Generalized Fuzzy Shapley Function for Fuzzy Games

A generalized fuzzy Shapley function for fuzzy games is proposed. First, a game with fuzzy characteristic function is introduced. Based on Hukuhara difference, the fuzzy Hukuhara-Shapley function is proposed as a solution concept to this class of fuzzy games. Some of its properties are shown. An equivalent axiomatic characterization of the fuzzy Hukuhara-Shapley function is given. Furthermore, a generalized fuzzy Shapley function for games with fuzzy coalition and fuzzy characteristic function is developed. It is shown that the simplified expression of the generalized fuzzy Shapley function can be regarded as the generalization of the fuzzy Shapley function defined for some particular games with fuzzy coalition and fuzzy characteristic function.


Introduction
The Shapley value [1] is a well-known solution concept in cooperative game theory, which has been investigated by a number of researchers.Most of them treat games with crisp coalitions.However, there are some situations where some agents do not fully participate in a coalition, but to a certain extent.In a class of production games, partial participation in a coalition means offering a part of resources, while full participation means offering all of resources.A coalition including some players who participate partially can be treated as a so-called fuzzy coalition, introduced by Aubin [2,3].It shows to what extent a player transfers his/her representability [4] and is also called a rate of participation.Thus it can describe different participation levels of different players in different game situations, varying from noncooperation to full cooperation.
After the pioneering work by Aubin [3], the Shapley function for games with fuzzy coalition has received more and more attentions.Butnariu [5] gave the expression of the Shapley function on a limited class of fuzzy games with proportional values form.However, most games with proportional values are neither monotone nondecreasing nor continuous with regard to rates of players' participation.In order to overcome this limitation, Tsurumi et al. [6] defined a new Shapley value on a new class of fuzzy games with Choquet integral form.This class of fuzzy games is both monotone nondecreasing and continuous with regard to rates of players' participation.Branzei et al. [7] also introduced a concept of Shapley value for games with fuzzy coalition, which was defined by the associated crisp game corresponding to fuzzy game.Butnariu and Kroupa [8] extend this kind of fuzzy games with proportional values to fuzzy games with weighted function, and the corresponding Shapley function was given.Li and Zhang [9] proposed a simplified expression of the Shapley function for games with fuzzy coalition, which can be regarded as the generalization of Shapley functions defined in some particular games with fuzzy coalition.
On the other hand, by using cooperative game theory, Owen considered linear production programming problems in which multiple decision-makers pool resources to produce some goods [10].An objective function of the linear production programming problem was represented as total revenue from selling some kinds of goods, and the problem was formulated as a linear programming problem in which, subject to the resource constraints, the revenue is maximized.In many decision-making situations, imprecision and uncertainty are often present due to (a) incomplete information, (b) conflicting evidence, (c) ambiguous information, and (d) subjective information.It can be seen that the possible values of parameters of this kind of production games model are often only imprecisely or ambiguously known to decisionmakers.With this observation in mind, it would be certainly more appropriate to interpret the decision-makers' understanding of the parameters as fuzzy numerical data which can be represented by means of fuzzy sets of the real line known as fuzzy numbers.It reflects the decision-makers' ambiguous or fuzzy understanding of the nature of the parameters in the problem-formulation process [11].The resulting production games problem involving fuzzy parameters would be viewed as a more realistic version than the conventional one [12,13].
From fuzzy mathematical programming perspective, Nishizaki and Sakawa [11] investigated cooperative game problems with fuzzy characteristic functions.Mares [14,15] and Mares and Vlach [16] were also concerned with the uncertainty in the value of the characteristic function associated with a game, where the characteristic function was expressed by fuzzy number.At the same time, they discussed the fuzzy Shapley values of this kind of fuzzy game.Borkotokey [17] considered a cooperative game with fuzzy coalitions and fuzzy characteristic function simultaneously.A Shapley function in the fuzzy sense was proposed as a solution concept to this class of fuzzy games.Yu and Zhang [18] studied a class of particular games with fuzzy coalitions and a fuzzy characteristic function with Choquet integral form and gave the explicit form of the Shapley value for this class of fuzzy games.However, as Li and Zhang [9] pointed out, there were other many approaches to extend cooperative games to fuzzy games besides Choquet integral method [6] and proportional values method [5].Which one is more natural?The specific game situation is needed to be considered, because each fuzzy game may only be suitable for a certain case.Therefore, as a kind of special fuzzy game with Choquet integral form, the fuzzy Shapley function of the fuzzy game was only suitable for a certain case.In order to improve this limit, in this paper, we pay more attentions to generalized fuzzy game with fuzzy coalitions and fuzzy characteristic function, and a generalized expression of the fuzzy Shapley function is proposed for the generalized fuzzy game.
In order to do this, this paper is organized as follows.In Section 2, we briefly review some concepts of interval numbers and fuzzy numbers and introduce the Hukuhara difference on interval numbers and fuzzy numbers.In Section 3, a game with fuzzy characteristic functions is introduced, and its fuzzy Hukuhara-Shapley function is proposed, and some of its properties are investigated.Furthermore, an applicable example is given.In Section 4, we investigate a new class of games with fuzzy coalitions and fuzzy characteristic function and give a simplified expression of the generalized fuzzy Shapley function for the new fuzzy games.It is shown that it can be regarded as a generalization of the fuzzy Shapley function defined in the proposed fuzzy games with some particular games with fuzzy coalition and fuzzy characteristic function.Finally, in Section 5, we summarize the main conclusions of the paper.

