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Qualitative Decision Rules Under Uncertainty

  • Didier Dubois
  • Hélène Fargier
Part of the International Centre for Mechanical Sciences book series (CISM, volume 472)

Abstract

This paper is a survey of qualitative decision theory focusing on available decision rules under uncertainty, and their properties. It is pointed out that two main approaches exist according to whether degrees of uncertainty and degrees of utility are commensurate (that is, belong to a unique scale) or not. Savage-like axiom systems for both approaches are surveyed. In such a framework, acts are functions from states to results, and decision rules are derived from first principles, bearing on a preference relation on acts. It is shown that the emerging uncertainty theory in qualitative settings is possibility theory rather than probability theory. However these approaches lead to criteria that are either little decisive due to incomparability, or too adventurous because focusing on the most plausible states, or yet lacking discrimination because or the coarseness of the value scale. Some suggestions to overcome these defects are pointed out.

Keywords

Weak Order Qualitative Decision Certainty Equivalent Possibility Distribution Likelihood Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Arrow K. (1951). Social Choice and Individual Values. New York, N.Y.: Wiley.MATHGoogle Scholar
  2. Arrow, K. Hurwicz L. (1972). An optimality criterion for decision-making under ignorance. In: C.F. Carter, J.L. Ford, eds., Uncertainty and Expectations in Economics. Oxford, UK: Basil Blackwell & Mott Ltd.Google Scholar
  3. Bacchus F. and Grove A. (1996). Utility independence in a qualitative decision theory. In: Proc. Of the 5rd Inter. Conf. on Principles of Knowledge Representation and Reasoning (KR’96), Cambridge, Mass., 542–552.Google Scholar
  4. Benferhat S., Dubois D., Prade H. (1999) Possibilistic and standard probabilistic semantics of conditional knowledge bases. J. Logic and Computation, 9, 873–895.MathSciNetCrossRefMATHGoogle Scholar
  5. Boutilier C. (1994). Towards a logic for qualitative decision theory. In: Proc. of the 4rd Inter. Conf. on Principles of Knowledge Representation and Reasoning (KR’94), Bonn, Germany, May. 2427, 75–86.Google Scholar
  6. Bouyssou D., Marchant T., Pirlot M., Perny P., Tsoukias A. and Vincke P. (2000). Evaluation Models: a Critical Perspective. Kluwer Acad. Pub. Boston.Google Scholar
  7. Brafman R.I., Tennenholtz M. (1997). Modeling agents as qualitative decision makers. Artificial Intelligence, 94, 217–268.MathSciNetCrossRefMATHGoogle Scholar
  8. Brafman R.I., Tennenholtz M. (2000). On the Axiomatization of Qualitative Decision Criteria, J. ACM, 47, 452–482MathSciNetCrossRefMATHGoogle Scholar
  9. Buckley J. J. (1988) Possibility and necessity in optimization, Fuzzy Sets and Systems, 25, 1–13.MathSciNetCrossRefMATHGoogle Scholar
  10. Cayrol M., Farreny H. (1982). Fuzzy pattern matching. Kybernetes, 11, 103–116.CrossRefGoogle Scholar
  11. Cohen M. and Jaffray J.-Y. (1980) Rational behavior under complete ignorance. Econometrica, 48, 1280–1299.MathSciNetCrossRefGoogle Scholar
  12. Doyle J., Thomason R. (1999). Background to qualitative decision theory. The AI Magazine, 20 (2), 1999, 55–68Google Scholar
  13. Doyle J., Wellman M.P. (1991). Impediments to universal preference-based default theories. Artificial Intelligence, 49, 97–128.MathSciNetCrossRefMATHGoogle Scholar
