The Stable Set and Weak Stable Set For n-person Repeated Fuzzy Cooperative Games

Zuofeng Gao Department of Science, Yanshan University, Qinhuangdao 066004, China Suting Zhang (Corresponding author) Department of Science, Yanshan University, Qinhuangdao 066004, China E-mail: zhangsuting2002@yahoo.com.cn Hongxin Bai Department of Science, Yanshan University, Qinhuangdao 066004, China Chunyan Han Department of Science, Yanshan University, Qinhuangdao 066004, China Sheng Zhao Department of Science, Yanshan University, Qinhuangdao 066004, China Supported by the Foundation for the natural science of He Bei province of China (A2005000301) Abstract In this paper,based on the fuzzy games,we define the imputation sequences of the n -person repeated games,and the domination,weak domination for the imputation sequences.Further,based on this theory,we define the core ,the weak core,the stable set,and the weak stable set of the n -person repeated fuzzy cooperative games.At last,some properties of the stable set and the weak stable set are given.


Introduction
In the former literature about the cooperative game, how to allocate the total benefit among the the players have been studied widely.In 1974, Aubin, JP introduced Fuzzy games, and combine cooperative game with fuzzy coalition for the first time (Aubin, 1981, PP. 1-13).And in 1999, Jorge Oviedo developed the n -person repeated cooperative game, the core of this game was discussed in his paper.And based on the these theory, we develoepde n -person Repeated fuzzy cooperative games.Further,we define the weak core,the stable set and the weak stable set for n -person Repeated fuzzy cooperative games.Thus,we extend the study for the solution of Repedted fuzzy games.

Basic definitions
be the set whose elements are called players. ,where is a fuzzy cooperative game with player set N .We call v fuzzy game in short.
We dnote by N FG the set of Fuzzy Games.And ) ,... , ( is a fuzzy coalition.The i th coordinate is called the participation degree of player i to fuzzy coalition d .And denote N F all the possible fuzzy coalitions.
We denote i e n R ∈ the i th coordinate is 1, and 0 otherwise.And we denote We repedte the fuzzy game v m times( m may be ∞ ),and we denote ( So we denote the core of the fuzzy cooperative game as be the coalition sequences when t from 0 to t .We denote )) ( ( = the all possible coalitions when t th stage.where ) (u L is all the possible fuzzy coalitions.We denote H the set of all the coalition sequences.
, and Then we call x dominates y through θ ( y .where t x is the imputation of t th stage game.thenwe define the core of the repeated fuzzy cooperative game as: Definition 7. Let y x, be two imputations of the repeated fuzzy cooperative game.θ is a coalition sequence,if it satisfies Then we call x dominate y weakly through θ , we denote by  , there is no weak dominationship between x and y .(innerstability) , there is a coalition sequences θ and an imputation sequences L y ∈ ,satisfy: x y w θ φ .(outerstability)

The properties of the stable set for n -person repeated fuzzy cooperative game
Theorem 1.To the same n -person repeated fuzzy cooperative game, ,we know from the definition of the weak core that there exist no coalition sequence θ and imputation sequence ỹ ,satisfy: ,then there exist a coalition θ and an imputation sequence y ,and ,we make sum to each side respectively.So we have Theorem 2. If the stable set of the n -person repeated fuzzy cooperative game L Φ ≠ , then ,then there exist coalition sequence θ and imputation sequence L y ∈ ,satisfy: , then there exist a coalitionθ and an imputation sequence ) ( ~w I x ∈ ,and To (6),we make sum in both sides respecr to t ,and we have .
All of the set of imputation sequences for repeated fuzzy cooperative game,if they are not dominated,then we call them the weak core of the repeated fuzzy cooperative game.And we denote by) stable set and the weak stable set for n -person repeated fuzzy cooperative games Definition 9. Let L be the set of some imputation sequences of repeated fuzzy game

−
We make sum in both sides respect to t, and we have ≥ th period,we call it the stage game of the n -person repeated fuzzy game.In order to make the stage-game and the repedted-game characteristic function to be measured in the same unities,we use a discount factor t t Definition 3. Let H be the set of coalition sequence,and w is the characteristic function of repeated fuzzy cooperative ~=θ Definition 10.Let L are the set of some imputation sequences of repeated fuzzy cooperative game,i.e.L