Preliminaries
In this section, we start by providing a summary of some concepts of interval numbers and fuzzy numbers, which will be used throughout the paper.

Interval Numbers
Obviously, when  − =  + , the interval number  reduces to a real number  − or  + .We say that a real number  is a member of an interval number , written as  ∈ , if  − ≤  ≤  + .Let us denote by IR the class of all closed and bounded intervals in R. Throughout this paper, when we say that  is a closed interval, we implicitly mean that  is also bounded in R.
In the following, we will briefly review the order relation and basic operation of interval numbers [19].

Furthermore, any family {𝑈
To quantify fuzzy concepts, we use the following fuzzy numbers [25].Definition 5. A fuzzy number, denoted by ã, is a fuzzy subset of R with membership function  ã : R → [0, 1] satisfying the following conditions.
(i) There exists at least one number  0 ∈ R such that  ã( 0 ) = 1; (ii)  ã() is nondecreasing on (−∞,  0 ] and nonincreasing on [ 0 , +∞); An important type of fuzzy numbers in common use is the trapezoidal fuzzy number [21] whose membership function has the form where It follows from the properties of the membership function of a fuzzy number ã that each of its cuts ã is an interval number, denoted by ã = [ −  ,  +  ].
Definition 6 (see [25]).Let ã be a fuzzy number; ã = ⋃ ∈[0,1] ã  (decomposition theory), where Let ã, b ∈ FR, and let * be a binary operation on R. cuts of the fuzzy number ã * b can be calculated because of the following: Employing the -cut representation, arithmetic operations on fuzzy numbers are defined in terms of the wellestablished arithmetic operations on interval numbers [25].
If, for any  1 ,  2 ∈ [0, 1], we have then  is called a nested set of .
Proof.For all  ∈ [0, 1], we have If  ≤ , according to Definition 7, we have Thus for any  ∈ [0, 1], () is a nested set.This completes the proof of Proposition 11.

Fuzzy Hukuhara-Shapley Function for
Games with Fuzzy Characteristic Function The set of all superadditive games is denoted by  0 ().For the sake of simplicity, the bracket is often omitted when a set is written in this paper.For example, we write  instead of {, , }.
As an important solution concept for crisp cooperative games, the Shapley value is defined as follows.
Definition 13.The Shapley value   (V) of player  with respect to a game V ∈  0 () is a weighted average value of the marginal contribution V( ∪ ) − V() of player  alone in all combinations, which is defined by Equation ( 17) is a unique expression which satisfies three axiomatic characterizations of Shapley value (see the study by Shapley [1]).Proposition 14 (see [1]).For any ,  ⊆  and  ̸ = ⌀, if the -unanimity game   is denoted by then, {  } ∈2  \⌀ is a basis in the linear space  0 () of all games, and, for V, it can be uniquely written as where where ,  denote the cardinality of crisp coalitions  and , respectively.

