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

Abstract

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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Acknowledgments

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). https://doi.org/10.1134/S0005117919060122

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Keywords

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