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Fuzzy Sets and Systems 198 (2012) 46–58www.elsevier.com/locate/fss

Bi-cooperative games with fuzzy bi-coalitions

Surajit Borkotokeya,∗, Pranjal Sarmahb

a Department of Mathematics, Dibrugarh University, Dibrugarh 786004, Indiab Department of Mathematics, Gauhati University, Guwahati 781014, India

Received 17 May 2010; received in revised form 6 October 2011; accepted 7 October 2011Available online 15 October 2011

Abstract

In this paper, we introduce the notion of a bi-cooperative game with fuzzy bi-coalitions and discuss the related properties. In realgame theoretic decision making problems, many criteria concerning the formation of coalitions have bipolar motives. Our modeltries to explore such bipolarity in fuzzy environment. The corresponding Shapley axioms are proposed. An explicit form of theShapley value as a possible solution concept to a particular class of such games is also obtained. Our study is supplemented with anillustrative example.© 2011 Elsevier B.V. All rights reserved.

Keywords: Fuzzy sets; Bi-cooperative Games; Bi-coalitions; Shapley function

1. Introduction

Given a finite set N of n players, and Q(N ), the set of all pairs (S, T ) with S, T ⊆ N , and S ∩T = ∅, a bi-cooperativegame is defined as a function b : Q(N ) → R satisfying b(∅, ∅) = 0. The pairs (S, T ) are called bi-coalitions. Playersin S are usually called positive contributors while those of T are negative contributors to the game and the membersof N \ (S ∪ T ) are the neutral players. Thus for each (S, T ) ∈ Q(N ), b(S, T ) represents the worth when playersin S contribute positively, those in T contribute negatively and the remaining players do not participate. Note thathere, worths may be both positive and negative. Bi-cooperative games have been introduced by Bilbao et al. [3] as ageneralization of TU cooperative games. In the literature on game theory, however, the first works on the introductionof several levels of cooperations are those of multi-choice games [16]. The major domain of application of games withseveral “levels of participations” is the “Cost Allocation Problems” [24]. The idea of choosing between two mutuallyexclusive roles by the players in a cooperative situation was originated from Felsenthal and Machover [9] who definedternary voting games.

A one point solution concept for bi-cooperative games is a function, which assigns to every game an n-dimensionalreal vector that represents a payoff distribution over the players. In [17], Labreuche et al. have axiomatized a valueintuitively. A value for bi-cooperative games has also been defined in [3] (and similarly in [13,14]), however the two

∗ Corresponding author. Tel.: +91 9954430677.E-mail address: surajitbor@yahoo.com (S. Borkotokey).

0165-0114/$ - see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2011.10.006

S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58 47

approaches differ by principle. While the later is isomorphic to multi-choice games with three levels of participations,the former is a bipolar extension of the power set of N .

Theory of fuzzy sets since its inception in 1965, by Zadeh [31], has become an important tool to model humanbehaviour in decision sciences. A large amount of research works has been carried out in the theory of cooperativegames in fuzzy environment, see for example [1,2,4–6,18,19,25]. When players partially participate in a coalition withsome membership degrees in the interval [0, 1], we call it a fuzzy coalition. In a similar fashion, when players participatepartially in a bi-coalition (i.e. in either of the mutually exclusive roles), we can call it a fuzzy bi-coalition. As shownin the following example, there are situations where the notion of a fuzzy bi-coalition essentially makes sense.

Example (The issue of mega dam construction). The issue of constructing mega dams over some of the rivers inIndia has been creating mass concern both in support and against. Mega dams are built mainly for generating hydro-electricity. The bigger is the dam, the more electricity is produced. Environmentalists and other pressure groups usuallyoppose such drives citing to adverse ecological impact on the downstream areas and rather advocate for small ones.A substantial amount of water stored in those dams can be used for irrigation, for maintaining a steady flow in the riverto control flood and other welfare projects. A reduction of the height of dam walls would lower the possibility of largemass destruction during disaster. The agencies, involved in construction and functioning of the dams, on the other hand,benefit more from mega projects, as the production and maintenance costs for the small ones and those involving otherwelfare components are comparatively high. Meanwhile, there are also agencies, who would mobilize the commonilliterate people against such projects for their own political motives. Thus, the issue of mega dam creates three disjointgroups of activists: the supporters’ group, the opposition group and the neutral group whose members are indifferent tothe issue. However, in this case, members of either groups (except the neutral group) will not only rest by supporting oropposing the move and rather would try to exert pressure over the other group in support of their own stand by means ofscientific expertize, mass awareness and mobilization, shouldering financial responsibilities, forming political pressuregroups and so on to name a few. If we can measure the levels of the contributions of each individual in a group bysome suitable mean, then this situation, where participation of the players are bipolar yet layered in the sense that eachmember provides her contribution (ranging between 0 and 1) in running the project or to scrap it or remains indifferent(by contributing nothing to either of the groups) would require a rather generalized game theoretic framework. The gamerepresenting such situations would be called a bi-cooperative game with fuzzy bi-coalitions (or fuzzy bi-cooperativegame in short). Investment problems in Share markets, coalition formation among political parties, cost allocationsinvolved in sharing of technology and so on are some other pertinent examples. In the literature so far, no such studyon the use of fuzzyness for describing a bi-cooperative game is noticed.

