Automation and Remote Control

, Volume 80, Issue 6, pp 1148–1163 | Cite as

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

  • V. A. Vasil′evEmail author
Mathematical Game Theory and Applications


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.


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


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


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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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