The Cores for Fuzzy Games Represented by the Concave Integral

We propose a new fuzzy game model by the concave integral by assigning subjective expected values to random variables in the interval [0,1] . The explicit formulas of characteristic functions which are determined by coalition variables are discussed in detail. After illustrating some properties of the new game, its fuzzy core is defined; this is a generalization of crisp core. Moreover, we give a further discussion on the core for the new games. Some notions and results from classical games are extended to the model. The nonempty fuzzy core is given in terms of the fuzzy convexity. Our results develop some known fuzzy cooperative games.


Introduction
In crisp cooperative game, gains from a coalition are supposed to be of certainty so that the solution concepts are also definitive which determine allocations of the total benefit from cooperation to the players.However, the cooperation is full of uncertainty.
The fuzzy game theory also deals with the problems of how to describe fuzzy coalitions, how to represent available fuzzy payoffs, and how to divide it among the various players.In Aubin [1] and Butnariu [2] fuzzy game, the characteristic function was an aggregated worth of the coalitions profits, which depended on the degree of participations of players in a coalition.Tsurumi et al. [3] proposed a class of fuzzy games using the concept of Choquet integrals.
The cooperative games, which lack precision game data, had been investigated by stochastic theory (see Granot [4] and Fernandez et al. [5]) In some situations, there is no reliable information on probability distributions and other aspects of the problems.Consequently, it is reasonable to adopt the fuzzy theory to constructing various fuzzy game models (see Zadeh [6], Mareš [7], Dubois and Prade [8]).
Mareš [9,10] and Vlach [11] suggested that the uncertainty value of the characteristic function was also associated with a game.By this assumption, the values assigned to coalitions were also fuzzy quantities even though the domain of the characteristic function of fuzzy games remained to be the same as crisp games.
Borkotokey [12] considered a cooperative game with fuzzy coalitions and fuzzy characteristic function simultaneously, whose characteristic functions were a fuzzy value which mapped the set of real numbers to the closed interval [0, 1].
Nowadays, the fuzzy games are mainly introduced in two ways.One is games with fuzzy payoffs and the coalitions are still crisp game coalitions.Another is games with fuzzy coalitions in which players partly take part in a coalition so as to form a fuzzy coalition but exact benefits from the fuzzy coalition can be attained.
After fuzzy games were defined, their solution concepts have been studied by many scholars.The firstly defined fuzzy Shapley function by Butnariu [13] showed a specific formula 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 [14] gave an analogous Shapley function on fuzzy games with weight function.Following Butnariu's way, Tsurumi et al. defined Shapley function on the fuzzy game with Choquet integral form, which was both monotone nondecreasing and continuous with regard to rates of players' participation rates because of the advantage properties of Choquet integral.In fact, the core for fuzzy games is another important solution concept which was focused on 2 Journal of Function Spaces by Tijs et al. [15].With the development of fuzzy cooperative game theory, many extended solutions of some fuzzy games, which are homogeneous or one-to-one of crisp game, draw much more attention of researchers.
In many game activities, players estimate game utility according to what is referred to as an uncertainty aversion.In this way, integrals with fuzzy measures should be suitable to model fuzzy games.Dow and Werlang [16,17] applied the Choquet integral to game theory and finance.The integral theory had made use of the concavification of a cooperative game that appeared in Weber [18] and later in Azrieli and Lehrer [19].Lehrer [20] proposed a new integral for capacities which differs from the Choquet integral on nonconvex capacities and discussed its properties in detail.
In fuzzy game situations, fuzzy capacities assign subjective expected values to some coalitions but not to all.Inspired by Lehrer's new integral, we introduce a new cooperative game form by the new integral with respect to fuzzy capacity.Further, it has been shown that the game defined by Tsurumi et al. is a special case when the fuzzy capacity satisfies convex in our new class of games.We have also defined a fuzzy core in order to provide a solution concept for the new proposed game.