In this paper, we introduce the notion of a bi-cooperative game with fuzzy bi-coalitions which has significant analogywith cooperative games with fuzzy coalitions discussed by Aubin [2], Butnariu [5], Tsurumi et al. [25], Butnariu andKroupa [6], Zhang et al. [18] and so on. The fuzzy games hitherto found in the literature do not address the problemof putting memberships in either of the two mutually exclusive roles. It is indeed interesting to see that the existingmodels [2,5,6,18,25] do not address the effects of players participating in such roles, rather investigate how to distributethe gain among the members according to their rates of participation in a single fuzzy coalition. Our study is based ona generalization of the bi-cooperative games into its fuzzy counterpart. We propose the axioms of a Shapley value as apossible solution concept to a bi-cooperative game with fuzzy bi-coalitions. These axioms are applicable to any classof bi-cooperative games with fuzzy bi-coalitions. We further introduce a new family of fuzzy bi-cooperative gamessimilar to ones in cooperative settings, however, they differ from their counterparts by the formulation of the problemdomain. This class, whose members will be called fuzzy bi-cooperative game in Choquet Integral form, is an extensionof the one given by Tsurumi et al. [25]. Grabisch [12] gave a natural justification of using Choquet Integrals as a linearinterpolator between vertices of the [−1, 1]n hypercube using the least possible number of vertices, in the context ofmulti-criteria decision making. Since a fuzzy game can be seen as an extension of a crisp game (defined on the verticesonly), to the entire hypercube, and the Choquet Integral in both cooperative and bi-cooperative format is the best linearinterpolator, it becomes a natural choice for extending a crisp game to its fuzzy counterpart in this way. This class alsopossesses monotonicity (whenever the corresponding crisp game is monotone) and continuity with respect to players’memberships in both positive and negative roles. An explicit form of the Shapley function for this class of games hasbeen obtained and finally an illustrative example is provided to supplement our findings. We have used the propertiesof a signed Choquet integral discussed in [7,8,10,11,20–23,26–30] in obtaining the Shapley value of this class.

48 S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58

This paper is organized as follows: in Section 2, we discuss the relevant properties of a bi-cooperative game andthe corresponding solution concepts. In Section 3, we introduce the notion of a bi-cooperative game with fuzzybi-coalitions and propose the Shapley axioms for this game. In Section 4, a special class of bi-cooperative games withfuzzy bi-coalitions is defined and corresponding properties are investigated. A possible Shapley function for this classhas been investigated and characterized with the Shapley axioms. Sections 5 and 6, respectively, include an illustrativeexample and the concluding remarks. Throughout the paper, we denote by 2N , the set of coalitions of N . We will oftenomit braces for singletons, e.g. writing S ∪ i and S \ i instead of S ∪ {i} and S \ {i} respectively whenever S ⊆ N .Moreover cardinality of a subset S of N will be denoted by corresponding lower case letter, i.e. |S| =: s.

2. Bi-cooperative games and a solution concept

Let us denote byBGN the class of all bi-cooperative games on N . While in standard cooperative games, each coalitionS ∈ 2N can be identified with a {0, 1}-vector, in a bi-cooperative game, each bi-coalition (S, T ) can be identified withthe {−1, 0, 1}-vector 1S,T defined, for all i ∈ N , by

