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**Cooperative fuzzy games extended from ordinary cooperative games with restrictions on coalitions.**
*(English)*
Zbl 1249.91010

Summary: Cooperative games are very useful in considering profit allocation among multiple decision makers who cooperate with each other. In order to deal with cooperative games in practical situations, however, we have to deal with two additional factors. One is some restrictions on coalitions which has been taken into consideration using feasibility of coalitions; the other is the partial cooperation of players. In order to describe this second factor, we consider fuzzy coalitions which permit partial participation in a coalition to a player. In this paper, we take both of these factors into account, namely, we analyze and discuss cooperative fuzzy games extended from ordinary cooperative games with restrictions on coalitions under two approaches. For the purpose of a comparison of these two approaches, we define two special classes of extensions called \(U\)-extensions which satisfy linearity and \(W\)-extensions which satisfy \(U\)-extensions, and two additional conditions, namely, restriction invariance and monotonicity. Finally, we show sufficient conditions under which the games obtained under these two approaches coincide.

### MSC:

91A12 | Cooperative games |

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\textit{A. Moritani} et al., Kybernetika 42, No. 4, 461--473 (2006; Zbl 1249.91010)

### References:

[1] | Algaba E., Bilbao J. M., Lopez J.: A unified approach to restricted games. Theory and Decision 50 (2001), 333-345 · Zbl 1039.91003 · doi:10.1023/A:1010344404281 |

[2] | Aubin J. P.: Mathematical Methods of Game and Economic Theory. North-Holland, Amsterdam 1979 · Zbl 1152.91005 |

[3] | Bilbao J. M.: Cooperative Games on Combinatorial Structures. Kluwer Academic Publishers, Boston 2000 · Zbl 0983.91013 |

[4] | Brânzei R.: Convex fuzzy games and partition monotonic allocation schema. Fuzzy Sets and Systems 139 (2003), 267-281 · Zbl 1065.91006 · doi:10.1016/S0165-0114(02)00510-9 |

[5] | Lovász L.: Submodular functions and convexity. Mathematical Programming: The State of the Art (A. Bachem et al., Springer-Verlag, Berlin 1983, pp. 235-257 · Zbl 0566.90060 |

[6] | Mareš M.: Fuzzy Cooperative Games. Physica-Verlag, Heidelberg 2001 · Zbl 1037.91007 |

[7] | Moritani A., Tanino T., Kuroki, K., Tatsumi K.: Cooperative fuzzy games with restrictions on coalitions. Proc. Third Internat. Conference on Nonlinear Analysis and Convex Analysis 2004, pp. 323-345 · Zbl 1149.91016 |

[8] | Nishizaki I., Sakawa M.: Fuzzy and Multiobjective Games for Conflict Resolution. Physical-Verlag, Heidelberg 2001 · Zbl 0973.91001 |

[9] | Owen G.: Multilinear extensions of games. Management Sci. 18 (1972), 64-79 · Zbl 0239.90049 · doi:10.1287/mnsc.18.5.64 |

[10] | Tanino T.: Cooperative fuzzy games as extensions of ordinary cooperative games. Proc. 7th Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty (H. Noguchi, H. Ishii, M. Inuiguchi, Awaji Yumebutai ICC 2004, pp. 26-31 |

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