The paper will be organized as follows.In Section 2, we introduce the concepts of crisp cooperative game and its imputation.In Section 3, some basic concepts of games with fuzzy coalitions and several game models, such as Butnariu and Tsurumi fuzzy games, will be given.In Section 4, we define a new fuzzy game by the concave integral and its several equal integral representations which described the fuzzy characteristic functions of the cooperative game will also be given.Meanwhile, some properties are discussed.We propose fuzzy core concept for the new game in Section 5 and its nonempty condition is given based on the fuzzy convexity.Moreover, we will give a calculating way of a fuzzy core.Finally, some conclusions appeared in Section 6.

Crisp Cooperative Game and Its Imputation
We consider cooperative games with a finite set of players  = {1, 2, . . ., } who may consider different cooperation possibilities.The set  is the grand set and any subset  of  can be seen as the grand coalition relative to  ⊆ .The notation () is the family of all crisp subsets (subcoalitions) of  ⊆ .
A crisp cooperative game on player set  is denoted by V, where the characteristic function V : () →  + ∪ {0} with V() = 0 and V() ( ∈ ()) is the worth of coalition  which can be seen as the global utility when the players work in the coalition  together.The class of crisp games with player set  is denoted by (, V).
The game V ∈ (, V) is said to be convex when If  and  are disjoint crisp coalitions in convex game V ∈ (, V), that is, then the game is said to be superadditive and we denote all the superadditive crisp cooperative games by  0 (, V).
A set of imputation of V ∈ (, V) is nonempty, and then the game is a superadditive crisp cooperative game, for example, V ∈  0 (, V).
The important solutions, such as the core and Shapley value, are imputation for a crisp cooperative game.The core on crisp game V ∈  0 (, V) is a convex set including all undominated imputations such that The Shapley value of player  is a probabilistic value and has a unique expression, which can be considered as the expectation of his marginal contribution V( ∪ ) − V() to any coalition  ∈  ( − ), which is where | ⋅ | is the cardinality of a coalition.Note that when V ∈  0 (, V), then the Shapley vector

Some Concepts of Game with Fuzzy Coalitions
A fuzzy coalition S is a fuzzy subset of the finite set  = {1, 2, . . ., }, which assigned a real valued function from  to [0, 1].In other words, a fuzzy coalition can be represented by a vector  = ( 1 ,  2 , . . .,   ), where   ∈ [0, 1] is a constant denoting the membership grade of player  in the fuzzy coalition S. Of course, the set  is the grand coalition and  is the empty coalition.It corresponds to the situation where the players participate fully in S; that is, each element of the level vector  = ( 1 ,  2 , . . .,   ) has participation level 1, and the players outside S are not involved at all; that is, they have participation level 0.
The set of all fuzzy coalitions in  is denoted by ().The support set of fuzzy coalition S ∈ () defined by Supp( S) = { ∈  |   > 0} which is a subset of ; its level subset denoted by S is [ S]  = { ∈  |   ≥ }.For  ∈ [0, 1], -section for S is the set S = { |  ∈ ,   = } which means a player set with the same level.Suppose the fuzzy coalition S and Ũ have vector  and  respectively, then   ≤   (∀ ∈ ) means that S ⊆ Ũ.A fuzzy cooperative game Ṽ is the function Ṽ : () →  + ∪ {0} with V() = 0. G(, Ṽ) denotes the class of all fuzzy games Ṽ.
In this paper, we assume that every of the fuzzy coalitions maps into the lattice ([0, 1], ∧, ∨), where ∧ and ∨ are the minimum and maximum operators, respectively.
For any fuzzy coalition S, Ũ ∈ (), the union of two fuzzy coalitions S and Ũ is denoted as S ∪ Ũ which satisfies Similarly, the intersection S ∩ Ũ satisfies Corresponding to the convex game in crisp game, the convex fuzzy game is defined as follows.
In fuzzy game literature, there are several game models which were aggregated function on fuzzy level coalitions, such as Butnariu game, Butnariu and Kroupa game, and Tsurumi game.Based on the definition of -section, Butnariu [13] proposed a fuzzy game with proportional values which was weighted by the participation level .Definition 4. The game V  ∈ G(, Ṽ) is said to be with proportional values if and only if It should be noted that there is a one-to-one correspondence between a crisp game and a fuzzy game with proportional values, because the characteristic function is a linear aggregation function which is a weighted average on the sets with the same participation levels.For the sake of simplicity, we will denote the fuzzy game with proportional values as the notation   (, V).
