The Shapley value of fuzzy bi-eooperative game is developed based on the conventional Shapley value of bi-cooperative game. From the viewpoint that the players can participate in the coalitions to a certain extent and...The Shapley value of fuzzy bi-eooperative game is developed based on the conventional Shapley value of bi-cooperative game. From the viewpoint that the players can participate in the coalitions to a certain extent and there are at least two independent cooperative projects for every player to choose, Shapley value which is introduced by Grabisch is extended to the case of fuzzy bi-cooperative game by Choquet integral. Moreover, the explicit fuzzy Shapley value is given. The explicit fuzzy Shapley function can be used to allocate the profits among players in supply-chain under the competitive and uncertain environment.展开更多
Fuzzy Shapley values are developed based on classical Shapley values and used to allocate profit among partners in virtual enterprises （VE）. Axioms of the classical Shapley value are extended to Shapley values with ...Fuzzy Shapley values are developed based on classical Shapley values and used to allocate profit among partners in virtual enterprises （VE）. Axioms of the classical Shapley value are extended to Shapley values with fuzzy payoffs by using fuzzy sets theory. Fuzzy Shapley function is defined based on these extended axioms. From the viewpoint the allocation for each partner should be a crisp value rather a fuzzy membership function at the end of cooperation, a crisp allocation scheme based on fuzzy Shapley values is proposed.展开更多
In a cooperative transferable utilities game, the allocation of the win of the grand coalition is an Egalitarian Allocation, if this win is divided into equal parts among all players. The Inverse Set relative to the S...In a cooperative transferable utilities game, the allocation of the win of the grand coalition is an Egalitarian Allocation, if this win is divided into equal parts among all players. The Inverse Set relative to the Shapley Value of a game is a set of games in which the Shapley Value is the same as the initial one. In the Inverse Set, we determined a family of games for which the Shapley Value is also a coalitional rational value. The Egalitarian Allocation of the game is efficient, so that in the set called the Inverse Set relative to the Shapley Value, the allocation is the same as the initial one, but may not be coalitional rational. In this paper, we shall find out in the same family of the Inverse Set, a subfamily of games with the Egalitarian Allocation is also a coalitional rational value. We show some relationship between the two sets of games, where our values are coalitional rational. Finally, we shall discuss the possibility that our procedure may be used for solving a very similar problem for other efficient values. Numerical examples show the procedure to get solutions for the efficient values.展开更多
Shapley value is one of the most fundamental concepts in cooperative games.This paper investigates the calculation of the Shapley value for cooperative games and establishes a new formula via carrier.Firstly,a necessa...Shapley value is one of the most fundamental concepts in cooperative games.This paper investigates the calculation of the Shapley value for cooperative games and establishes a new formula via carrier.Firstly,a necessary and sufficient condition is presented for the verification of carrier,based on which an algorithm is worked out to find the unique minimum carrier.Secondly,by virtue of the properties of minimum carrier,it is proved that the profit allocated to dummy players(players which do not belong to the minimum carrier)is zero,and the profit allocated to players in minimum carrier is only determined by the minimum carrier.Then,a new formula of the Shapley value is presented,which greatly reduces the computational complexity of the original formula,and shows that the Shapley value only depends on the minimum carrier.Finally,based on the semi-tensor product(STP)of matrices,the obtained new formula is converted into an equivalent algebraic form,which makes the new formula convenient for calculation via MATLAB.展开更多
Fuzzy Shapley values are developed based on conventional Shapley value. This kind of fuzzy cooperative games admit the representation of rates of players＇ participation to each coalition. And they can be applicable t...Fuzzy Shapley values are developed based on conventional Shapley value. This kind of fuzzy cooperative games admit the representation of rates of players＇ participation to each coalition. And they can be applicable to both supperadditive and subadditvie cooperative games while other kinds of fuzzy cooperative games can only be superadditive. An explicit form of the Shapley function on fuzzy games with λ-fuzzy measure was also proposed.展开更多
基金Sponsored by the National Natural Science Foundation of China(70771010)the Second Phase of "985 Project" of China (107008200400024)the Graduate Student’s Science and Technology Innovation Project of Beijing Institute of Technology (GB200818)
文摘The Shapley value of fuzzy bi-eooperative game is developed based on the conventional Shapley value of bi-cooperative game. From the viewpoint that the players can participate in the coalitions to a certain extent and there are at least two independent cooperative projects for every player to choose, Shapley value which is introduced by Grabisch is extended to the case of fuzzy bi-cooperative game by Choquet integral. Moreover, the explicit fuzzy Shapley value is given. The explicit fuzzy Shapley function can be used to allocate the profits among players in supply-chain under the competitive and uncertain environment.
