The Shapley Values on Fuzzy Coalition Games with Concave Integral Form

A generalized form of a cooperative game with fuzzy coalition variables is proposed. The character function of the new game is described by the Concave integral, which allows players to assign their preferred expected values only to some coalitions. It is shown that the new game will degenerate into the Tsurumi fuzzy game when it is convex. The Shapley values of the proposed game have been investigated in detail and their simple calculation formula is given by a linear aggregation of the Shapley values on subdecompositions crisp coalitions.


Introduction
There are many solution concepts in crisp cooperative game which are allocations for the profits of a cooperative game, such as Shapley value [1], Banzhaf value [2], and -value [3]. However, there are many possible or vague factors that might influence players' decisions such that the cooperation is full of uncertainty. As a result, these solutions are not suitable to games with vague factors.
In fuzzy environment, fuzzy cooperative games theory extends crisp games' results by using fuzzy set theory (such as Zadeh [4], Mareš [5], and Dubois and Prade [6]). It also focuses on the problems of how to express fuzzy coalitions, how to evaluate fuzzy payoffs, and how to distribute it among players.
Nowadays the fuzzy games mainly consist of two types. One is games with fuzzy coalitions, in which players partly take part in a coalition, but exact profits of fuzzy coalitions can be gained. For examples, Aubin [7] and Butnariu [8] defined a fuzzy game whose character function was the aggregated worth of the coalitions profits with respect to players' participation degree. The other is games with fuzzy payoffs, a game with fuzzy payoffs but the coalitions are still crisp game coalitions. For examples, Mareš [9,10] and Mareš and Vlach [11] suggested that the values assigned to coalitions were fuzzy quantities even though the domain of the character function of fuzzy games remained to be accurate like crisp games.
In games with fuzzy coalitions literature, Tsurumi et al. [12] pointed out shortcomings of the game proposed by Aubin and Butnariu and proposed a class of fuzzy games by the Choquet integral. Borkotokey [13] took a cooperative game with fuzzy coalitions and fuzzy character functions into consideration simultaneously, where character functions were fuzzy value which mapped the set of real numbers to the closed interval [0, 1].
At present, the Shapley values of fuzzy games have been studied by many scholars after Butnariu [14] who firstly defined fuzzy Shapley function on a limited class of fuzzy games with proportional values. But it was neither monotone nondecreasing nor continuous with regard to rates of players' participation. Later, Butnariu and Kroupa [15] similarly gave the Shapley values on fuzzy games with weighted function. Tsurumi et al. also discussed the Shapley values on their fuzzy game, which was both monotone nondecreasing and continuous with regard to players' participation rates because of the advantageous properties of the Choquet integral. Borkotokey [13] discussed its Shapley value on games whose payoffs and coalitions are both fuzzy.

Crisp Cooperative Game and the Shapley Value
A finite set of players = {1, 2, . . . , } is a nonempty set, in which players may take part in different feasible subcoalition of . The greatest coalition is the grand set and the smallest coalition is . The power set ( ) is the family of all crisp subcoalitions of ⊆ . A crisp cooperative game on player set is denoted by V where the character function V : ( ) → + ∪ {0} with V( ) = 0. For any a ∈ ( ), the worth V( ) assigning to can be regarded as the maximal worth or cost of the coalition which is obtained when players in work together. We take the notation ( , V) to express the class of all crisp games with player set .
For a nonempty subset ∈ ( ), the simple games are defined by And each cooperative game V ∈ ( , V) can be represented by as follows: where The game V ∈ ( , V) is said to be convex when The convex game V ∈ ( , V) is said to be superadditive, if any disjoint crisp coalitions and satisfy The notation 0 ( , V) represents all superadditave crisp cooperative games. For the game V ∈ ( , V), players and are said to be symmetric, if ⊆ \ { , } such that The player ∈ is a dummy player of the game, if for any The player ∈ is a null player in a coalition for a game It is obvious that V ∈ 0 ( , V) when imputation for a crisp cooperative game V ∈ ( , V) is nonempty.
∈ ( ); then ∈ ( ) is called a carrier in a coalition for a game V if If the set of all carriers in coalition for V is denoted by ( | V), then A well-known solution for cooperative game V ∈ 0 ( , V), the Shapley value is a mathematical expectation on [1] defined the function satisfying the following 4 axioms.

Definition 3. A function
: 0 ( , V) → ( + ) ( ) is said to be a Shapley value on 0 ( , V) if it satisfies the following four axioms.
where (V)( ) is the th element of (V)( ) ∈ + . Axiom 2. If V ∈ 0 ( , V), ∈ ( ), and ∈ ( | V), then Shapley [1] also gave the uniquely explicit form of a Shapley value on 0 ( , V) which was obtained by extending the Shapley value for the grand coalition as below.

Fuzzy Coalition Games
Let us start by presenting some general definitions related to fuzzy coalition games.

