Approximations of One-dimensional Expected Utility Integral of Alternatives Described with Linearly-Interpolated p-Boxes

Part of the Studies in Computational Intelligence book series (SCI, volume 502)


In the process of quantitative decision making, the bounded rationality of real individuals leads to elicitation of interval estimates of probabilities and utilities. This fact is in contrast to some of the axioms of rational choice, hence the decision analysis under bounded rationality is called fuzzy-rational decision analysis. Fuzzy-rationality in probabilities leads to the construction of x-ribbon and p-ribbon distribution functions. This interpretation of uncertainty prohibits the application of expected utility unless ribbon functions were approximated by classical ones. This task is handled using decision criteria Q under strict uncertainty—Wald, maximax, Hurwicz\({}_{\alpha }\), Laplace—which are based on the pessimism-optimism attitude of the decision maker. This chapter discusses the case when the ribbon functions are linearly interpolated on the elicited interval nodes. Then the approximation of those functions using a Q criterion is put into algorithms. It is demonstrated how the approximation is linked to the rationale of each Q criterion, which in three of the cases is linked to the utilities of the prizes. The numerical example demonstrates the ideas of each Q criterion in the approximation of ribbon functions and in calculating the Q-expected utility of the lottery.


Quantitative decision making Rational choice Interval estimates Bounded rationality Fuzzy-rationality  Decision criteria under strict uncertainty Expected utility 