  14. Dubois D. (1986). Belief structures, possibility theory and decomposable confidence measures on finite sets. Computers and Artificial Intelligence (Bratislava), 5 (5), 403–416.MATHGoogle Scholar
  15. Dubois D. (1987) Linear programming with fuzzy data, In: The Analysis of Fuzzy Information — Vol. 3: Applications in Engineering and Science ( J.C. Bezdek, ed.), CRC Press, Boca Raton, Fl., 241–263Google Scholar
  16. Dubois D., Fargier H., and Prade H. Fuzzy constraints in job-shop scheduling. J. of Intelligent Manufacturing, 64: 215–234, 1995.CrossRefGoogle Scholar
  17. Dubois D., Fargier H., and Prade H. (1996). Refinements of the maximin approach to decision-making in fuzzy environment. Fuzzy Sets and Systems, 81, 103–122.MathSciNetCrossRefMATHGoogle Scholar
  18. Dubois D., Fargier H., and Prade H. (1997). Decision making under ordinal preferences and uncertainty. In: Proc. of the 13th Conf. on Uncertainty in Artificial Intelligence ( D. Geiger, P.P. Shenoy, eds.), Providence, RI, Morgan & Kaufmann, San Francisco, CA, 157–164.Google Scholar
  19. Dubois D., Fargier H., and Perny P. (2001). Towards a qualitative multicriteria decision theory. In: Proceedings of Eurofuse Workshop on Preference Modelling and Applications, Granada, Spain, 121–129, 25–27 avril 2001. To appear in Int. J. Intelligent Systems, 2003.Google Scholar
  20. Dubois D., Fargier H., Perny P. and Prade H. (2002a). Qualitative Decision Theory: from Savage’s Axioms to Non-Monotonic Reasoning. Journal of the ACM, 49, 455–495.MathSciNetCrossRefGoogle Scholar
  21. Dubois D., Fargier H., and Perny P. (2002b). On the limitations of ordinal approaches to decision-making. Proc. of the 8th International Conference, Principles of Knowledge Representation and Reasoning (KR2002), Toulouse, France. Morgan Kaufmann Publishers, San Francisco, California, 133–144.Google Scholar
  22. Dubois D., Fargier H., and Perny P. (2003). Qualitative models for decision under uncertainty: an axiomatic approach. Artificial Intell., to appear.Google Scholar
  23. Dubois D., Fortemps P. (1999). Computing improved optimal solutions to max-min flexible constraint satisfaction problems. European Journal of Operational Research, 118, p. 95–126.CrossRefMATHGoogle Scholar
  24. Dubois D., Grabisch M., Modave F., Prade H. (2000) Relating decision under uncertainty and multicriteria decision making models. Int. J. Intelligent Systems, 151, 967–979.CrossRefGoogle Scholar
  25. Dubois D., Marichal J.L., Prade H., Roubens M., Sabbadin R. (2001) The use of the discrete Sugeno integral in decision-making: a survey. Int. J. Uncertainty, Fuzziness and Knowledge-based Systems, 9, 539–561.MathSciNetMATHGoogle Scholar
  26. Dubois D., Prade H. (1988) Possibility Theory — An Approach to the Computerized Processing of Uncertainty. Plenum Press, New YorkMATHGoogle Scholar
  27. Dubois D., Prade H. (1995a) Numerical representation of acceptance. In: Proc. of the 11th Conf. on Uncertainty in Articicial Intelligence, Montréal, August, 149–156.Google Scholar
  28. Dubois D., Prade H.(1995ó) Possibility theory as a basis for qualitative decision theory. In: Proc. of the Inter. Joint Conf. on Artificial Intelligence (IJCAI’95), Montréal, August, 1924–1930.Google Scholar
  29. Dubois D., Prade H. (1998). Possibility theory: qualitative and quantitative aspects. P. Smets, (Eds), In: Handbook on Defeasible Reasoning and Uncertainty Management Systems — Volume 1: Quantified Representation of Uncertainty and Imprecision. Kluwer Academic Publ., Dordrecht, The Netherlands, 169–226Google Scholar
