Decision-theoretic foundations of qualitative possibility theory

Abstract This paper presents a justification of two qualitative counterparts of the expected utility criterion for decision under uncertainty, which only require bounded, linearly ordered, valuation sets for expressing uncertainty and preferences. This is carried out in the style of Savage, starting with a set of acts equipped with a complete preordering relation. Conditions on acts are given that imply a possibilistic representation of the decision-maker uncertainty. In this framework, pessimistic (i.e., uncertainty-averse) as well as optimistic attitudes can be explicitly captured. The approach thus proposes an operationally testable description of possibility theory.

[1]  Hung T. Nguyen,et al.  Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference , 1994 .

[2]  Representation of preferences on fuzzy measures by a fuzzy integral , 1996 .

[3]  S. Weber ⊥-Decomposable measures and integrals for Archimedean t-conorms ⊥ , 1984 .

[4]  Madan M. Gupta,et al.  Fuzzy automata and decision processes , 1977 .

[5]  G. Klir,et al.  Fuzzy Measure Theory , 1993 .

[6]  Daniel Lehmann,et al.  Generalized Qualitative Probability: Savage revisited , 1996, UAI.

[7]  Michio Sugeno,et al.  Fuzzy integral representation , 1996, Fuzzy Sets Syst..

[8]  M. Sugeno,et al.  Fuzzy measure of fuzzy events defined by fuzzy integrals , 1992 .

[9]  Dov M. Gabbay,et al.  Handbook of defeasible reasoning and uncertainty management systems: volume 2: reasoning with actual and potential contradictions , 1998 .

[10]  D. Ellsberg Decision, probability, and utility: Risk, ambiguity, and the Savage axioms , 1961 .

[11]  Didier Dubois,et al.  Fuzzy constraints in job-shop scheduling , 1995, J. Intell. Manuf..

[12]  C. Kraft,et al.  Intuitive Probability on Finite Sets , 1959 .

[13]  Rakesh K. Sarin,et al.  A SIMPLE AXIOMATIZATION OF NONADDITIVE EXPECTED UTILITY , 1992 .

[14]  Didier Dubois,et al.  Qualitative Decision Theory with Sugeno Integrals , 1998, UAI.

[15]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[16]  Didier Dubois,et al.  A class of fuzzy measures based on triangular norms , 1982 .

[17]  L. Zadeh Fuzzy sets as a basis for a theory of possibility , 1999 .

[18]  Luis M. de Campos,et al.  Characterization and comparison of Sugeno and Choquet integrals , 1992 .

[19]  Jean-Luc Marichal,et al.  Aggregation operators for multicriteria decision aid , 1998 .

[20]  B. Jones DETERMINATION OF RECONSTRUCTION FAMILIES , 1982 .

[21]  Didier Dubois,et al.  Decision-Making under Ordinal Preferences and Comparative Uncertainty , 1997, UAI.

[22]  D. Dubois,et al.  Aggregation of decomposable measures with application to utility theory , 1996 .

[23]  Wolfgang Spohn,et al.  Ordinal Conditional Functions: A Dynamic Theory of Epistemic States , 1988 .

[24]  D. Schmeidler Subjective Probability and Expected Utility without Additivity , 1989 .

[25]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[26]  R. Yager On the specificity of a possibility distribution , 1992 .

[27]  P G rdenfors,et al.  Knowledge in flux: modeling the dynamics of epistemic states , 1988 .

[28]  S. Moral,et al.  A unified approach to define fuzzy integrals , 1991 .

[29]  Yiyu Yao,et al.  Axiomatization of qualitative belief structure , 1991, IEEE Trans. Syst. Man Cybern..

[30]  D Dubois,et al.  Belief structures, possibility theory and decomposable confidence measures on finite sets , 1986 .

[31]  Thomas Whalen,et al.  Decisionmaking under uncertainty with various assumptions about available information , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[32]  Hidetomo Ichihashi,et al.  Possibilistic Linear Programming with Measurable Multiattribute Value Functions , 1989, INFORMS J. Comput..

[33]  Simon Grant,et al.  Weakening the Sure-Thing Principle: Decomposable Choice under Uncertainty , 1997 .

[34]  Didier Dubois,et al.  Possibility Theory: Qualitative and Quantitative Aspects , 1998 .

[35]  Abraham Wald,et al.  Statistical Decision Functions , 1951 .

[36]  Didier Dubois,et al.  Possibility Theory as a Basis for Qualitative Decision Theory , 1995, IJCAI.

[37]  M. Sugeno FUZZY MEASURES AND FUZZY INTEGRALS—A SURVEY , 1993 .

[38]  I. Gilboa Expected utility with purely subjective non-additive probabilities , 1987 .

[39]  E. Rowland Theory of Games and Economic Behavior , 1946, Nature.

[40]  M. Allais Le comportement de l'homme rationnel devant le risque : critique des postulats et axiomes de l'ecole americaine , 1953 .