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Integral Representations of Cooperative Game with Fuzzy Coalitions
Classical extensions of fuzzy game models are based on various integrals, such as Butnariu game and Tsurumi game. A new class of symmetric extension of fuzzy game with fuzzy coalition variables is put forward with Concave integral, where players’ expected values are on a partial set of coalitions. Some representations and properties of some limited models are compared in this paper. The explicit formula of characteristic function determined by coalition variables is given. Moreover, a calculation approach of imputations is discussed in detail. The new game could be regarded as a general form of cooperative game. Furthermore, the fuzzy game introduced by Tsurumi is a special case of the proposed game when game is convex.
Cooperation in game is often full of vagueness because of players’ partial participation and players’ preference to some coalitions. Games without precise information have been investigated by stochastic framework [1, 2]. Generally speaking, probability distributions on game events cannot be obtained perfectly. It is reasonable to adopt fuzzy mathematics to model fuzzy games [3–5].
The main ways of describing cooperative game with fuzzy coalitions aim to extend crisp game theory, which is to construct a one-to-one correspondence between a crisp game and a fuzzy game. At present, there are three views of fuzzy games including games with fuzzy payoffs, games with fuzzy coalitions, and games with both fuzzy payoffs and fuzzy coalitions. Aubin  firstly studied games with fuzzy coalitions and proposed a line fuzzy game, and a review of this line fuzzy game can be found in Branzei et al. . Butnariu  extended the domain of fuzzy coalitions and defined a fuzzy game. Tsurumi et al.  introduced a class of fuzzy games with Choquet integral. In addition, Branzei et al.  have done lots of work to model fuzzy cooperative game, which was mainly defined by the associated crisp game corresponding to fuzzy game. Borkotokey  investigated a class of cooperative games with fuzzy coalitions and fuzzy characteristic function simultaneously. He proposed another class of fuzzy games different from that defined by Butnariu and Branzei, and so forth, where the characteristic value mapping the set of real numbers to the closed interval was of fuzzy quantity.
No matter what game model it is, the representation of characteristic function of games with fuzzy coalitions is an expression of integral, such as linear integral in Butnariu game and Choquet integral in Tsurumi et al. game. In this paper, several integral representations which described the fuzzy characteristic function of the cooperative game will be listed and compared mutually. It has been proved that the Concave integral is concave with respect to fuzzy capacities, which might be interpreted as uncertainty aversion . Consequently, it is a better way to model games with fuzzy coalition variables by Concave integral. We extend characteristic function in fuzzy game on a partial set of coalitions via its decompositions of corresponding crisp coalitions. The classical model is a special case of our approach.
In Section 2, we recall some basic concepts of crisp games and fuzzy games. Integral representations for several games with fuzzy coalitions are introduced in Section 3. Some properties and relationships of the several games will also be discussed in detail. We show the differences among these games by numerical examples. Section 4 studies an important case of game in which players’ expected values are on a partial set of coalitions. A new class of games which depends on decompositions of coalitions will be proposed by Concave integral. In particular, the so-called game with Choquet integral form is a special case of game with Concave integral form. Moreover, in Section 5, an imputation of the new game is investigated and a calculation approach of imputations is proposed. Finally, some conclusions appear in Section 6.
2. Crisp Cooperative Game and Fuzzy Cooperative Game
A crisp cooperative game on player set is the characteristic function with and is the worth of coalition . The class of crisp games with player set is denoted by .
The game is convex if , . is superadditive if , , . Denote all the superadditve crisp cooperative games by .
For a nonempty set , the unanimity games are defined by Each cooperative game can be represented by as follows: where .
Definition 1. An imputation for a crisp cooperative game is a vector satisfying(1),
Denote all of imputations for a crisp cooperative game as a set . When , is nonempty.
The Shapley value of player has a unique expression given by where is the cardinality of a coalition.
When , the Shapley vector .
We call a fuzzy coalition with coalition variable , where is a constant participation level of player . The set of fuzzy coalitions in is denoted by . If is a fuzzy subset of , its support is defined by and its level subset is denoted by as . If , it means that for any . The identical level set is the players set with the same participation level .
A characteristic function of cooperative game is a function such that . The real value function associates to each coalition with worth , which measures the utility of forming coalition . We take the notation as a fuzzy game on with the characteristic function .
For any fuzzy coalition , is the union of two fuzzy coalitions and , and when , the coalition variable and otherwise .
Similarly, is the intersection of two fuzzy coalitions and with the coalition variable for any .
