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.
The Shapley value  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  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 , the Shapley function for games with fuzzy coalition has received more and more attentions. Butnariu  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.  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.  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  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  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 . 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 decision-makers. 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 . 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  investigated cooperative game problems with fuzzy characteristic functions. Mares [14, 15] and Mares and Vlach  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  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  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  pointed out, there were other many approaches to extend cooperative games to fuzzy games besides Choquet integral method  and proportional values method . 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.
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.
2.1. Interval Numbers
Let be , that is, the set of all real numbers. Given , and , the closed interval defines an interval number . 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 the class of all closed and bounded intervals in . Throughout this paper, when we say that is a closed interval, we implicitly mean that is also bounded in .
In the following, we will briefly review the order relation and basic operation of interval numbers .
Definition 1. For any interval number and , , .
Consider the following:(1) iff , ,(2) iff , ,(3) iff , .
Definition 2. For any interval numbers , , the arithmetic operations on interval numbers are defined as follows:(1),(2),(3), ,(4).
It is known that scalar multiplication is a special case of interval numbers multiplication operation. Scalar multiplication can be expressed by , if ; , if .
In general, according to Definition 2, for , does not mean that . For example, (unless is a singleton). To overcome this situation, next we introduce H-difference of interval numbers, called Hukuhara difference [20–22], which is denoted by in this paper. Let and be two closed intervals in . For a closed interval such that , is called the Hukuhara difference. Since , it is easy to see that and ; that is, and . Therefore, the closed interval exists if . In this case, and we also denote . When we say that the Hukuhara difference exists, we implicitly mean that . An important property of is that , and if exists, H-difference is unique . For any positive number , . In general, .
Definition 3. For any , let and define
It is easy to prove the following conclusions.
(i) Let , , , and ; then
(ii) Let , for any ; then
2.2. Fuzzy Numbers
A general fuzzy set over a given set (or space) of elements (the universe) is usually defined by its membership function and a fuzzy (sub)set of is uniquely characterized by the pairs for each ; the value is the membership grade of to the fuzzy set and is the membership function of a fuzzy set over [23, 24] for the origins of fuzzy set theory. The support of is the (crisp) subset of points of at which the membership grade is positive: . For , the -level-cut of (or simply the -cut) is defined by and for (or ) by the closure of the , where denotes the closure of sets.
The following properties characterize the normal, convex, and upper semicontinuous fuzzy sets (in terms of the level-cuts):(B1) is the spaces of nonempty compact and compact convex sets of for all ;(B2) for ;(B3) for all increasing sequences converging to .
Furthermore, any family satisfying conditions (B1)–(B3) represents the level-cuts of a fuzzy set having .
To quantify fuzzy concepts, we use the following fuzzy numbers .
Definition 5. A fuzzy number, denoted by , is a fuzzy subset of with membership function satisfying the following conditions.(i)There exists at least one number such that ;(ii) is nondecreasing on and nonincreasing on ;(iii) is upper semicontinuous; that is, if and
(iv) is compact, where .
An important type of fuzzy numbers in common use is the trapezoidal fuzzy number  whose membership function has the form where with and the trapezoidal fuzzy number is simply denoted by . It is called nonnegative if . 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 is denoted by .
Let any fuzzy number have membership function , and the level set (or -cut) is defined as , . It follows from the properties of the membership function of a fuzzy number that each of its -cuts is an interval number, denoted by .
Definition 6 (see ). Let be a fuzzy number; (decomposition theory), where is the -cut set of for any . For all ,(i) iff ;(ii) iff ;(iii) iff ;(iv)if , then .
Let , and let be a binary operation on . -cuts of the fuzzy number can be calculated because of the following:
Employing the -cut representation, arithmetic operations on fuzzy numbers are defined in terms of the well-established arithmetic operations on interval numbers .
Definition 7. Given any pair of fuzzy numbers, , the basic operations on the -cuts of and are defined for all by the general formula
However, similar to the interval number operations, in the fuzzy contexts, equation is not equivalent to or . This has motivated the introduction of the following Hukuhara difference .
Definition 8. Given , if there exists such that , then is called the Hukuhara difference (H-difference), denoted by .
Clearly, ; if exists, it is unique. And the -cuts of H-difference are ; and , for , . 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 .
Proposition 9. Let . The Hukuhara difference exists if and only if
Definition 10. Let be a nonempty set; denotes its power set. For mapping
If, for any , we have then is called a nested set of .
Proposition 11. Given , for any , let ; then is a nested set.
Proof. For all , we have
If , according to Definition 7, we have
Thus for any , is a nested set.
This completes the proof of Proposition 11.
For , we define
According to representation theorem  and Proposition 4, it is easy to obtain the following conclusions.
(i) Let , , , and ; then
(ii) Given fuzzy number , for , then
3. Fuzzy Hukuhara-Shapley Function for Games with Fuzzy Characteristic Function
3.1. Crisp Cooperative Games and Shapley Value
A finite transferable utility cooperative game (from now on, simply a game or cooperative game) is a pair , where is a finite set of players and is called characteristic function satisfying . is the family of crisp subsets of ; that is, is equivalent to . We will refer to a subset of as a coalition or crisp coalition and to 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 and a coalition , we write for the subgame obtained by restricting to subsets of only (i.e., to ). If there is no fear of confusion, a (cooperative) game is replaced by . For any , , if , then is called superadditive game. The set of all superadditive games is denoted by . 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 of player with respect to a game is a weighted average value of the marginal contribution 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 ).
