Event-Based Transformations of Set Functions and the Consensus Requirement

  • Andrey G. BronevichEmail author
  • Igor N. Rozenberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11144)


Non-additive measures, capacities or generally set functions are widely used in decision models, data processing and game theory. In these applications we can find many structures identified as linear transformations or linear operators. The most remarkable of them are Choquet integral, Möbius transform, interaction transform, Shapley value. The main goal of the presented paper is to study some of them recently called event-based linear transformations. We describe them considering the set of all possible linear operators as a linear space w.r.t. their linear combinations and compute the dimensions of its some subspaces. We also study the consensus requirement, i.e. we analyze the condition when the linear operator maps one family of non-additive measures to other family.


Set functions Event-based transformations Linear operators Consensus requirement 



This work has been supported by the grant 18-01-00877 of RFBR (Russian Foundation for Basic Research).


  1. 1.
    Grabisch, M., Roubens, M.: Application of the Choquet integral in multicriteria decision making. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals: Theory and Applications. Studies on Fuzziness and Soft Computing, pp. 415–434. Physica-Verlag, Heidelberg (2000)Google Scholar
  2. 2.
    Grabisch, M.: Set Functions, Games and Capacities in Decision Making. Springer, Cham (2016). Scholar
  3. 3.
    Klir, G.J.: Uncertainty and Information: Foundations of Generalized Information Theory. Wiley-Interscience, Hoboken (2006)zbMATHGoogle Scholar
  4. 4.
    Denneberg, D.: Non-additive Measure and Integral. Kluwer, Dordrecht (1997)zbMATHGoogle Scholar
  5. 5.
    Chateauneuf, A., Jaffray, J.Y.: Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion. Math. Soc. Sci. 17, 263–283 (1989)CrossRefGoogle Scholar
  6. 6.
    Grabisch, M.: The interaction and Möbius representations of fuzzy measures on finite spaces, k-additive measures: a survey. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals: Theory and Applications. Studies on Fuzziness and Soft Computing, pp. 70–93. Physica-Verlag, Heidelberg (2000)zbMATHGoogle Scholar
  7. 7.
    Borkotokey, S., Mesiar, R., Li, J.: Event-based transformations of capacities. In: Torra, V., Narukawa, Y., Honda, A., Inoue, S. (eds.) MDAI 2017. LNCS (LNAI), vol. 10571, pp. 33–39. Springer, Cham (2017). Scholar
  8. 8.
    Mesiar, R., Borkotokey, S., Jin, L., Kalina, M.: Aggregation functions and capacities, Fuzzy Sets and Systems, 24 August 2017. Scholar
  9. 9.
    Kouchakinejad, F., Šipošová, A.: On some transformations of fuzzy measures. Tatra Mt. Math. Publ. 69, 75–86 (2017)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bronevich, A.G.: On the closure of families of fuzzy measures under eventwise aggregations. Fuzzy Sets Syst. 153(1), 45–70 (2005)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia
  2. 2.JSC “Research and Design Institute for Information Technology, Signalling and Telecommunications on Railway Transport”MoscowRussia

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