  30. Dubois D., Prade H., and Sabbadin R. (1998). Qualitative decision theory with Sugeno integrals.Proceedings of 14th Conference on Information Processing and Management of Uncertainty in Artificial Intelligence (UAI’98), Madison, WI, USA. Morgan Kaufmann, San Francisco, CA, p. 121–128.Google Scholar
  31. Dubois D., Prade H., and Sabbadin R. (2001). Decision-theoretic foundations of possibility theory. European Journal of Operational Research, 128, 459–478.MathSciNetCrossRefMATHGoogle Scholar
  32. Dubois D., Prade H., Testemale C. (1988). Weighted fuzzy pattern matching. Fuzzy Sets and Systems, 28, 313–331.MathSciNetCrossRefMATHGoogle Scholar
  33. Fargier H., Lang J. and Schiex T. (1993) Selecting preferred solutions in Fuzzy Constraint Satisfaction Problems, In: Proc. of the 1st Europ. Conf. on Fuzzy Information Technologies (EUFIT’93), Aachen, Germany, 1128–1134.Google Scholar
  34. Fargier H., Perny P. (1999). Qualitative models for decision under uncertainty without the commensurability assumption. In: Proc. of the 15th Conf. on Uncertainty in Artificial Intelligence ( K. Laskey, H. Prade, eds.), Providence, RI, Morgan & Kaufmann, San Francisco, CA, 157–164.Google Scholar
  35. Fargier H., Lang J., Sabbadin R. (1998). Towards qualitative approaches to multi-stage decision making. International Journal of Approximate Reasoning, 19, 441–471.MathSciNetCrossRefMATHGoogle Scholar
  36. Fargier H., Sabbadin R., (2000) Can qualitative utility criteria obey the surething principle? Proceedings IPMU2000, Madrid, 821–826.Google Scholar
  37. Fargier H. Sabbadin R. (2003) Qualitative decision under uncertainty: back to expected utility, Proc. IJCAI’03, Acapulco, Mexico.Google Scholar
  38. Fishburn P. (1975). Axioms for lexicographic preferences. Review of Economical Studies, 42, 415–419CrossRefMATHGoogle Scholar
  39. Fishburn P. (1986). The axioms of subjective probabilities. Statistical Science 1, 335–358.MathSciNetCrossRefGoogle Scholar
  40. Friedman N., Halpern J. (1996). Plausibility measures and default reasoning. Proc of the 13th National Conf on Artificial Intelligence (AAAI’96), Portland, 1297–1304.Google Scholar
  41. Giang P., Shenoy P. (2000). A qualitative utility theory for Spohn’s theory of epistemic beliefs. In: Proc. of the 16th Conf. on Uncertainty in Artificial Intelligence, 220–229.Google Scholar
  42. Giang P., Shenoy P. (2001). A comparison of axiomatic approaches to qualitative decision-making using possibility theory. Proc. 17 th Int. Conf. on Uncertainty in Artificial Intelligence, 162–170.Google Scholar
  43. Grabisch M., Murofushi T., Sugeno M., Eds. (1999) Fuzzy Measures and Integrals Physica-Verlag, Heidelberg, Germany.Google Scholar
  44. Grabisch M., De Baets B., and Fodor J. (2002) On symmetric pseudo-additions and pseudo-multiplications: is it possible to build rings on [-1, +1]? In: Proc. 9th Int Conf. on Information Processing and Management of Uncertainty in Knowledge based Systems (IPMU2002), Annecy, France, pp 1349–1355.Google Scholar
  45. Grant S., Kajii A., Polak B. (2000) Decomposable Choice under Uncertainty, J. Economic Theory. Vol. 92, No. 2, pp. 169–197.MathSciNetCrossRefMATHGoogle Scholar
  46. Jaffray J.-Y. (1989) Linear utility theory for belief functions. Operations Research Letters, 8, 107–112.MathSciNetCrossRefMATHGoogle Scholar
  47. Inuiguichi M., Ichihashi H., and Tanaka, H. (1989). Possibilistic linear programming with measurable multiattribute value functions. ORSA J. on Computing, 1, 146–158.CrossRefGoogle Scholar