  1. 1.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior, 2nd edn. Princeton University Press, Princeton (1947)MATHGoogle Scholar
  2. 2.
    Tenekedjiev, K.: Decision problems and their place among operations research. Autom. Inform. XXVIII(1), 7–10 (2004)Google Scholar
  3. 3.
    Ramsay, F.P.: Truth and Probability, The Logical Foundations of Mathematics and Other Essays. Kegan Paul, London (1931) (Reprinted: Kyburg, H.E. Jr., Smolker, H.E. (eds.). (1964). Studies in Subjective Probability. Wiley, 61–92)Google Scholar
  4. 4.
    Savage, L.J.: The Foundations of Statistics, 1st edn. Wiley, New York (1954)MATHGoogle Scholar
  5. 5.
    Pratt, J.W., Raiffa, H., Schlaifer, R.: Introduction to Statistical Decision Theory. MIT Press, Cambridge (1995)Google Scholar
  6. 6.
    De Finetti, B.: La prevision: ses lois logiques, ses sorces subjectives. Annales de l’Institut Henri Poincare 7, 1–68 (1937) (Translated in Kyburg, H.E. Jr., Smokler, H.E. (eds.): Studies in Subjective Probability, pp. 93–158. Wiley, New York (1964))Google Scholar
  7. 7.
    De Groot, M.H.: Optimal Statistical Decisions. McGraw-Hill, New York (1970)Google Scholar
  8. 8.
    Villigas, C.: On qualitative probability\(\sigma \)-algebras. Ann. Math. Stat. 35, 1787–1796 (1964)CrossRefGoogle Scholar
  9. 9.
    French, S., Insua, D.R.: Statistical Decision Theory. Arnold, London (2000)Google Scholar
  10. 10.
    Keeney, R.L., Raiffa, H.: Decisions with Multiple Objectives: Preference and Value Tradeoffs. Cambridge University Press, Cambridge (1993)Google Scholar
  11. 11.
    Bernstein, P.L.: Against the Gods—The Remarkable Story of Risk. Wiley, New York (1996)Google Scholar
  12. 12.
    De Cooman, G., Zaffalon, M.: Updating beliefs with incomplete observations. Artif. Intell. 159, 75–125 (2004)CrossRefMATHGoogle Scholar
  13. 13.
    Cano, A., Moral, S.: Using probability trees to compute marginals with imprecise probabilities. Int. J. Approximate Reasoning 29, 1–46 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Guo, P., Wang, Y.: Eliciting dual interval probabilities from interval comparison matrices. Inf. Sci. 190, 17–26 (2012)CrossRefMATHGoogle Scholar
  15. 15.
    Guo, P., Tanaka, H.: Decision making with interval probabilities. Eur. J. Oper. Res. 203, 444–454 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Walley, P.: Inferences from multinomial data: learning about a bag of marbles. J. Royal Stat. Soc. B, 58, 3–57 (1996)Google Scholar
  17. 17.
    Miranda, E., de Cooman, G., Couso, I.: Lower previsions induced by multi-valued mappings. J. Stat. Plann. Infer. 133, 173–197 (2005)CrossRefMATHGoogle Scholar
  18. 18.
    Knetsch, J.: The endowment effect and evidence of nonreversible indifference curves. In: Kahneman, D., Tversky, A. (eds.) Choices, Values and Frames, pp. 171–179. Cambridge University Press, Cambridge (2000)Google Scholar
  19. 19.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991)CrossRefMATHGoogle Scholar
  20. 20.
    Troffaes, M.C.M.: Decision making under uncertainty using imprecise probabilities. Int. J. Approximate Reasoning 45, 7–29 (2007)Google Scholar
  21. 21.
    Kozine, I., Utkin, L.: Constructing imprecise probability distributions. Int. J. General Syst. 34(4), 401–408 (2005)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Tenekedjiev, K., Nikolova, N.D., Toneva, D.: Laplace expected utility criterion for ranking fuzzy rational generalized lotteries of I type. Cybern. Inf. Technol. 6(3), 93–109 (2006)Google Scholar
  23. 23.
    Nikolova, N.D., Shulus, A., Toneva, D., Tenekedjiev, K.: Fuzzy rationality in quantitative decision analysis. J. Adv. Comput. Intell. Intell. Inf. 9(1), 65–69 (2005)Google Scholar
  24. 24.
    Yager, R.R.: Decision making using minimization of regret. Int. J. Approximate Reasoning 36, 109–128 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yager, R.R.: Generalizing variance to allow the inclusion of decision attitude in decision making under uncertainty. Int. J. Approximate Reasoning 42, 137–158 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Fabrycky, W.J., Thuesen, G.J., Verma, D.: Economic Decision Analysis. Prentice Hall, Englewood Cliffs (1998)Google Scholar
  27. 27.
    Hackett, G., Luffrum, P.: Business Decision Analysis—An Active Learning Approach. Blackwell, New York (1999)Google Scholar
  28. 28.
    Milnor, J.: Games against nature. In: Thrall, R., Combs R.D. (eds.) Decision Processes, pp. 49–59. Wiley, Chichester (1954)Google Scholar
  29. 29.
    Szmidt, E., Kacprzyk, J.: Probability of intuitionistic fuzzy events and their application in decision making. In: Proc. EUSFLAT-ESTYLF joint conference, September 22–25, Palma de Majorka, Spain, pp. 457–460 (1999)Google Scholar
  30. 30.
    Atanassov, K.: Intuitionistic Fuzzy Sets. Springer, Heidelberg (1999)Google Scholar
  31. 31.
    Atanassov, K.: Review and New Results on Intuitionistic Fuzzy Sets, Preprint IM-MFAIS, vol. 1–88. Sofia (1988)Google Scholar
  32. 32.
    Atanassov, K.: Four New Operators on Intuitionistic Fuzzy Sets, Preprint IM-MFAIS, vol. 4–89. Sofia (1989)Google Scholar
  33. 33.
    Utkin, L.V. (2007). Risk analysis under partial prior information and non-monotone utility functions. Int. J. Inf. Technol. Decision Making. 6(4), 625–647 (2007)Google Scholar
  34. 34.
    Nikolova, N.D.: Three criteria to rank x-fuzzy-rational generalized lotteries of I type. Cybern. Inf. Technol. 7(1), 3–20 (2007)Google Scholar
  35. 35.
    Dantzig, G.B., Orden, A., Wolfe, Ph: The generalized simplex method for minimizing a linear form under linear inequality restraints. Pacific J. Math. 5(2), 183–195 (1955)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    French, S.: Decision Theory: An Introduction to the Mathematics of Rationality. Ellis Horwood, Chichester (1993)Google Scholar
  37. 37.
    Rapoport, A.: Decision Theory and Decision Behaviour-Normative and Descriptive Approaches. USA, Kluwer Academic Publishers (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Nikola Vaptsarov Naval AcademyVarna Bulgaria

Personalised recommendations