Let ; the game is superadditive if for all and . The game is fuzzy convex if . is a fuzzy carrier in fuzzy coalition if , for all .
Let be a nonempty set and let be the power set of . The function is called a capacity on if and whenever , where indicates the weight of the elements .
Definition 2. Let be a capacity on ; the Choquet integral of a nonnegative function with respect to is defined by
The Choquet integral with a discrete capacity can be rewritten as follows: where indicates a permutation of such that .
Definition 3. Let be a capacity on . Fix a nonnegative random variable , and define
where the minimum is taken over all concave and homogeneous functions satisfying for every and is an indicator of .
It has been proved that if and only if is convex (see ).
3. Integral Representations for Some Limited Games with Fuzzy Coalitions
Butnariu  defined a characteristic function of cooperative game with proportional values which was weighted by participation level set.
Definition 4. The game is said to be a game with proportional values if and only if
The fuzzy game with proportional values corresponds to a crisp game as the associated fuzzy game. For the sake of simplicity, we will denote the fuzzy game with proportional values as .
Example 5. Let and let be a characteristic function on which is an output of joint workers; values on crisp coalitions are listed in Table 1.
Consider the fuzzy game and the fuzzy coalition corresponding to coalition variable , by Definition 4, Butnariu and Kroupa  proposed a class of games with weight function which was extension of games with proportional value.
Definition 6. The game satisfying is called a fuzzy game with weight function if and only if
where is a function with the properties and .
The set of games with weight functions is denoted by . If ; then the fuzzy game is equivalent to the game .
Example 7. Let the game be the same as Example 5, and suppose players set the weight function , ; then , , , , and
From Example 7, we know that the game value is much greater to depend on the weight function. In spite of the fact that crisp games are often considered to be monotone nondecreasing, Tsurumi et al. thought that most of this class of games are neither monotone nondecreasing nor continuous with regard to rates of players' participation. They introduced a class of fuzzy games, simply denoted by as follows.
Definition 8. Given , let and let be the cardinality of . The elements in are rewritten by the increasing order as . Then a game is said to be a fuzzy game with Choquet integral form if and only if the following holds:
for any , where .
It is apparent that Choquet integral  is an integral form for a general class of fuzzy measures, so the fuzzy game model proposed by Tsurumi is Choquet integral of the function with respect to derived from level set. Because the case that implies , so the worth of coalition is the maximum sum on all subsets which make an including chain.
A fuzzy game with Choquet integral form has the following properties .
Proposition 10. Let , for any and ; then the following holds:
Proposition 11. Let ; define the distance for any ; then is continuous.
Proposition 12. Let and such that if and only if
Borkotokey  defined the following extended class of fuzzy games, when both coalitions and expectation are fuzzy.
Definition 13. Let and let be a real number, for any a fuzzy coalition ; define a function satisfying(i)if , then where , and ,(ii)if , then .
We note that this game , is an extended fuzzy game with fuzzy coalitions and vague expectation. For a sufficiently small , is continuous for every fuzzy coalition . The corresponding class of such games is denoted by .
Example 14. We continue to consider Example 9; for any and with fuzzy coalition variable , let ; then
In game , when players’ expected values are among and , they will set their maximum contribution level to the fuzzy coalition . By contrast, the maximum contribution of keeps sufficiently close to .
Proposition 15. Let ; then is monotone nondecreasing when the game is monotone nondecreasing. That is, for every pair of fuzzy coalitions and such that it would imply
Proposition 16. Let ; then the game is(1)superadditive if when and is superadditive,(2)convex if when is convex.
4. A New Fuzzy Games with Concave Integral
As mentioned above, the present forms for game with fuzzy coalitions are only limited to some special games. Next, we will consider another extended game with fuzzy coalitions, that is, the fuzzy game with Concave integral, where Tsurumi game can be taken as a special case for the proposed new game. Firstly, we recall the fuzzy capacity and the Concave integral.
Definition 17. Let ; the pair is said to be a fuzzy capacity game if is monotonic, continuous, and there is a positive such that for every .
Definition 18. Let ; the pair is said to be an additive fuzzy capacity game, for every fuzzy coalition variable , if there is a nonnegative constant vector such that
It is obvious that Butnariu game is an additive fuzzy capacity game. However, the fuzzy coalition probably contains only some extremes or discrete points of the domain of , and . Therefore, might be partially non-additive or non-additive on its domains.
Let be a random variable; a subdecomposition of is a finite summation that satisfies where is an indicator of .
