axiomatization for likelihood gambles. In: Dechter, R., Richardson, T.S. (eds.) Uncertainty in Artificial Intelligence: Proceedings of the 22nd Conference, pp. 192–199. AUAI Press, (2006)Google Scholar
  11. 11.
    Giang, P.H.: Dynamic consistency and decision making under vacuous belief. In: Cozman, F.G., Pfeffer, A. (eds.) Uncertainty in Artificial Intelligence: Proceedings of the 27nd Conference (UAI-2011), pp. 230–237. AUAI Press, (2011)Google Scholar
  12. 12.
    Giang, P.H.: Decision with Dempster-Shafer belief functions: decision under ignorance and sequential consistency. Int. J. Approximate Reasoning 53(1), 38–53 (2012)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Giang, P.H., Shenoy, P.P.: Two axiomatic approaches to decision making using possibility theory. Eur. J. Oper. Res. 162(2), 450–467 (2005)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Giang, P.H., Shenoy, P.P.: A decision theory for partially consonant belief functions. Int. J. Approximate Reasoning 52(3), 375–394 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Giang, P.H., Shenoy, P.P.: A decision theory for partially consonant belief functions. Int. J. Approximate Reasoning 52, 375–394 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gilboa, I.: Theory of Decision Under Uncertainty. Cambridge University Press, New York (2009)Google Scholar
  17. 17.
    Glimcher, P.W.: Foundations of Neuroeconomic Analysis. Oxford University Press, New York (2011)Google Scholar
  18. 18.
    Halpern, J.Y.: Reasoning About Uncertainty. MIT Press, Cambridge (2003)MATHGoogle Scholar
  19. 19.
    Jaffray, J.-Y.: Linear utility theory for belief functions. Oper. Res. Lett. 8, 107–112 (1989)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Jaffray, J.-Y., Wakker, P.: Decision making with belief functions: compatibility and incompatibility with the sure-thing principle. J. Risk Uncertainty 8(3), 255–271 (1994)Google Scholar
  21. 21.
    Jensen, N.E.: An introduction to Bernoullian utility theory I: utility functions. Swed. J. Econ. 69, 163–183 (1967)CrossRefGoogle Scholar
  22. 22.
    Savage, L.J.: The Foundations of Statistics, 2nd edn. Dover, New York (1972)MATHGoogle Scholar
  23. 23.
    Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press, Princeton (1976)MATHGoogle Scholar
  24. 24.
    Smets, P.: The transferable belief model for quantified belief representation. In: Gabbay, D.M., Smets, P. (eds.) Handbook of Defeasible Reasoning and Uncetainty Management System, vol. 1, pp. 267–301. Kluwer, Doordrecht (1998)Google Scholar
  25. 25.
    Smets, P., Kennes, R.: The transferable belief model. Artif. Intell. 66(2), 191–234 (1994)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Wakker, P.P.: Prospect Theory for Risk and Ambiguity. Cambridge University Press, Cambridge (2010)Google Scholar
  27. 27.
    Walley, P.: Belief function representation of statistical evidence. Ann. Stat. 15(4), 1439–1465 (1987)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)CrossRefMATHGoogle Scholar
  29. 29.
    Yager, R.: Decision making under Dempster-Shafer uncertainties. Int. J. Gen Syst 20, 223–245 (1992)Google Scholar
  30. 30.
    Zadeh, L.: Fuzzy set as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.George Mason UniversityFairfax, VAUSA

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