Derivation of some results on monotone capacities by Mobius inversion

  • Alain Chateauneuf
  • Jean-Yves Jaffray
Section II Approaches To Uncertainty A) Evidence Theory
Part of the Lecture Notes in Computer Science book series (LNCS, volume 286)


Monotone capacities are characterized by properties of their Möbius inverses. A necessary property of probabilities dominating a given capacity is given. It is shown to be also sufficient if and only if the capacity is monotone of infinite order. A characterization of dominating probabilities specific to capacities or order 2 is also proved.

Key words

Decision theory Lower probabilities Belief functions capacities Möbius inversion representation of uncertainty 


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  1. BERGE, C. (1965). Espaces topologiques, fonctions multivoques. Dunod, Paris.Google Scholar
  2. BIXBY, R.E., CUNNINGHAM, W.H. and TOKPIS, D.M. (1985). The partial order of a polymatroīd extreme point, Math. Oper. Res. 10, 367–378.Google Scholar
  3. CHATEAUNEUF, A., and JAFFRAY, J.Y. (1986). Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion (working paper).Google Scholar
  4. CHOQUET, G. (1953). Théorie des capacités. Ann. Inst. Fourier. (Grenoble) V 131–295.Google Scholar
  5. COHEN, M. and JAFFRAY, J.Y. (1985). Decision making in a case of mixed uncertainty: A normative model. J. of Math. Psych. 29, No4.Google Scholar
  6. DELLACHERIE, C. (1971). Quelques commentaires sur les prolongements de capacités. Lect. Notes Math. 191 (Sem. Prob. V), 77–81Google Scholar
  7. DEMPSTER, A.P. (1967) Upper and lower probabilities induced by a multivalued mapping. Ann. of Math. Statist. 38, 325–339.Google Scholar
  8. EDMONDS, J. (1970). Submodular functions, matroïds and certain polyhedra. Combinatorial structures and their applications (Proc. Calgary Internat. Conf. 1969). R.K. Guy & al, eds, Gordon and Breach, New York, 69–87.Google Scholar
  9. GALE, D. (1960). The theory of linear economic models. Mc Graw Hill, New York.Google Scholar
  10. HUBER, P.J. (1973). The use of Choquet capacities in statistics. Bull. Intern. Statist. Inst. XLV, Book 4, 181–188.Google Scholar
  11. HUBER, P.J. (1976). Kapazitäten statt Wahrscheinlichkeiten. Gedanken zur Grundlegung der Statistik, J. der Dt Math. Verein. 78, 81–92.Google Scholar
  12. HUBER, P.J. and STRASSEN, V. (1973). Minimax tests and the Neyman-Pearson lemma for capacities. Ann. Statist. 1, 251–263.Google Scholar
  13. KARLIN, S. (1959). Mathematical Methods and Theory in Games, Programming and Economics, Vol. I, Pergamon Press, London, Paris.Google Scholar
  14. REVUZ, A. (1955). Fonctions croissantes et mesures sur les espaces topologiques ordonnés. Ann. Inst. Fourier (Grenoble) VI, 187–169.Google Scholar
  15. SHAFER, G. (1976). A mathematical theory of evidence. Princeton University Press, Princeton, New Jersey.Google Scholar
  16. SHAFER, G. (1979). Allocations of Probability, Ann. Prob. 7, 827–839.Google Scholar
  17. SHAFER, G. (1981). Constructive Probability, Synthese, 48, 1–59.Google Scholar
  18. SHAPLEY, L.S. (1971). Cores of Convex Games, Internat. J. Game Theory 1, 11–26.Google Scholar
  19. WALD, A. (1971). Statistical decision functions. Chelsea Publishing Company, Bronx, New York.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Alain Chateauneuf
    • 1
  • Jean-Yves Jaffray
    • 2
  1. 1.Université Paris IParis
  2. 2.Université Paris VIParis

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