A subdecomposition is a representation of a random variable as a positive linear combination of indicators.
Remark 19. For any random variable , there are different subdecompositions of . Moreover, for maximum decomposition of , it may be different.
Example 20. Let players’ set ; define a fuzzy coalition variable on . For one subdecomposition and . We take , , and then we get
Hence, the subdecomposition is a maximum decomposition of .
On the other hand, if we consider other subdecompositions , , and ; take , , ; and then
We note that this subdecomposition is also a maximum decomposition of and .
However, let , , and .
It is not hard to find that the value of additive capacity on different maximum decomposition can be different.
Definition 21. Let be a capacity over , and let be a nonnegative random variable; define the Concave integral as where is an indicator of .
Definition 22. Let be a fuzzy capacity game, and let be a random fuzzy coalition with nonnegative variable ; define a game with Concave integral form by Let be an indicator of that And the minimum is taken all over concave and homogeneous functions such that for every .
By Definition 22, is based on decomposition of random variables of crisp game and can be gained by values on crisp coalitions which correspond to subdecompositions of . We denote all fuzzy games defined by Concave integral as . It is easy to prove the following lemma.
Lemma 23. Let be a fuzzy capacity game, for every random fuzzy coalition with nonnegative variable ; then The game can also be calculated by
Remark 24. When ,
It can be easily seen that . It means that the fuzzy game with Concave integral extends crisp game.
Since crisp game , can be represented by simple unanimity games; it is easy to prove the following lemma.
Lemma 25. Let be a fuzzy capacity game, and let be a random fuzzy coalition with nonnegative variable ; then where .
Example 26. Let and let be a characteristic function on . , , , , and . Let with the fuzzy coalition variable .
Suppose , we have Then , , and .
If , rearrange elements of as ; then by (11),
We note that .
We know that is the maximum of the values among all possible decompositions of with the coalition variable .
In the fuzzy game given by Tsurumi et al., the Choquet integral of nonnegative with respect to a capacity is defined by , where is the set of all rates of participation such that and is a permutation of .
Let ; note that is a decomposition of . That is, the Choquet integral is defined under the special decomposition of . By contrast, all possible decompositions are allowed in the Concave integral. In this way, it implies that .
Lemma 28. Let be a fuzzy capacity game, and let be a random fuzzy coalition with nonnegative variable ; then where .
Proof. Let be a fuzzy capacity game, and ; by Definition 8, we get
where . Take and , since , .
On the other hand, by , is a maximum decomposition with the fuzzy coalition variable .
Therefore, when , then .
For any two nonnegative random fuzzy coalitions and with variables and , define distance on by .
Theorem 29. Let be a fuzzy capacity game, then is continuous with respect to fuzzy coalition variables.
Proof. Let and be any two fuzzy coalitions and nonnegative variables and ; we will prove that if , then . By the definitions,
Note that is finite such that and is a sufficient great constant, that is,
Therefore, is continuous with respect to fuzzy coalition variables.
Remark 31. When is convex, and ; it is apparent .
Theorem 32. Let be a fuzzy capacity game; if is convex, then if and only if for any .
5. Imputation of Fuzzy Games with the Concave Integral
Imputation is an important concept; players will get rational share from a cooperative team work, where he gets more shares than he gets by working alone, and the sum of all players’ shares in a team is equal to the team value. Next, we will extend the crisp imputation to fuzzy game.
Definition 33. Let ; a fuzzy coalition with variable ; a vector function is called an imputation if (1),
where and .
We denote the set of all imputations of the fuzzy game on the fuzzy coalition as .
Theorem 34. Let a vector for crisp cooperative game , and then is an imputation for game given by Definition 21, where and .
Proof. For crisp cooperative game , define fuzzy coalition by . For , if , then . So , .
Since , we have On the other hand,
Corollary 35. Let a vector for crisp cooperative game , and fuzzy coalition has fuzzy coalition variable , then can be calculated by
Remark 36. When and is nonempty, then is nonempty for fuzzy game , where fuzzy coalition is determined by .
Corollary 37. Let ; fuzzy coalition is determined by , and then where is the Shapley vector for crisp game .
Example 38. Let , , , , , and .
Suppose a decomposition , , , , , and .
Then the fuzzy coalition defined by .
The crisp Shapley values are obtained as in Table 2 In fact, , so .
Let us consider an another decomposition , , , , and .
Note that , so these two decomposition have the same coalition variable In addition, .