Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2014, Special Issue (2013), Article ID 639684, 9 pages.

Integral Representations of Cooperative Game with Fuzzy Coalitions

Jinhui Pang and Xiang Chen

Full-text: Open access

Abstract

Classical extensions of fuzzy game models are based on various integrals, such as Butnariu game and Tsurumi game. A new class of symmetric extension of fuzzy game with fuzzy coalition variables is put forward with Concave integral, where players’ expected values are on a partial set of coalitions. Some representations and properties of some limited models are compared in this paper. The explicit formula of characteristic function determined by coalition variables is given. Moreover, a calculation approach of imputations is discussed in detail. The new game could be regarded as a general form of cooperative game. Furthermore, the fuzzy game introduced by Tsurumi is a special case of the proposed game when game is convex.

Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2013), Article ID 639684, 9 pages.

Dates
First available in Project Euclid: 1 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1412177643

Digital Object Identifier
doi:10.1155/2014/639684

Mathematical Reviews number (MathSciNet)
MR3178966

Zentralblatt MATH identifier
07010706

Citation

Pang, Jinhui; Chen, Xiang. Integral Representations of Cooperative Game with Fuzzy Coalitions. J. Appl. Math. 2014, Special Issue (2013), Article ID 639684, 9 pages. doi:10.1155/2014/639684. https://projecteuclid.org/euclid.jam/1412177643


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References

  • D. Granot, “Cooperative games in stochastic characteristic function form,” Management Science, vol. 23, no. 6, pp. 621–630, 1977.
  • F. R. Fernández, J. Puerto, and M. J. Zafra, “Cores of stochastic cooperative games with stochastic orders,” International Game Theory Review, vol. 4, no. 3, pp. 265–280, 2002.
  • L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, no. 3, pp. 338–353, 1965.
  • M. Mareš, “Weak arithmetics of fuzzy numbers,” Fuzzy Sets and Systems, vol. 91, no. 2, pp. 143–153, 1997.
  • D. Dubois and H. Prade, Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Stes, Kluver Academic, Boston, Mass, USA, 2000.
  • J. P. Aubin, “Coeur et valeur des jeux flous à paiements latéraux,” Comptes Rendus Hebdomadaires Des Séances De 1'Académie Des Sciences A, vol. 279, pp. 891–894, 1974.
  • R. Branzei, D. Dimitrov, and S. Tijs, Models in Cooperative Game Theory: Crisp, Fuzzy and Multichoice Games, Springer, Berlin, Germany, 2005.
  • D. Butnariu, “Fuzzy games: a description of the concept,” Fuzzy Sets and Systems, vol. 1, no. 3, pp. 181–192, 1978.
  • M. Tsurumi, T. Tanino, and M. Inuiguchi, “A Shapley function on a class of cooperative fuzzy games,” European Journal of Operational Research, vol. 129, no. 3, pp. 596–618, 2001.
  • S. Borkotokey, “Cooperative games with fuzzy coalitions and fuzzy characteristic functions,” Fuzzy Sets and Systems, vol. 159, no. 2, pp. 138–151, 2008.
  • E. Lehrer, “A new integral for capacities,” Economic Theory, vol. 39, no. 1, pp. 157–176, 2009.
  • M. Sugeno and T. Murofushi, “Choquet's integral as an integral form for a general class of fuzzy measures,” in Proceedings of the 2nd IFSA Congress, vol. 1, pp. 408–411, 1987.
  • Y. Azrieli and E. Lehrer, “Extendable cooperative games,” Journal of Public Econ Theory, vol. 9, no. 6, pp. 1069–1078, 2007.
  • D. Butnariu, “Stability and Shapley value for an $n$-persons fuzzy game,” Fuzzy Sets and Systems, vol. 4, no. 1, pp. 63–72, 1980.
  • D. Butnariu and T. Kroupa, “Shapley mappings and the cumulative value for $n$-person games with fuzzy coalitions,” European Journal of Operational Research, vol. 186, no. 1, pp. 288–299, 2008. \endinput