As the extension of games with proportional value, Butnariu and Kroupa [14] proposed a class of games with weight function.
The set of games with weight functions is denoted by   (, V).If () = , then the fuzzy game V ∈   (, V) is equivalent to the game V ∈   (, V).
It is obvious that the characteristic function of a cooperative game with proportional values or weight functions is a linear aggregation function.For any fuzzy game V ∈   (, V), there is no excess of any two players with different participation levels, and the payoffs of a fuzzy coalition are only a simple accumulation of utility created by players with the same participation level.These two fuzzy game models defined by Butnariu cannot embody the interaction among players with different participation levels.
After considering Butnariu's approach, Tsurumi et al. thought that most of this class games were neither monotone nondecreasing nor continuous with regard to rates of players' participations although crisp games are often considered to be monotone nondecreasing.In other words, these games cannot be regarded as quite natural.Tsurumi defined a payoff function and a fuzzy population monotonic allocation scheme as extensions of imputation and population monotonic allocation scheme.The following definitions and theorems were introduced.Definition 6.Given S ∈ (), let ( S) = {  |   > 0,  ∈ } and let ( S) be the cardinality of ( S).The elements in ( S) are rewritten by the increasing order as ℎ 1 < ℎ 2 < ⋅ ⋅ ⋅ < ℎ ( S) .
Then, a game V ℎ : () →  is said to be a fuzzy game with Choquet integral form if and only if the following holds: for any S ∈ (), where ℎ 0 = 0.
It is apparent that the fuzzy game model proposed by Tsurumi, which incorporated the notion of vague expectation along with fuzzy coalitions, is a Choquet integral of the function ℎ with respect to V derived from level set.We note that there is also a one-to-one correspondence between a crisp game and a fuzzy game with Choquet integral form.For the sake of simplicity, a fuzzy game with Choquet integral form is denoted by  Ch (, V).
In the above definition, let a set Tsurumi et al. had proved that the game V ∈  Ch (, V) has the following properties.Proposition 7. Let V ∈  Ch (, V), for any K, Ũ ∈ () and K ⊆ Ũ; then the following holds: Proposition 8. Let V ∈  Ch (, V); define the distance ( K, Ũ) = max ∈ |  −   | for any K, Ũ ∈ (); then V is continuous.
Proposition 9. Let V ∈  Ch (, V) and S, T ∈ () such that V( S) = V( T) if and only if

The Concave Integral Representation for Fuzzy Cooperative Game
The characteristic function of fuzzy game with Choquet integral form defined according to the Choquet integral is an expected value of utility with respect to a nonadditive probability distribution.Players may choose the act that maximizes the expected utility so that the one that achieves the maximum of the respective value is chosen.Since a particular decomposition of ( S) rather than all possible decompositions is used for the calculation of the Choquet integral, the value V Ch ( S) may not be a more reasonable outcome, while the method related to the concave integral seems to be more suitable to measuring the productivity of a coalition.
Let  be a finite set (|| = ); a capacity  over  is a function  : () →  + ∪ {0} such that  ⊆  ⊆  implies () ≤ () with () = 0.A random variable over  is a function  :  →  and a random variable is nonnegative if () ≥ 0 for every  ∈ .Definition 10.Let  be a random variable; a subdecomposition of  is a finite summation Definition 11.Let  be a capacity over  and let  be a nonnegative random variable; define the concave integral as where the minimum is taken all over concave and homogeneous functions  :   + →  such that (1  ) ≥ V(), for every  ⊆ , where 1  is the indicator of  which is the random variable that takes the value 1 over  and the value 0 otherwise.
Let  and  be two capacities, if  ≥  implies that () ≥ () for every  ⊆ .Lehrer had proved the concave integral properties.
Proposition 12.For every nonnegative  defined over , where  is additive and  ≥ .
Note that the capacities  need not be a probability distribution and () = () is not necessary.
