Aggregation of decomposable measures with application to utility theory

This paper investigates the eventwise aggregations of decomposable measures preserving the same decomposable property. These operations are obtained by solving a functional equation closely related to the bisymmetry property. Known results for probability as well as possibility measures can be derived as particular cases of our approach. In addition, the unicity of weighted consensus functions is proved in the Archimedean case. An extension of Von Neumann-Morgenstern utility theory is outlined, where probabilities are changed into decomposable measures.

[1]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .

[2]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[3]  J. Milnor,et al.  AN AXIOMATIC APPROACH TO MEASURABLE UTILITY , 1953 .

[4]  L. J. Savage,et al.  The Foundations of Statistics , 1955 .

[5]  G. Choquet.  Theory of capacities , 1954 .

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

[7]  H. Bandemer.  Savage, L. J.: Foundations of Statistics. Dover Publ., Inc,. New York 1972. 310 S. , 1974 .

[8]  Ralph L. Keeney,et al.  Multiplicative Utility Functions , 1974, Oper. Res..

[9]  M. J. Frank.  On the simultaneous associativity ofF(x, y) andx+y−F(x, y) , 1978 .

[10]  K. McConway.  Marginalization and Linear Opinion Pools , 1981 .

[11]  C. Wagner,et al.  Rational Consensus in Science and Society , 1981 .

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

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

[14]  C. Alsina.  On a family of connectives for fuzzy sets , 1985 .

[15]  Didier Dubois,et al.  Weighted minimum and maximum operations in fuzzy set theory , 1986, Inf. Sci..

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

[17]  Didier Dubois,et al.  Possibility Theory - An Approach to Computerized Processing of Uncertainty , 1988 .

[18]  Carl G. Wagner.  Consensus for belief functions and related uncertainty measures , 1989 .

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

[20]  Didier Dubois,et al.  Aggregation of Possibility Measures , 1990 .

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

[22]  Judea Pearl,et al.  From Conditional Oughts to Qualitative Decision Theory , 1993, UAI.