Basic Concepts.
We consider cooperative fuzzy games with the player set = {1, 2, . . . , }. A fuzzy coalitioñis a fuzzy subset of the finite set , which is a vector = ( 1 , 2 , . . . , ) where ∈ [0, 1] describes the membership grade of player in the fuzzy coalitioñ. We note that a different coalition has different vector = ( 1 , 2 , . . . , ), so we also call it as fuzzy coalition variable. If element = 1 when fully take part iñand others = 0, then the coalitioñis a crisp coalition.
In this paper, we assume that every fuzzy coalition variables maps into the lattice ([0, 1], ∧, ∨), where ∧ and ∨ are the minimum and maximum operators, respectively. For any fuzzy coalitioñ,̃∈ ( ), we adopt the usual definition of the union and intersection of fuzzy subsets given by Similarly to crisp convex game, for all̃,̃∈ ( ), V ∈ ( ,Ṽ) is said to be fuzzy convex, if it satisfies andṼ ∈ ( ,Ṽ) is said to be superadditive such that with̃∩̃= .

Definition 6.
Let V ∈ ( ,Ṽ); the player ∈ is said to be a dummy player on fuzzy coalitioñ∈ ( ) with fuzzy coalition variable = ( 1 , 2 , . . . , ), if for any fuzzy coalitioñ, and if The player is called a null player on fuzzy coalitioñ.
is called a fuzzy carrier in a coalitioñif for any ∀̃∈ (̃) such that The set of all carriers in fuzzy coalitioñfor V ∈ ( ,Ṽ) is denoted by (̃| V); it is obvious that

The Present Forms for Fuzzy Coalition Games.
In the field of fuzzy cooperative games with fuzzy coalitions, there were several definitions given by aggregating function on fuzzy coalition variables, such as Butnariu game, Butnariu and Kroupa game, and Tsurumi game. In Butnariu game, V(̃) was an aggregated worth of the crisp coalitions̃where the players have the same participation level , defined by It is obvious that the game value is a linear aggregation function which is a weighted average on the sets with the same participation levels, namely, a fuzzy game with proportional values as the associated crisp game. We denote the fuzzy game with proportional values as the notation ( ). It is a one-toone correspondence between a crisp game and a fuzzy game with proportional values.
Similarly, it is also a simple linear aggregation function which cannot embody the interaction among players with different participation levels. Moreover, if ( ) = implies that the game is equivalent to the proportional game, we denote it as ( ).
Tsurumi et al. introduced another form definition based on the Choquet integral, which was not only monotone nondecreasing but also continuous with regard to rates of players' participation. Let̃∈ ( ), (̃) = { | > 0, ∈ } and rearrange elements in (̃) such that 0 = ℎ 0 ≤ ℎ 1 < ℎ 2 < ⋅ ⋅ ⋅ < ℎ (̃) ; then for anỹ∈ ( ), a game V : ( ) → is defined by where (̃) is the cardinality of (̃). The fuzzy game given by Tsurumi et al. is simply denoted by Ch ( ). It is apparent that the fuzzy game V ∈ Ch ( ) is a Choquet integral of the function ℎ with respect to V derived from level set. We note that ℎ −1 < ℎ implies that [̃] ℎ ⊆ [̃] ℎ −1 , so the worth of coalitioñis the maximum sum on all subsets which is an including chain.
Suppose that the fuzzy coalition = (1, 0.4, 0.6); rearrange it as 0.4 < 0.6 < 1; thus, the value of this fuzzy coalition is evaluated by (27) as follows: However, there are another linear aggregation values which are greater than that of Tsurumi's form. For example, we make a linear sum as Hence, Tsurumi's class cannot be considered as an optimal product on ( ), for fuzzy variables assign subjective expected values to some coalitions but not to all in Tsurumi game. As a result Tsurumi fuzzy game is not suitable in some situations.