  48. Kraus K., Lehmann D, Magidor M. (1990). Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44, 167–207.MathSciNetCrossRefMATHGoogle Scholar
  49. Lang J. (1996). Conditional desires and utilities: an alternative logical approach to qualitative decision theory. Proc. of the 12th European Conf. on Artificial Intelligence (ECAI’96), Budapest, 318–322.Google Scholar
  50. Lehmann D. (1996). Generalized qualitative probability: Savage revisited. Proc. of the 12th Conf. on Uncertainty in Artificial Intelligence, Portland, August, Morgan & Kaufman, San Mateo, CA, 381–388.Google Scholar
  51. Lehmann D. (2001). Expected Qualitative Utility Maximization, J. Games and Economic Behavior. 35, 54–79CrossRefMATHGoogle Scholar
  52. Lewis D. (1973). Counterfactuals. Basil Blackwell, London.Google Scholar
  53. Moulin H. (1988). Axioms of Cooperative Decision Making. Cambridge University Press, Cambridge, MA.Google Scholar
  54. Roubens M., Vincke P. (1985) Preference Modelling. Lecture Notes in Economics and Mathematical Systems, Vol. 250, Springer Verlag, Berlin.Google Scholar
  55. Sabbadin R. (2000), Empirical comparison of probabilistic and possibilistic Markov decision processes algorithms. Proc. 14 th Europ. Conf. on Artificial Intelligence (ECAI’00), Berlin, Germany, 586–590.Google Scholar
  56. Savage L.J. (1972). The Foundations of Statistics. Dover, New York.MATHGoogle Scholar
  57. Schmeidler D. (1989) Subjective probability and expected utility without additivity, Econometrica, 57, 571–587.MathSciNetCrossRefMATHGoogle Scholar
  58. Sen A.K. (1986). Social choice theory. In K. Arrow, M.D. Intrilligator, Eds., Handbook of Mathematical Economics, Chap. 22, Elsevier, Amsterdam, 1173–1181.Google Scholar
  59. Shackle G.L.S. (1961) Decision Order and Time In Human Affairs Cambridge University Press, Cambridge, U.K.(2nd edition, 1969).Google Scholar
  60. Snow P. (1999) Diverse confidence levels in a probabilistic semantics for conditional logics. Artificial Intelligence 113, 269–279.MathSciNetCrossRefMATHGoogle Scholar
  61. Tan S.W., Pearl J. (1994). Qualitative decision theory. Proc. 11th National Conf. on Artificial Intelligence (AAAI-94), Seattle, WA, pp. 70–75.Google Scholar
  62. Thomason R. (2000), Desires and defaults: a framework for planning with inferred goals. In: Proc. of the Inter. Conf. on Principles of Knowledge Representation and Reasoning (KR’00), Breckenridge, Col.., Morgan & Kaufmann, San Francisco, 702–713.Google Scholar
  63. VonNeumannJ. and Morgenstern O. (1944): Theory of Games and Economic Behaviour (Princeton Univ. Press, Princeton, NJ).Google Scholar
  64. Vincke P. Multicriteria Decision-Aid, J. Wiley & Sons, New York, 1992Google Scholar
  65. Wald A. (1950), Statistical Decision Functions. J. Wiley & Sons, New York.Google Scholar
  66. Whalen T. (1984). Decision making under uncertainty with various assumptions about available information. IEEE Trans. on Systems, Man and Cybernetics, 14: 888–900.MathSciNetCrossRefGoogle Scholar
  67. Yager.R.R. (1979). Possibilistic decision making. IEEE Trans. on Systems, Man and Cybernetics, 9: 388–392.MathSciNetCrossRefGoogle Scholar
  68. L.A. Zadeh L.A. (1978). Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3–28.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • Didier Dubois
    • 1
  • Hélène Fargier
    • 1
  1. 1.IRIT-CNRSUniversité Paul SabatierToulouseFrance

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