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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 231508, 13 pages
http://dx.doi.org/10.1155/2014/231508
Research Article

The Shapley Values on Fuzzy Coalition Games with Concave Integral Form

1Library, Beijing Institute of Technology, Beijing 100081, China
2Patent Examination Cooperation Center of the Patent Office, State Intellectual Property Office of P.R.C, Beijing 100088, China
3Department of Information Management, The Central Institute for Correctional Police, Baoding 071000, China

Received 15 November 2013; Accepted 15 December 2013; Published 27 January 2014

Academic Editor: Pu-yan Nie

Copyright © 2014 Jinhui Pang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. 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 .

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.

Actually, players mostly prefer to estimate cooperative profits on their vague knowledge of the game. Based on this, integrals with fuzzy measures or fuzzy capacities should be suitable to model fuzzy games. The integral theory had made use of a cooperative game that appeared in Weber [16] and Azrieli and Lehrer [17]. Besides Tsurumi et al. [12], Dow and Werlang [18, 19] applied the Choquet integral to game theory and finance. In many cases, fuzzy capacities assign decision-subjective expected values to some coalitions but not to all. Therefore, we will at first introduce a new cooperative game form by the Concave integral with respect to fuzzy capacity, which has been discussed in detail by Lehrer [20], and will investigate the Shapley values on the new fuzzy cooperative games in detail.

The lecture will be organized as follows. In Section 2, we will recall the concepts of crisp cooperative game and its Shapley values will also be recalled. In Section 3, some games with fuzzy coalitions and several game forms, such as Butnariu and Tsurumi fuzzy game, will be given. Moreover, we define a new fuzzy game by the Concave integral and its several properties will also be provided. Meanwhile, we illustrate that the proposed game is an extension of Tsurumi fuzzy game. In Section 4, we discuss the Shapley values for the new game, which can be gained by a linear aggregation of the Shapley values on subdecompositions crisp coalitions. Finally, some conclusions appear in Section 5.

2. Crisp Cooperative Game and the Shapley Value

A finite set of players 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 where the character function with . For any a , the worth 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 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 can be represented by as follows: where .

The game is said to be convex when

The convex game is said to be superadditive, if any disjoint crisp coalitions and satisfy

The notation represents all superadditave crisp cooperative games. For the game , players and are said to be symmetric, if such that The player is a dummy player of the game, if for any such that The player is a null player in a coalition for a game , if

Definition 1. An imputation for a crisp cooperative game is a vector satisfying(1),(2), .
It is obvious that when imputation for a crisp cooperative game is nonempty.

Definition 2. Let , ; then is called a carrier in a coalition for a game if If the set of all carriers in coalition for is denoted by , then
A well-known solution for cooperative game , the Shapley value is a mathematical expectation on with regard to marginal contribution where . Shapley [1] defined the function satisfying the following 4 axioms.

Definition 3. A function is said to be a Shapley value on if it satisfies the following four axioms.
Axiom 1. If and , then
where is the th element of .
Axiom 2. If , , and , then
Axiom 3. If , , , and for any , then
Axiom 4. For any , define a game by for any . If and , then
Shapley [1] also gave the uniquely explicit form of a Shapley value on which was obtained by extending the Shapley value for the grand coalition as below.

Theorem 4. Define a function by where , and is the number of players in set . Then the function is the unique Shapley function on .
It is obvious that is an imputation of the cooperative game . Meanwhile, if is convex, then and imply that .

3. Fuzzy Coalition Games

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

3.1. Basic Concepts

We consider cooperative fuzzy games with the player set . A fuzzy coalition is a fuzzy subset of the finite set , which is a vector where describes the membership grade of player in the fuzzy coalition . We note that a different coalition has different vector , so we also call it as fuzzy coalition variable. If element when fully take part in and others , then the coalition is a crisp coalition.