Cooperative Games with Fuzzy Characteristic Function.
In a crisp cooperative game, characteristic function describes a cooperative game and associates a crisp coalition  with the worth V(), which is interpreted as the payoff that the coalition  can acquire only through the action of .The cooperative crisp game is based on the assumption that all players and coalitions know the payoff value V before the cooperation begins.
The traditional cooperative game assumes that all data of a game are known exactly by players.However, in real game situations often the players are not able to evaluate exactly some data of the game due to a lack of information or/and imprecision of the available information on the environment or on the behavior of the other players.Taking imprecision of information in decision-making problems into account, this assumption is not realistic because there are many uncertain factors during negotiation and coalition formation.In many situations, the players can have only vague ideas about the real payoff value.Therefore, it is more suitable to incorporate fuzzy characteristic function, represented by fuzzy numbers, into cooperative games.In this section, the fuzzy Hukuhara-Shapley function for games with a fuzzy characteristic function is proposed.
Definition 15.A cooperative game with fuzzy characteristic function form is an ordered pair (, Ṽ) where Ṽ : () → Fuzzy characteristic function Ṽ() can be interpreted as the maximal fuzzy worth or cost savings that the members of  can obtain when they cooperate.Often we identify the game (, Ṽ) with its fuzzy characteristic function Ṽ.The class of games with fuzzy characteristic function is denoted by ().In this paper, we mainly discuss the superadditive games with fuzzy characteristic function; that is, for any two crisp coalitions ,  ∈ () such that  ∩  = ⌀, for any  ∈ (0, 1], Ṽ ( ∪ ) ≥ Ṽ () + Ṽ ().Then, for any crisp coalitions ,  ∈ (),  ⊆  and any ,  ∈ (0, 1] such that  > , the superadditive game (, Ṽ) also satisfies According to Proposition 9 it is easy to see that the Hukuhara difference Ṽ() − H Ṽ() exists for the superadditive games with fuzzy characteristic function.The set of all superadditive games with fuzzy characteristic functionis denoted by  0 ().Definition 16.Given Ṽ ∈  0 (), a carrier of Ṽ is any set  ⊆  with Ṽ() = Ṽ( ∩ ), ∀ ⊆ .
Obviously, players outside any carrier have no influence on the play since they contribute nothing to any coalition.Such a player that does not contribute anything to any coalition is called a null player; that is, for any  ⊆  \ , Ṽ( ∪ ) = Ṽ().
Lemma 20.If Ṽ ∈ (), for any , ⌀ ̸ =  ⊆ , then Ṽ is a linear combination of   : where  is a carrier of Ṽ.The coefficients c are independent of  and are given by where , denote the cardinality of coalitions  and , respectively.
Proof.If  ⊆ , then In general, according to the definition of carrier, we have This completes the proof of Lemma 20.
Remark 21.For Ṽ ∈  0 () and any  ∈ (0, 1], it is obvious that crisp games V −  , V +  ∈  0 ().Assume According to Proposition 14, it is seen that V −  and V +  can be uniquely written as Obviously, According to representation theorem of fuzzy set [25], it is easy to obtain the same conclusion as Lemma 20.This completes the proof of Proposition 22.
Proof.According to Lemmas 19 and 20 and Axiom A 3 , we have This completes the proof of Lemma 23.
Proof.According to Lemmas 23 and 19, we have Since Therefore, From Lemma 20, it is known that c (Ṽ) is well defined by  and Ṽ.Thus according to Lemma 23, φ (Ṽ) is well defined by Ṽ, , and ; that is, φ (Ṽ) is well defined by Ṽ and .
In the following, we prove that the function defined by (44) satisfies Axioms A 1 -A 3 in Definition 17.
According to Lemmas 19 and 20, it is easy to verify that φ (Ṽ) satisfies Axioms A 1 and A 2 .

Proposition 26. (i) For any pair
Proof.(i) According to (49), it is easy to prove the conclusion.
According to Theorem 24, it is easy to obtain the following conclusion.
Symmetric players have the same contribution to any coalition, and therefore it seems reasonable that they should obtain the same payoff according to the value.A dummy player only contributes his/her own worth to every coalition, and that is what he/she should be paid.A null player does not contribute anything to any coalition; in particular also Ṽ() = 0.So it seems reasonable that such a player obtains zero according to the value.
The proof is completed.
According to Theorem 24, it is easy to obtain the following conclusions, too.
(i) Let z ∈  0 () that is identically zero.In this fuzzy game, all players are symmetric, so Axiom F 2 and Axiom F 1 together imply ψ(z) = 0.
The proof is completed.
Example 32.Consider three economic companies, named 1, 2, and 3 ( = 1, 2, 3).They possess different resources.Now they want to do a joint project by means of pooling their resources.It is natural for these three decision-makers to try to evaluate the revenue of the joint project in the early period of the project in order to decide whether the project can be realized or not.However, the average profit of the joint project is dependent on a number of companies.
And the average profit of the joint project is an approximate evaluation, which is represented by fuzzy numbers as follows: According to (44), we can calculate fuzzy Hukuhara-Shapley function of each decision-maker as follows: By judging the allocations of fuzzy Hukuhara-Shapley function of each decision-maker, decision-makers can conclude whether the joint project can be realized or not.According to fuzzy Hukuhara-Shapley function, it is very convenient for us to obtain their interval Hukuhara-Shapley values of different -level as shown in Table 1.
By Corollary 29, it is obvious that φ (w)([] ℎ  ) = 0 for any ℎ  > ℎ  .Then  According to Lemma 41, the remaining part of the proof of Theorem 42 is similar to that of Theorem 4.3 in Li and Zhang [9]; therefore, here it is omitted.
Remark 44.From the above analysis, itis easily seen that the simplified expression of the generalized fuzzy Shapley function given by (70) can be regarded as the generalization of fuzzy Shapley functions defined in some particular fuzzy games with fuzzy coalition.The simplified expression of the fuzzy Shapley function is equivalent to Yu and Zhang's definition [18] when fuzzy characteristic function is a fuzzy game with Choquet integral forms and is equivalent to that when fuzzy characteristic function is a fuzzy game with proportional values.
Remark 45.The generalized fuzzy Shapley function given by (70) can be applied to crisp games, games with fuzzy coalitions, and games with fuzzy characteristic functions by restricting the domain.Equation (70) coincides with (44) or (52) when restricted to a crisp coalition; (70) is equal to (17) when restricted to a real-valued characteristic function and a crisp coalition.Briefly, the generalized fuzzy Shapley function defined by (70) is a general Shapley function, which serves as the connection between games with fuzzy coalitions and games with fuzzy characteristic functions.The most important aspect is that we do not need to transform the Shapley function when dealing with different kinds of cooperative games.