基金the National Natural Science Foundation of China (70471063 , 70171036)the Second Phase of "985"Project of China(107008200400024) the Main/Subject Project of Beijing of China(XK100070534)
文摘Fuzzy Shapley values are developed based on classical Shapley values and used to allocate profit among partners in virtual enterprises （VE）. Axioms of the classical Shapley value are extended to Shapley values with fuzzy payoffs by using fuzzy sets theory. Fuzzy Shapley function is defined based on these extended axioms. From the viewpoint the allocation for each partner should be a crisp value rather a fuzzy membership function at the end of cooperation, a crisp allocation scheme based on fuzzy Shapley values is proposed.
文摘In a cooperative transferable utilities game, the allocation of the win of the grand coalition is an Egalitarian Allocation, if this win is divided into equal parts among all players. The Inverse Set relative to the Shapley Value of a game is a set of games in which the Shapley Value is the same as the initial one. In the Inverse Set, we determined a family of games for which the Shapley Value is also a coalitional rational value. The Egalitarian Allocation of the game is efficient, so that in the set called the Inverse Set relative to the Shapley Value, the allocation is the same as the initial one, but may not be coalitional rational. In this paper, we shall find out in the same family of the Inverse Set, a subfamily of games with the Egalitarian Allocation is also a coalitional rational value. We show some relationship between the two sets of games, where our values are coalitional rational. Finally, we shall discuss the possibility that our procedure may be used for solving a very similar problem for other efficient values. Numerical examples show the procedure to get solutions for the efficient values.
基金supported by the National Natural Science Foundation of China(No.62073202,No.61873150)the Young Experts of Taishan Scholar Project(No.tsqn201909076)the Natural Science Fund for Distinguished Young Scholars of Shandong Province(No.JQ201613).
文摘Shapley value is one of the most fundamental concepts in cooperative games.This paper investigates the calculation of the Shapley value for cooperative games and establishes a new formula via carrier.Firstly,a necessary and sufficient condition is presented for the verification of carrier,based on which an algorithm is worked out to find the unique minimum carrier.Secondly,by virtue of the properties of minimum carrier,it is proved that the profit allocated to dummy players(players which do not belong to the minimum carrier)is zero,and the profit allocated to players in minimum carrier is only determined by the minimum carrier.Then,a new formula of the Shapley value is presented,which greatly reduces the computational complexity of the original formula,and shows that the Shapley value only depends on the minimum carrier.Finally,based on the semi-tensor product(STP)of matrices,the obtained new formula is converted into an equivalent algebraic form,which makes the new formula convenient for calculation via MATLAB.
基金the National Natural Science Foundation of China(70771010)the Second Phase of"985 Project"of China (107008200400024)the Graduate Student s Science and Technology Innovation Project of Beijing Institute of Technology (GB200818)
文摘Fuzzy Shapley values are developed based on conventional Shapley value. This kind of fuzzy cooperative games admit the representation of rates of players＇ participation to each coalition. And they can be applicable to both supperadditive and subadditvie cooperative games while other kinds of fuzzy cooperative games can only be superadditive. An explicit form of the Shapley function on fuzzy games with λ-fuzzy measure was also proposed.
