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

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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.

Original languageEnglish
Pages (from-to)1148-1163
Number of pages16
JournalAutomation and Remote Control
Volume80
Issue number6
DOIs
Publication statusPublished - 1 Jun 2019

Keywords

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

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