Solidarity Value and Solidarity Share Functions for TU Fuzzy Games

TU games under both crisp and fuzzy environments describe situations where players make full (crisp) or partial (fuzzy) binding agreements and generate worth in return. The challenge is then to decide how to distribute the profit among them in a rational manner: we call this a solution. In this paper, we introduce the notion of solidarity value and the solidarity share function as a suitable solution toTU fuzzy games. Two special classes of TU fuzzy games, namely, TU fuzzy games inChoquet integral formand in multilinear extension form, are studied and the corresponding solidarity value and the solidarity share functions are characterized.


Introduction
A cooperative game with transferable utility, or simply a TU game, is a pair (, V), where  is a set of  players, called the grand coalition, and V is the characteristic function defined on 2  that assigns to every subset (coalition) a real number called its worth which gives zero worth to the empty coalition.Let  0 denote the class of TU games.A solution for any TU game is a function on  0 which assigns to the TU game a distribution of payoffs for its players.If there is no ambiguity on the player set , we denote by  0 () the class of all TU games with the fixed .
Among the various one-point solutions for TU games, the Shapley value [1] and the solidarity value [2] are perhaps the most popular ones.The Shapley value builds on the axioms of efficiency, linearity, anonymity, and the null player.The solidarity value on the other hand is characterized by efficiency, linearity, anonymity, and the axiom of ]-null player.The null player axiom of the Shapley value rewards nothing to the nonperforming players.However in recent years solidarity has been considered as an important human attribute influencing both rationality (limited rationality) and social preference for fairness [3][4][5].Therefore the role of solidarity in TU games is essentially discussed in the literature and the notion of the solidarity value was proposed as an alternative to the Shapley value.It follows that, unlike the Shapley value, the solidarity value expresses solidarity to both the nonperforming and performing players; see [2].
As an alternative to the values, the share functions are proposed in [6] as useful solution concepts for TU games that assign to every game a vector whose components add up to one.A share function determines how much share a player can get from the worth of the grand coalition and therefore is devoid of the efficiency requirement as opposed to the other standard value functions.Therefore a share function simplifies the model formulation to a great extent.In [7], it is shown that, on a ratio scale, meaningful statements can be made for a certain class of share functions, whereas all statements with respect to the value functions are meaningless.The share function corresponding to the solidarity value is called the solidarity share function.It is obtained by dividing the solidarity value of each player by the sum of the solidarity values of all the players.In [8] the solidarity share function for TU games is studied in detail.
Cooperative games with fuzzy coalitions or simply TU fuzzy games are a generalization of the ordinary TU games 2 Advances in Fuzzy Systems in the sense that participation of the players in a fuzzy coalition belongs to the interval [0, 1]; see [9].A fuzzy coalition is a fuzzy subset of the player set  which assigns a membership grade to its members.This membership of a player in a coalition represents her rate of participation in it.When distinction between the two classes of games is needed, we call the standard TU game the crisp TU game or simply the TU game.TU fuzzy games derived from their crisp counterparts are found in the literature; see, for example, [9][10][11][12][13][14][15].The Shapley share function for TU fuzzy games is studied in [16].The relevance of the solidarity value and the corresponding share function for TU fuzzy games can be realized in situations where players with partial participations are marginally unproductive but being the part of the cooperative endeavor may be rewarded with some nonzero payoffs.In this paper, we introduce the notion of solidarity value and solidarity share functions for TU fuzzy games.A set of axioms to characterize these functions is proposed.We define two classes of TU fuzzy games, namely, the TU fuzzy games in Choquet integral form due to [15] and TU fuzzy games in multilinear extension form due to [17].These two classes are continuous with respect to the standard metric and also monotonic when the associated crisp game is monotonic.Moreover they build on the idea of nonadditive interactions among the players; for more details we refer to [15][16][17].
The rest of the paper proceeds as follows.In Section 2, we compile the related definitions and results from the existing literature.Section 3 discusses the solidarity share functions for TU fuzzy games.In Section 4 we discuss the solidarity share functions for TU fuzzy games in Choquet integral form followed by some illustrative example.Section 5 concludes the paper.

Preliminaries
In this section we compile the definitions and results necessary for the development of the present study from [2,6,8,9,11,12,15,16,18].We start with the notion of solidarity values and share functions in crisp games.

The Solidarity Value and the Share Function for TU Games.
Let the player set  be fixed so that the class of TU games can be taken as  0 ().We define the following.
] is called the average marginal contribution of a player of the coalition .Definition 2. Given a game V ∈  0 (), player  ∈  is called a ]-null player if ] V () = 0, for every coalition  ⊆  containing .
Proof.We refer to [8] for a detailed proof of Theorem 4.
From now onward we denote the solidarity value by Φ sol (, V) where  ⊆  and V ∈  0 ().Definition 5. Let  ⊆  0 () be a set of TU games, and let  :  → R be a given function.A -share function on a set of games  ⊆  0 () is a function Ψ  :  → (R  + ) 2  that satisfies the following Axioms CS 1 , CS 2 , and CS 3 and either Axiom CS 4 or CS 5 : Axiom CS 4 (-Additivity).For any pair Axiom CS 5 (-Linearity).For any pair V 1 , V 2 of games in  and for any pair of real numbers  and  such that