Conclusion
Shapley value is a well-known solution concept in cooperative game theory.In crisp game, it has been applied widely in many cases.In fuzzy game, Shapley functions of game with fuzzy coalition or with fuzzy characteristic function, as an important solution concept, have been paid more attention.Games, subject to fuzzy coalitions as well as those pertaining to fuzzy characteristic function, are separately investigated in the literature.In this paper, based on the Hukuhara difference, a fuzzy Hukuhara-Shapley function on the class of games with fuzzy characteristic function is investigated.
And some interesting properties are shown.Further, a new fuzzy game model that admits both fuzzy coalitions and fuzzy characteristic functions is proposed.A simplified expression of the generalized fuzzy Shapley function for the generalized fuzzy games is given.It is shown that the simplified expression of the generalized fuzzy Shapley function is equivalent to that of fuzzy games with indeterminate integral form in the study by Yu and Zhang [18].When the domain of the characteristic function or coalition is restricted, the generalized fuzzy Shapley function can be applicable to crisp games, games with fuzzy coalitions, and games with fuzzy characteristic functions.Thus, the simplified expression of the generalized fuzzy Shapley function for this kind of fuzzy games can be widely applied in many cases to address more realistic situations.

Definition 7 .Definition 8 .
Given any pair of fuzzy numbers, ã, b ∈ FR, the basic operations on the -cuts of ã and b are defined for all  ∈ [0, 1] by the general formula ã * b = ⋃ ∈[0,1]  (ã  * b ) .(7) However, similar to the interval number operations, in the fuzzy contexts, equation ã = b + c is not equivalent to c = ã− b = ã+(−1) b or b = ã−c = ã+(−)c.This has motivated the introduction of the following Hukuhara difference [22].Given ã, b ∈ FR, if there exists c ∈ FR such that ã = b + c ∈ FR, then c is called the Hukuhara difference (H-difference), denoted by ã − H b = c.Clearly, ã − H ã = 0; if ã − H b exists, it is unique.And the -cuts of H-difference are (ã − H b)  = [ −  −  −  ,  +  −  +  ]; and (ã − H b) = ã − H  b, for  ∈ R,  > 0. The H-difference inverts the addition of fuzzy numbers.But the Hukuhara difference between two fuzzy numbers does not always exist.Regarding the existence of the Hukuhara difference, there is an extensive literature described in the study by Dubois et al. in [21].Proposition 9. Let ã, b ∈ FR.The Hukuhara difference ã − H b exists if and only if
Proposition 4. (i) Let  = {1, 2, . . ., },   = [ −  ,  +  ],  1 ∩  2 = ⌀, and  1 ∪  2 = ; then ,   ,   ,   ∈ R with   ≤   ≤   ≤   and the trapezoidal fuzzy number ã is simply denoted by (  ,   ,   ,   ).It is called nonnegative if   ≥ 0. The trapezoidal fuzzy number degenerates to be a triangular fuzzy number when   =   , while it becomes an interval number (i.e., a rectangular fuzzy) when   =   and   = .Any crisp real number  can be regarded as a special trapezoidal fuzzy number with   =   =   =   = .In this paper, the set of all fuzzy numbers on R is denoted by FR.Let any fuzzy number ã ∈ FR have membership function  ã, and the level set (or -cut) is defined as ã 3.1.CrispCooperative Games and Shapley Value.A finite transferable utility cooperative game (from now on, simply a game or cooperative game) is a pair (, V), where  = {1, 2, . . ., } is a finite set of players and V : () → R + is called characteristic function satisfying V(⌀) = 0. () is the family of crisp subsets of ; that is, () is equivalent to 2 || .We will refer to a subset  of  as a coalition or crisp coalition and to V() as the worth of , which can be seen as the amount of utility the coalition obtains when the players in  work together.The class of all crisp games with player set  is denoted by ().|| denotes the cardinality of .Given a game (, V) and a coalition , we write (, V) for the subgame obtained by restricting V to subsets of  only (i.e., to 2 || ).If there is no fear of confusion, a (cooperative) game

Table 1 :
Interval Hukuhara-Shapley values for different -level of decision-makers.