Consider the crisp coalition where the th element is 1 and others are zero. The fuzzy coalition variable has the regular form

For the fuzzy coalition and with fuzzy coalition variables vector and , if and only if . The class of all fuzzy coalitions in is denoted by ; that is, . The level set of fuzzy coalition is the set ; its -section is the set which is a player set with the same level , and its support set is the set .

A fuzzy coalition game is the function with . We take the notation as the class of all fuzzy coalition games . We call continuous, for any two fuzzy coalitions and with variables and , respectively, if where . A game is said to be monotonic, for every fuzzy coalition and , implies .

In this paper, we assume that every fuzzy coalition variables maps into the lattice , where and are the minimum and maximum operators, respectively. For any fuzzy coalition , we adopt the usual definition of the union and intersection of fuzzy subsets given by

Similarly to crisp convex game, for all , is said to be fuzzy convex, if it satisfies and is said to be superadditive such that with .

Definition 5. A function is said to be an imputation for a fuzzy game in fuzzy coalition with fuzzy coalition variable , if (1), ,(2),(3),where .

Definition 6. Let ; the player is said to be a dummy player on fuzzy coalition with fuzzy coalition variable , if for any fuzzy coalition , and if The player is called a null player on fuzzy coalition .

Example 7. Let and for any fuzzy coalition variable , define by Suppose a fuzzy coalition with the fuzzy variable value ; then for any fuzzy coalition with fuzzy variable value , player 3 is a dummy player when and he is a null player when .

Definition 8. Let , ; then is called a fuzzy carrier in a coalition if for any such that The set of all carriers in fuzzy coalition for is denoted by ; it is obvious that

3.2. 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, was an aggregated worth of the crisp coalitions where 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-to-one correspondence between a crisp game and a fuzzy game with proportional values.

Butnariu and Kroupa [15] proposed a fuzzy game model with weight function as follows: where is a function with the properties and .

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. Let , and rearrange elements in such that ; then for any , a game is defined by where is the cardinality of .

The fuzzy game given by Tsurumi et al. is simply denoted by . It is apparent that the fuzzy game is a Choquet integral of the function with respect to derived from level set. We note that implies that , so the worth of coalition is the maximum sum on all subsets which is an including chain.

Example 9. Let and let be a character function on which is joint workers’ output. , , , , and .
Suppose that the fuzzy coalition ; rearrange it as ; 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.

3.3. 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 such that implies with . A random variable over is a function and a random variable is nonnegative if for every .

We proposed fuzzy capacity game concept defined by the following way.

Definition 10. Let ; the pair is said to be a fuzzy capacity game if is monotonic and continuous, and there is a positive such that for every .

Definition 11. 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 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 coalition might contain only extreme or discrete points of the domain of where such as and , therefore may be partially nonadditive or non-additive 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 be a fuzzy capacity game, let be a random fuzzy coalition with nonnegative variable , and define a game by where the minimum is taken all over concave and homogeneous functions and for every , and is the indicator of which is the random variable that takes the value 1 over and the value 0, otherwise.
By Definition 12, 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 .
From the above definition, the function is defined on all over concave coalitions. It is easy to prove the following lemma.

Lemma 13. Let be a fuzzy capacity game; for every random fuzzy coalition with nonnegative variable , The game can also be calculated by

Remark 14. When , . It is apparent that the fuzzy game with the concave integral extends the crisp game.

Example 15. Consider again Example 9; for the fuzzy coalition , by inequality (33), we have Hence, So
We know that is the maximum of the values among all possible decompositions of with 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 is defined by where is a permutation on such that and .
Let ; note that 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 for any . In addition, it has been proven that if and only if is convex (see [17]).

Lemma 16. Let be a fuzzy capacity game, and let be a random fuzzy coalition with nonnegative variable ; then where ().

Theorem 17. Let a fuzzy capacity game ; then is continuous with respect to the fuzzy coalition variables.

Proof. For any two nonnegative random fuzzy coalitions and with variables and , define a distance on by We have Since there exists a constant such that , we get Therefore, when then .