TU Games with Fuzzy
Coalitions.Now we make a brief discourse of TU games with fuzzy coalitions or simply TU fuzzy games with the player set .A fuzzy coalition is a fuzzy subset of , which is identified with a characteristic function from  to [0, 1].Let () be the set of all fuzzy coalitions in .For a fuzzy coalition  ∈ () and player  ∈ , () represents the membership grade of  in .The empty fuzzy coalition denoted by 0 is one where all the players provide zero membership.If no ambiguity arises we use the same notations to represent crisp and fuzzy coalitions as crisp coalitions are special fuzzy coalitions with memberships 0 or 1.
For  ∈  and  ∈ (), the fuzzy coalitions   and  − are given by the following: Definition 7. A TU game with fuzzy coalitions or simply a TU fuzzy game is a pair (, V) where  ∈ () and V : () → R are a set function, satisfying V(0) = 0.
Let FG() denote the class of TU fuzzy games with player set . Now we define the two classes of TU fuzzy games, namely, the TU fuzzy games in Choquet integral form due to [15] and TU fuzzy games in multilinear extension form due to [17].As an a priori requirement, the following definition is given.
The set of all TU fuzzy games in Choquet integral form is denoted by FG  ().
Definition 10 (see [12]).For any given  ∈ () and V  ∈  0 (), a TU fuzzy game V ∈ FG() generated by V  and given by is said to be a TU fuzzy game "in multilinear extension form."The set of all TU fuzzy games with form is denoted by multilinear extension form FG  ().
Following section includes the main contribution of the present study.

Solidarity Value for TU Fuzzy Games
We now discuss the notion of a solidarity value to the class FG() of TU fuzzy games with player set  along the line of [2].Begin with the following definition.
is called the average marginal contribution of a player of the fuzzy coalition .
Note that Axioms F 1 -F 4 are standard axioms derived from their crisp counterparts and therefore can be applied to any class of fuzzy games.Moreover, when we revert back to the class of crisp games these axioms become the standard Axioms CS 1 -CS 4 .

Solidarity Value for the Class 𝐹𝐺 𝐶 (𝑁).
We now find the solidarity value for the class FG  () by use of the following theorem. ), defined by

Theorem 14. Let V ∈ 𝐹𝐺 𝐶 (𝑁) and 𝑈 ∈ 𝐿(𝑁). A function
is a solidarity value function in  for V ∈   (), where Proof.Recall from Theorem 4 that there exists a unique function Φ sol satisfying Axioms CS 1 -CS 4 .We use this to prove that the function Ω sol satisfies Axioms F 1 -F 4 .

Solidarity Value for the Class 𝐹𝐺 𝑀 (𝑁)
Theorem 15.For 0 ̸ =  ⊆  ∈ (), the game û , that is, has the following properties: and every player  ∈ supp \supp  is ]-null in the game û .
Proof.Let us construct any value function  on FG  () satisfying efficiency, symmetry, additivity, and ]-null player axioms by Now, we know that V ∈ FG  () can be expressed by where clearly Ω sol given by ( 21) and ( 22) satisfies symmetry and ]null player axioms.Moreover, Ω sol is a linear mapping.Hence additivity is satisfied.Using now linearity of Ω sol , we get which proves that Ω sol is efficient.It is obvious that (, û ) = Ω sol  (, û ) for each game û .Thus (, V) = Ω sol  (, V) for every V ∈ FG  ().

Solidarity Share Functions for TU Fuzzy Games
We now extend the notion of a share function to the class FG() of TU fuzzy games with player set .In the line of its crisp counterpart we assume here also that the share function assigns to each player her share in the payoff V() of the fuzzy coalition  ∈ ().Therefore we provide the following definitions as an extension to their crisp versions.

Advances in Fuzzy Systems
Definition 17.A real valued function  : () × FG() → R is called -additive if, for  ∈ () and any pair ) () that satisfies the following axioms, that is, Axioms FS 1 -FS 3 along with Axiom FS 4 or Axiom FS 5 .
Note that Axioms FS 1 -FS 5 are intuitive of their crisp counterparts in the sense that reverting back to the crisp formulation we get the standard axioms of share functions.It follows that for any V ∈ FG() a solidarity -share function Ψ  gives a payoff Ψ  (, V)⋅V() to player  when she is involved in the fuzzy coalition  and satisfies the above-mentioned axioms.
4.1.Solidarity Share Functions for   ().In this section we prove the existence and uniqueness of the solidarity -share function for the class FG  () of fuzzy games in Choquet integral form.To discuss the existence and uniqueness of the solidarity -share function for TU fuzzy game in FG  () we have to use some classical results from [1,2].Recall that, given a coalition  ⊂  ∈ (), the game   is defined as follows: , if  ⊃  0, otherwise.
Following similar procedure as in Lemma 3.2. of [10], we can have, for V ∈ FG  (), (32) It follows from the above discussion that V() can be rewritten as (V)   ())
Proof.The proof proceeds in the line of [6].First we suppose that Ψ  satisfies -efficiency and additivity.It follows that Ψ  is -additive on  0 ().Thus we have

)
Theorem 6.Let  :  → R be a positive function on .Then on the subclass  there exists a unique solidarity -share function Ψ  :  → (R +  ) 2  satisfying the axioms CS 1 -CS 4 if and only if  is additive on .