Proposition 14 (see ). For any and , if the -unanimity game is denoted by then, is a basis in the linear space of all games, and, for , it can be uniquely written as where where denote the cardinality of crisp coalitions and , respectively.
3.2. 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 , 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 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 where with .
Fuzzy characteristic function 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 with its fuzzy characteristic function . 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 , . Then, for any crisp coalitions , and any such that , the superadditive game also satisfies
According to Proposition 9 it is easy to see that the Hukuhara difference exists for the superadditive games with fuzzy characteristic function. The set of all superadditive games with fuzzy characteristic functionis denoted by .
Definition 16. Given , a carrier of is any set with , .
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 , .
Let ; for define . It is easily seen that .
Definition 17. Let ; for any , a function , where , is said to be a fuzzy Hukuhara-Shapley function on if it satisfies the following three axioms.
Axiom A1. If is any carrier of , then
Axiom A2. If is a permutation of the player set, that is, , then, for any ,
Axiom A3. If , then
It is easy to obtain the following conclusion.
Proposition 18. If , for any , then .
Lemma 19. For any , , , , fuzzy Hukuhara-Shapley value of is
Proof. It is obvious that is a carrier of . For , since and are also carriers of , according to Axiom A1, we have
For and , let be a permutation of such that , , , . First, we prove that, for any ,
For , . We have
For , if , then . If , then . If and , then . So .
According to Axiom A2, we have
According to the above analysis and Axiom A1, we obtain that
This completes the proof of Lemma 19.
Lemma 20. If , for any , then is a linear combination of : where is a carrier of . The coefficients 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 and any , it is obvious that crisp games . Assume
According to Proposition 14, it is seen that and can be uniquely written as
Obviously, . According to representation theorem of fuzzy set , it is easy to obtain the same conclusion as Lemma 20.
Proposition 22. If and , then
Proof. Let ; then . According to Axiom A3, we have
So ; that is, .
This completes the proof of Proposition 22.
From Proposition 18, if , then ; otherwise, .
Lemma 23. If , then
Proof. According to Lemmas 19 and 20 and Axiom A3, we have
This completes the proof of Lemma 23.
Theorem 24. Given , there exists a unique fuzzy Hukuhara-Shapley function , satisfying Axioms A1–A3, with element where , denote the cardinality of crisp coalitions and , respectively.
Proof. According to Lemmas 23 and 19, we have
From Lemma 20, it is known that is well defined by and . Thus according to Lemma 23, is well defined by , , and ; that is, is well defined by and .
In the following, we prove that the function defined by (44) satisfies Axioms A1–A3 in Definition 17.
According to Lemmas 19 and 20, it is easy to verify that satisfies Axioms A1 and A2.
Let and ; then
These complete the proof of Proposition 14.
Remark 25. Given , from Theorem 24, for any and , there exists a unique interval Hukuhara-Shapley function with element
For the interval Hukuhara-Shapley function , , and it has the following properties.
(i) For any pair satisfying , .
(ii) For any , .
Proof. (i) According to (49), it is easy to prove the conclusion.
(ii) Let ; from (49), for any , we have
These complete the proof of Proposition 26.
Call players and symmetric in the game if for every coalition . Call a player in a game a dummy (player) if for all .
According to Theorem 24, it is easy to obtain the following conclusion.
Corollary 27. Given , if and are symmetric in game , then . If is a dummy in game , then . If is a null player in game , then .
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 . So it seems reasonable that such a player obtains zero according to the value.
Corollary 28. Given , for all ,
Proof. Let be a carrier of . If and , then .
From , we have .
From Corollary 27, we have , so .
If and , since , we have .
Hence, according to (44), we easily have .
The proof is completed.
According to Theorem 24, it is easy to obtain the following conclusions, too.
Corollary 29. Define a function by
Then the function is the unique fuzzy Shapley function on .
Corollary 30. For any two , if for any , then, for any ,
According to the above analysis, we can use another equivalent axiomatic system to define the fuzzy Hukuhara-Shapley function on .
Theorem 31. Given , let be a value. Then satisfies the following three axioms, if and only if is the fuzzy Hukuhara-Shapley function .
Axiom F1. If , then
Axiom F2. If players and are symmetricin the game , then
Axiom F3. For any two , if for any , then, for ,
Proof. Obviously, satisfies the three axioms. Conversely, suppose satisfies the three axioms.(i)Let that is identically zero. In this fuzzy game, all players are symmetric, so Axiom F2 and Axiom F1 together imply .(ii)Let be a null player in . Then the condition in Axiom F3 applies to and with all inequalities being equalities. So Axiom F3 yields and . Hence, by (i), .(iii)Let and . According to Lemma 20, we have
Hence, , where satisfying when and when .(iv)According to Lemma 23 we have
The proof of is completed by induction on the number of terms in with .
From (i), if and, from (iii), if because of . Suppose for all with , where . Let with . Then there exist coalitions and fuzzy numbers unequal to zero, such that . Let .
For , define . Because , the induction hypothesis implies Further, for every , So, by Axiom F2, it follows that . So, for all , we have Further, according to Axiom F1, combining with (61), we have Let ; then, for every , so and are symmetric. Hence, by Axiom F2, we have From (61), (62), and (64), we have .
The proof is completed.
Example 32. Consider three economic companies, named 1, 2, and 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: