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An Analog of the Bondareva-Shapley Theorem I. The Non-Emptiness of the Core of a Fuzzy Game


This paper deals with a generalization of the famous Bondareva-Shapley theorem [1, 9] on the core of TU cooperative games to the case of fuzzy blocking. The suggested approach is based on the concept of a balanced collection of fuzzy coalitions. Introduced by the author, this extension of the classical balanced collection of standard coalitions yields a natural analog of balancedness for the so-called fuzzy TU cooperative games. As established below, the general balancedness is a necessary and sufficient condition for the non-emptiness of the core of fuzzy TU cooperative games. The non-emptiness criterion of the core is further refined using the classical Helly's theorem on the intersection of convex sets. The S*-representation of a fuzzy game is studied, which simplifies the existence conditions for non-blocking imputations of this game in a series of cases.

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  1. 1.

    Bondareva, O.N., The Theory of the Core of an n-Player Game, Vest. Leningrad. Gos. Univ., Ser. Mat., Mekh., Astronom., 1962, vol. 13, no. 3, pp. 141–142

    Google Scholar 

  2. 2.

    Vasil'ev, V.A., An Extension of Scarf Theorem on the Nonemptiness of the Core, Preprint of Sobolev Math. Inst., Novosibirsk, 2012, no. 283, p. 41.

    Google Scholar 

  3. 3.

    Pecherskii, S.L. and Yanovskaya, E.B., Kooperativnye igry: resheniya i aksiomy (Cooperative Games: Solutions and Axioms), St. Petersburg: Europ. Univ., 2004.

    Google Scholar 

  4. 4.

    Rockafellar, R.T., Convex Analysis, Princeton: Princeton Univ. Press, 1970. Translated under the title Vypuklyi analiz, Moscow: Mir, 1973.

    Book  MATH  Google Scholar 

  5. 5.

    Ekeland, I., Éléments d'économie math'ematique, Paris: Hermann, 1979. Translated under the title Elementy matematicheskoi ekonomiki, Moscow: Mir, 1983.

    MATH  Google Scholar 

  6. 6.

    Aubin, J.-P., Optima and Equilibria, Berlin: Springer-Verlag, 1993.

    Book  MATH  Google Scholar 

  7. 7.

    Owen, G., Multilinear Extensions of Games, J. Manage. Sci., 1972, vol. 18, no. 5, pp. 64–79.

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Peleg, B. and Sudhölter, P., Introduction to the Theory of Cooperative Games, Boston: Kluwer, 2003.

    Book  MATH  Google Scholar 

  9. 9.

    Shapley, L.S., On Balanced Sets and Cores, Naval Res. Logist. Quart., 1967, vol. 14, no. 4, pp. 453–460.

    Article  Google Scholar 

  10. 10.

    Vasil'ev, V.A., A Fuzzy-Core Extension of Scarf Theorem and Related Topics, in Contributions to Game Theory and Management, vol. VIII: Collected Papers of the 8th Int. Conf. “Game Theory and Management,” Petrosyan, L.A. and Zenkevich, N.A., Eds., St. Petersburg: St. Petersburg Gos. Univ., 2015, pp. 300–314.

    Google Scholar 

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This work was supported by the Russian Foundation for Basic Research, project no. 16-06-00101.

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Correspondence to V. A. Vasil′ev.

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Russian Text © The Author(s), 2017, published in Matematicheskaya Teoriya Igr i Ee Prilozheniya, 2017, No. 1, pp. 3–26.

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Vasil′ev, V.A. An Analog of the Bondareva-Shapley Theorem I. The Non-Emptiness of the Core of a Fuzzy Game. Autom Remote Control 80, 1148–1163 (2019).

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  • fuzzy cooperative game
  • balanced family of fuzzy coalitions
  • V -balancedness
  • the core of a fuzzy game