A Class of Fuzzy Coalition
Games with the Concave Integral. As mentioned above, the present forms for fuzzy coalition games were only limited to some special games and will be invalid in many game situations. Next, we will consider another extended game with fuzzy coalitions, that is, the fuzzy game with the Concave integral, where Tsurumi game can be taken as a special case as the proposed new game. Firstly, we recall the fuzzy capacity and the Concave integral.
Let be a finite set (| | = ); a capacity over is a function : Journal of Applied Mathematics 5 a function : → and a random variable is nonnegative if ≥ 0 for every ∈ .
We proposed fuzzy capacity game concept defined by the following way.
Definition 11. Let̃∈ ( ); the pair (V,̃) is said to be an additive fuzzy capacity game, for every fuzzy coalition vari- . It is not hard to see that the limited game given by Butnariu is an additive fuzzy capacity game. A fuzzy capacity game assigns values (subjective expected value) to fuzzy coalition random variables, in which players express their preferences of some coalitions but not of all. The fuzzy coali-tioñmight contain only extreme or discrete points of the domain of (̃) wherẽ⊆ ( ) such as (1, 1, . . . , 1) and (0, 0, . . . , 0), therefore V may be partially nonadditive or nonadditive on its domains.
The integral aggregates all available fuzzy coalitions, including individual assessments of the likelihood of events and expected values of variables, into a comprehensive value. By this value, the players reevaluate their likely coalitions or expected values on random coalition variables.
Let be a random variable; a subdecomposition of is a finite summation ∑ ⊆ that satisfies Definition 12. Let V ∈ ( ,Ṽ) be a fuzzy capacity game, let̃∈ ( ) be a random fuzzy coalition with nonnegative variable = ( 1 , 2 , . . . , ), and define a game V Cav : where the minimum is taken all over concave and homogeneous functions : + → and (1 ) ≥ V( ) for every ⊆ , and 1 is the indicator of which is the random variable that takes the value 1 over and the value 0, otherwise.
By Definition 12, V Cav (̃) can be gained by the values on crisp coalitions which correspond with subdecompositions of .
We denote all fuzzy games defined by the concave integral as Cav ( ).
From the above definition, the function is defined on all over concave coalitions. It is easy to prove the following lemma.
Lemma 13. Let V ∈ ( ,Ṽ) be a fuzzy capacity game; for every random fuzzy coalitioñ∈ ( ) with nonnegative variable = ( 1 , 2 , . . . , ), The game V V : ( ) → + ∪ {0} can also be calculated by (33) It is apparent that the fuzzy game with the concave integral extends the crisp game. Hence, So We know that V Cav (̃) = ∫ Cav V is the maximum of the values ∑ =1 ( ) among all possible decompositions of̃with the coalition variable . The maximum focuses all possible decompositions rather than restrict viable decompositions like the fuzzy game given by the Choquet integral.
In the fuzzy game given by Tsurumi et al., the Choquet integral of nonnegative with respect to a capacity V is defined by ; note that = ∑ 1 is a decomposition of . That is to say that the Choquet integral is defined under the special decomposition of . By contrast, all possible decompositions are allowed in the concave integral.
By this way, it implies that ∫ Ch ≤ ∫ Cav for any .
In addition, it has been proven that ∫ Ch = ∫ Cav if and only if is convex (see [17]).

Theorem 19. Let V ∈ ( ,Ṽ) be a fuzzy capacity game; V is convex if and only if for any two nonnegative random fuzzy coalitions̃and̃with variables and , respectively,
Proof. If a fuzzy capacity game V ∈ ( ,Ṽ) is convex, then by the property of the concave integral, ∫ Conversely, if V is not convex, then there exits a non- there is at least a crisp coalition ⊆ and a nonnegative constant such that We have Similarly, where the crisp coalition ⊆ and is a nonnegative constant.

The Shapley Values on Fuzzy Coalition Games with the Concave Integral
The fuzzy Shapley value is one of the important solutions for fuzzy games. It is interesting to study the Shapley function for game V ∈ Cav ( ).

The Shapley Axioms for Games with Fuzzy
These axioms for the Shapley value are extensions of the crisp Shapley axioms and are suitable to games with fuzzy coalitions. It is unnecessary to transform the Shapley axioms to deal with our fuzzy cooperative games.
Hence, the Shapley value of player on the fuzzy coalitioñ will be Lemma 23. Let V ∈ V ( ) be a fuzzy capacity game, and let ∈ ( ) be a random fuzzy coalition with nonnegative variable = ( 1 , 2 , . . . , ); then the vector is an imputation of the fuzzy coalitioñ, where (̃)(̃) is defined as (65).
Lemma 24 suggests that on game V ∈ Cav ( ), (̃)(̃) is monotone nondecreasing with respect to fuzzy coalition variable when V ∈ Cav ( ) is convex. In fact, (̃)(̃) is also continuous with respect to fuzzy coalition variable when V ∈ Cav ( ).
The next lemma can easily be proved by the same manner as Lemma 24.
Note that the game is convex such that the Shapley 1 (̃)(̃) is the same as that of Tsurumi game.

The Shapley Values on Restricted
According to the above discussion, the existence of the Shapley values on restricted coalition is possible. It is not hard to prove the following theorem and corollary.
where is the function given in Theorem 4.

Conclusion
We have proposed an extension of fuzzy cooperative games with fuzzy coalitions by the concave integral so as to address the optimal profits from all subdecompositions of fuzzy coalitions. The proposed class of games is more realistic since it has continuity except for the other properties such as superadditivity and convexity. Meanwhile, the proposed fuzzy game is also an extension of the game with the Choquet integral form defined by Tsurumi et al.
In fuzzy game, the Shapley values of game with fuzzy coalitions are also an important solution concept. Inspired by Tsurumi et al., the general Shapley values for games with fuzzy coalitions are proposed, and the correlation with the crisp Shapley values is discussed. Further, we give the simplified expression by the crisp Shapley values. In fact, as long as the fuzzy coalition variables are given, the Shapley value of a player with a certain participation level can be completely obtained similarly to that of crisp cooperative game.