Abstract
In this paper, non-additivity of a set function is interpreted as a method to express relations between sets which are not modeled in a set theoretic way. Drawing upon a concept called “quasi-analysis” of the philosopher Rudolf Carnap, we introduce a transform for sets, functions, and set functions to formalize this idea. Any image-set under this transform can be interpreted as a class of (quasi-)components or (quasi-)properties representing the original set. We show that non-additive set functions can be represented as signed σ-additive measures defined on sets of quasi-components. We then use this interpretation to justify the use of non-additive set functions in various applications like for instance multi criteria decision making and cooperative game theory. Additionally, we show exemplarily by means of independence, conditioning, and products how concepts from classical measure and probability theory can be transfered to the non-additive theory via the transform.
Similar content being viewed by others
References
Brüning, M. (2003). Products of Monotone Measures, Möbius Transform and k-monotonicity. Dissertation thesis, Universität Bremen.
R. Carnap (1928) Der logische Aufbau der Welt Meiner Hamburg
A. Chateauneuf J.-Y. Jaffray (1989) ArticleTitleSome characterization of lower probabilities and other monotone capacities through the use of Möbius inversion Mathematical Social Sciences 17 263–283 Occurrence Handle10.1016/0165-4896(89)90056-5
G. Choquet (1953–1954) ArticleTitleTheory of Capacities Annales de l’Institut Fourier 5 131–295
Denneberg, D. (1994, 1997 2nd. ed.), Non-additive Measure and Integral Kluwer, Dordrecht.
D. Denneberg (1994) ArticleTitleConditioning (updating) non-additive measures Annals of Operations Research 52 21–42 Occurrence Handle10.1007/BF02032159
D. Denneberg (1997) ArticleTitleRepresentation of the Choquet integral with the σ-additive Möbius transform Fuzzy Sets and Systems 92 139–156 Occurrence Handle10.1016/S0165-0114(97)00166-8
D. Denneberg (2000) ArticleTitleTotally monotone core an products of monotone measures International Journal of Approximate Reasoning 24 273–281 Occurrence Handle10.1016/S0888-613X(00)00039-6
D. Denneberg (2002) ArticleTitleConditional expectation for monotone measures, the discrete case Journal of Mathematical Economics 37 105–121 Occurrence Handle10.1016/S0304-4068(02)00011-3
I. Gilboa D. Schmeidler (1989) ArticleTitleMaxmin expected utility with non-unique prior Journal of Mathematical Economics 18 141–153 Occurrence Handle10.1016/0304-4068(89)90018-9
I. Gilboa D. Schmeidler (1995) ArticleTitleCanonical representation of set functions Mathematics of Operations Research 20 197–212
M. Grabisch (1995) ArticleTitleFuzzy integral in multicriteria decision making Fuzzy Sets and Systems 69 279–298 Occurrence Handle10.1016/0165-0114(94)00174-6
M. Grabisch (1996) ArticleTitleThe application of fuzzy integrals in multicriteria decision making European Journal of Operational Research 89 445–456 Occurrence Handle10.1016/0377-2217(95)00176-X
E. Hendon H.J. Jacobsen B. Sloth T. Tranæs (1996) ArticleTitleThe product of capacities and belief functions Mathematical Social Sciences 32 95–108 Occurrence Handle10.1016/0165-4896(96)00813-X
G.A. Koshevoy (1998) ArticleTitleDistributive lattices and products of capacities Journal of Mathematical Analysis and Applications 219 427–441 Occurrence Handle10.1006/jmaa.1997.5830
M. Marinacci (1996) ArticleTitleDecomposition and representation of coalitional games Mathematics of Operations Research 21 1000–1015 Occurrence Handle10.1287/moor.21.4.1000
T. Murofushi M. Sugeno (1989) ArticleTitleAn interpretation of fuzzy measures and the Choquet integral as an integral with respect to a fuzzy measure Fuzzy Sets and Systems 29 201–227 Occurrence Handle10.1016/0165-0114(89)90194-2
T. Murofushi M. Sugeno (1991) ArticleTitleA theory for fuzzy measures: representations, the Choquet integral, and null sets Journal of Mathematical Analysis and Applications 159 535–549 Occurrence Handle10.1016/0022-247X(91)90213-J
G. Shafer (1976) A Mathematical Theory of Evidence Princeton Univ. Press Princeton, NJ
P. Walley T.L. Fine (1982) ArticleTitleTowards a frequentist theory of upper and lower probability Annals of Statistics 10 741–761
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Maaß, S. A Philosophical Foundation of Non-Additive Measure and Probability. Theor Decis 60, 175–191 (2006). https://doi.org/10.1007/s11238-005-4591-z
Issue Date:
DOI: https://doi.org/10.1007/s11238-005-4591-z