Let   =  () −  (−1) ; note that  = ∑   1   is a decomposition of .That is to say, the Choquet integral is defined under a special decomposition of .By contrast, all possible decompositions are allowed in the concave integral.

It means that ∫
Ch   ≤ ∫ Cav   for any .In addition, In fuzzy game literature, many researchers devote lots work to searching for a better expression of fuzzy game.But most of them were usually limited to the participation levels of players and the payoffs of crisp coalitions.
It should be noted that there are some fuzzy coalitions whose payoffs cannot be expressed by crisp coalition values and participation levels.As a result, their method of constructing fuzzy characteristic function, which is only limited to some special game, will be invalid in many game situations.Inspired by better properties of the concave integral, we follow the method of Tsurumi game to define a new class fuzzy game, where Tsurumi game can take a special case as the proposed new game.
We will define a new class of fuzzy games which has its good properties, as will be shown in what follows.
Remark 14.In Definition 13, min{()} is the characteristic function of the fuzzy games with concave integral form and the minimum is taken over all concave and homogeneous functions  :   + → .Since the domain of the minimum of the family function of concave and homogeneous functions  is   + , so V Cav ( S) is concave and homogeneous and the class of the fuzzy games with concave integral form is nonempty.
We denote all the fuzzy games with concave integral form as  Cav (, V).In fact, from Definition 10, the characteristic function of the fuzzy game with concave integral form can be gained by the subdecomposition of  and the characteristic function of subdecomposition crisp coalitions.
and V(1, 2, 3) = 10.Let the fuzzy coalition  = (1, 0.4, 0.6); then, by (21), we have that Hence, So However, by the method given by Tsurumi et al., we rearrange the factor of fuzzy coalition variable  = (1, 0.4, 0.6) such that 0.4 < 0.6 < 1.We get the level sets [ S] 0.4 = {1, 2, 3}, [ S] 0.6 = {1, 3}, and [ S] 1 = {1}; then we have It is easy to see that the outputs of the fuzzy game are different and V ℎ ( S) < V Cav ( S).Therefore, the characteristic function given by Tsurumi et al. is not the maximal product such that Tsurumi fuzzy game is not more suitable than the proposed method by the concave integral in some situations.
We know that V Cav ( S) = ∫ Cav  V is the maximum of the values ∑  =1    (  ) among all possible decompositions of S with the coalition variable .The fuzzy game given by the concave integral imposes no restriction over the decompositions being used; that is, all possible decompositions are taken into account when considering the maximum.Although the fuzzy game given by the Choquet integral can also be expressed in terms of decompositions, unlike the concave integral, Choquet integral instead does impose restrictions.
The fuzzy game defined by the Choquet integral is the maximum of ∑  =1    V(  ) over all decompositions in which every   and   are nested.It is evident that V Ch ( S) ≤ V Cav ( S).
The game V ∈  Cav (, V) has the following properties.
We take the notation Ẽ(V Cav )( Ũ) for the set of all imputations of the fuzzy game V ∈  Cav (, V) on the restricted fuzzy coalition Ũ ∈ ().
Note that the definition above is also suitable to crisp games.Butnariu [13] and Tsurumi et al. [3] have also proposed the imputation concepts, but these definitions are different from the definition above.
Generally, characteristic functions of games with fuzzy coalitions are difficult to describe clearly in practice so that many fuzzy game models constructed their characteristic function by aggregating one of the crisp games.It means that the game with fuzzy coalition can be represented by a mapping from the characteristic functions of the crisp games to that of the game with fuzzy coalitions, such as Owen fuzzy game, Butnariu fuzzy game, Tsurumi et al. fuzzy game, and the new fuzzy game proposed in this paper.
In the following, we will give another solution for games with fuzzy coalitions, that is, the fuzzy core.At first, we extend the core of crisp game as the imputations for game with fuzzy coalitions.
Definition 24.Let Ũ ∈ ().The fuzzy core for a game V ∈  Cav (, V) in the restricted fuzzy coalition Ũ is the convex set C(V Cav )( Ũ), that is, After defining the fuzzy core for a game V ∈  Cav (, V), we define an excess of a fuzzy coalition similar to that of a crisp coalition.