Example 18. Continuing with Example 9, for another fuzzy coalition with , by inequality (33), we have By (27), we also get From the above examples, we note that while .

Theorem 19. Let be a fuzzy capacity game; is convex if and only if for any two nonnegative random fuzzy coalitions and with variables and , respectively, whenever .

Proof. If a fuzzy capacity game is convex, then by the property of the concave integral, and , so when , it implies that .
Conversely, if is not convex, then there exits a nonnegative variable such that . Since , then ; 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.
Therefore, whenever , it cannot be confirmed that ; thus is convex.
Since if and only if is convex, conversely is convex if and only if , for any .

4. 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 .

4.1. The Shapley Axioms for Games with Fuzzy Coalitions

Tsurumi et al. defined the Shapley function which is based on the natural extension of carrier and null player to fuzzy games. Before introducing the definition, we provide some notations introduced by Tsurumi et al.

For any , , and , Obviously, , , and .

Definition 20. Let ; a function is said to be a Shapley value on if it satisfies the following four axioms.
Axiom 1. If and , then where is the ith element of .
Axiom 2. If , , and , then
Axiom 3. If , , , , and for any , then
Axiom 4. For any , define a game by for any . If and , then
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.

Theorem 21. If and , then , .

Proof. For , note that if and only if for any . Since is convex, implies that for any ; therefore, for any .

4.2. The Shapley Values for Simple Game with Fuzzy Coalitions

Following Shapley [1], with any nonempty coalition , we consider the fuzzy simple game defined by and the number where is a fuzzy coalition variable of .

Consider In addition, for any a cooperative game , Hence,

Lemma 22. Let ; then for any , a Shapley value on the simple game is as follows:

Proof. Let with fuzzy coalition variable ; for all , it is obvious that , and is a fuzzy carrier in fuzzy coalition .
Axiom 1. Consider That is,
Axiom 2. For any , . and ; then is also a fuzzy carrier in coalition , so For any , and .
Axiom 3. For any , let ; by the fuzzy simple definition, , so .
Axiom 4. For any two fuzzy simple games and , define , . If is also a fuzzy simple game, then It satisfies for any , .
It is evident that , .

4.3. The Shapley Values for Game with Concave Integral

For a fuzzy coalition with fuzzy coalition variable , we simply denote it by crisp game marks as follows:

If we let , it is obvious that is a linear sum with the concave integral on , and Therefore, where and .

Hence, the Shapley value of player on the fuzzy coalition will be

Lemma 23. Let be a fuzzy capacity game, and let be a random fuzzy coalition with nonnegative variable ; then the vector is an imputation of the fuzzy coalition , where is defined as (65).

Proof. For any , it is apparent that and . So .
Meanwhile, Therefore, we have for any , so is an imputation.

Lemma 24. Let be a convex game; for any fuzzy coalitions such that ; then ().

Proof. If is convex implies that . From Theorem 21, for fuzzy coalitions , then for any .
Therefore, for any , .
Lemma 24 suggests that on game , is monotone nondecreasing with respect to fuzzy coalition variable when is convex. In fact, is also continuous with respect to fuzzy coalition variable when .
The next lemma can easily be proved by the same manner as Lemma 24.

Lemma 25. Let ; then is also continuous with respect to fuzzy coalition variable for any .

Example 26. Let , with the fuzzy coalition variable , , , , , , , , , .
Now we consider the Shapley value of player 1 in fuzzy coalition by (65); we have
Note that the game is convex such that the Shapley is the same as that of Tsurumi game.

Theorem 27. Let be a fuzzy capacity game, and let be a random fuzzy coalition with nonnegative variable ; reorder its components such that ; then () can be calculated by where and ().

Proof. Let , ; reorder the elements in such that .
For the fuzzy variable , let , where if the set is a subdecomposition set, else and (). Then all the subdecomposition sets correspond with the order.
Let . For any a subdecomposition , by ,
It is obvious that ; for any , we get Therefore,