Advertisement

Cybernetics and Systems Analysis

, Volume 53, Issue 3, pp 432–440 | Cite as

Shapley Value of a Cooperative Game with Fuzzy Set of Feasible Coalitions

  • S. O. MashchenkoEmail author
  • V. I. Morenets
Article
  • 42 Downloads

Abstract

The paper investigates Shapley value of a cooperative game with fuzzy set of feasible coalitions. It is shown that the set of its values is a type 2 fuzzy set (a fuzzy set whose membership function takes fuzzy values) of special type. Furthermore, the corresponding membership function is given. Elements of the support of this set are defined as particular Shapley values. The authors also propose the procedure of constructing these elements with maximum reliability of their membership and reliability of non-membership, not exceeding a given threshold.

Keywords

fuzzy set type 2 fuzzy set Shapley value cooperative games 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. B. Myerson, “Graphs and cooperation in games,” Mathematics of Operations Research, Vol. 2, 225–229 (1977).MathSciNetCrossRefGoogle Scholar
  2. 2.
    P. Borm, G. Owen, and S. Tijs, “On the position value for communication situations,” SIAM J. on Discrete Mathematics, Vol. 5, 305–320 (1992).MathSciNetCrossRefGoogle Scholar
  3. 3.
    E. Algaba, J. M. Bilbao, P. Borm, and J. J. Lo’pez, “The Myerson value for union stable structure,” Mathematical Methods of Operations Research, Vol. 54, 359–371 (2001).MathSciNetCrossRefGoogle Scholar
  4. 4.
    E. Algaba, J. M. Bilbao, R. Van Den Brink, and A. Jiménez-Losada, “An axiomatization of the Banzhaf value for cooperative games on antimatroids,” Mathematical Methods of Operations Research, Vol. 59, 147–166 (2004).MathSciNetCrossRefGoogle Scholar
  5. 5.
    I. V. Katsev and E. B. Yanovskaya, “Solutions to cooperative games, intermediate between pre-k- and pre-n-kernels,” Upravlenie Bol’shimi Sistemami, Vol. 26-1, 32–54 (2009).zbMATHGoogle Scholar
  6. 6.
    A. Charnes, B. Golany, M. Keane, and J. Rousseau, “Extremal principle solutions of games in characteristic function form: Core, Chebyshev and Shapley value generalizations,” in: Econometrics of Planning and Efficiency. Kluwer Academic Publisher, Dordrecht (1988), pp. 123–133.Google Scholar
  7. 7.
    L. M. Ruiz, F. Valenciano, and J. M. Zarzuelo, “The family of least square values for transferable utility games,” Games and Economic Behavior, Vol. 24, 109–130 (1998).MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Derks and H. Peters, “A Shapley value for games with restricted coalitions,” Int. J. Game Theory, Vol. 21, 351–360 (1993).MathSciNetCrossRefGoogle Scholar
  9. 9.
    I. V. Katsev, “The least square values for games with restricted cooperation,” July 22, 2013.Google Scholar
  10. 10.
    L. S. Shapley, “A value for N-person games,” in: Contributions to the Theory of Games, Vol. II, Princeton University Press, Princeton (1953), pp. 307–317.Google Scholar
  11. 11.
    S. O. Mashchenko, “A mathematical programming problem with the fuzzy set of indices of constraints,” Cybern. Syst. Analysis, Vol. 49, No. 1, 62–68 (2013).MathSciNetCrossRefGoogle Scholar
  12. 12.
    N. V. Semenova, L. N. Kolechkina, and A. M. Nagirna, “Vector optimization problems with linear criteria over a fuzzy combinatorial set of alternatives,” Cybern. Syst. Analysis, Vol. 47, No. 2, 250–259 (2011).MathSciNetCrossRefGoogle Scholar
  13. 13.
    L. A. Zadeh, “The concept of a linguistic variable and its application to approximate reasoning-I,” Inf. Sci., Vol. 8, 199–249 (1975).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Taras Shevchenko National University of KyivKyivUkraine

Personalised recommendations