1S,T (i) =

⎧⎪⎨⎪⎩

1 if i ∈ S

−1 if i ∈ T

0 otherwise

It is indeed interesting to note that the bi-cooperative games given in [3,13,14] are isomorphic to multi-choice gamesgiven in [16] with three levels of participation. This is because the order relation denoted by in Q(N ) is the oneimplied by monotonicity (i.e. for (S, T ), (S′, T ′) ∈ Q(N ), (S, T ) (S′, T ′) iff S ⊆ S′ and T ′ ⊆ T ). This makesthe two elements (∅, N ) and (N , ∅) as the bottom and top elements of Q(N ). Nevertheless, in [17], the order relationis taken as the product order (i.e. for (S, T ), (S′, T ′) ∈ Q(N ), (S, T ) (S′, T ′) iff S ⊆ S′ and T ⊆ T ′) so that(∅, ∅) becomes the bottom element and all (S, N \ S), S ⊆ N , the top elements. This idea adheres to the notion ofbi-cooperative games as it incorporates bipolarity, a crucial concept for such games. Therefore, in this paper, we shalladopt this later approach.

In [17], a value on BGN is defined as a function � : BGN → (Rn)Q(N ) which associates each bi-cooperative game ba vector (�1(b), �2(b), . . . , �n(b)) representing a payoff distribution to the players in the game. Note that the elements�i (b) belong to RQ(N ) and hence for (S, T ) ∈ Q(N ), �i (b)(S, T ) ∈ R, ∀i ∈ N . It is further interesting to note thatthere are two more definitions of a value found in the literature (see [17]). In [3,13,14], the value is defined as a function� : BGN → Rn . In the same references, a refined value is also defined as a function � : BGN → (Rn)2. However,we shall adopt here, the definition given in [17] as it endorses bipolarity of a bi-cooperative game in a more naturalway. There are different proposals of solution concepts hitherto found in the literature as already mentioned in theIntroduction. Among them the first proposal was put forward by Felsenthal and Machover [9] especially for ternaryvoting games. Their notion applies mainly to voting games and the corresponding expressions seem to be too technical.In proposals made by Labreuche and Grabisch [13,14] and Bilbao et al. [3], only the largest and least elements of thelattice (Q(N ), ) are of primary concerns. This would require all players to agree to a certain proposal or oppose it toform the bi-coalitions (N , ∅) and (∅, N ) respectively. However this requirement is too harsh to impose in most situationsand as already mentioned, in particular, does not comply with the notion of bipolarity present in a bi-cooperative game.Following [17], we have the next definition.

Definition 2.1. Let b ∈ BGN . A player i is called positively monotone with respect to b if

∀(S, T ) ∈ Q(N \ i), b(S ∪ i, T ) ≥ b(S, T )

A player i is negatively monotone with respect to b if

∀(S, T ) ∈ Q(N \ i), b(S, T ∪ i) ≤ b(S, T )

The bi-cooperative game b is monotone if all players are positively and negatively monotone with respect to b.

Let BGM (N ) denote the class of monotone games over Q(N ).

S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58 49

Remark 2.2. The expression b(S ∪ i, T ) − b(S, T ) (respectively b(S, T ) − b(S, T ∪ i)) may be called the marginalcontribution of player i with respect to (S, T ) ∈ Q(N \ i) when she is a positive contributor (respectively a negativecontributor).

Now, for (S, T ) ∈ Q(N ), set,

Q(S,T )(N ) = {(S′, T ′) ∈ Q(N ), S′ ⊆ S, T ′ ⊆ T } (2.1)

Prior to defining the value in a bi-cooperative game we define the following:

Definition 2.3. Let (S, T ) ∈ Q(N ) and b ∈ BGN . Player i ∈ N is a null player in (S, T ) for b, if it satisfies

b(S′ ∪ i, T ′) = b(S′, T ′) = b(S′, T ′ ∪ i) (2.2)

for every (S′, T ′) ∈ Q(S,T )(N \ i).

Note that the above definition of a null player is specified not only by the game, but also by a bi-coalition and is aslight variation of the one given in [17]. However it will not change the formulation of a value given there. This willbe evidenced as a simple consequence of Theorem 2.5, that the only terms of a bi-cooperative game b that are used todetermine the value belong to the set Q(S,T )(N ), defined in (4.1). Thus keeping this in mind, the following definitionof a value on BGN and its corresponding characterization are derived from that given in [17].

Definition 2.4. A function �′ : BGN → (Rn)Q(N ) defines a value due to Labreuche and Grabisch, for every (S, T ) ∈Q(N ), if it satisfies the following axioms:

Axiom b1 (Efficiency). If b ∈ BGN , it holds,∑i∈N

�′i (b)(S, T ) = b(S, T )

Axiom b2 (Linearity). For all �, � ∈ R and b, v ∈ BGN ,

�′i (�b + �v) = ��′

i (b) + ��′i (v)

Axiom b3 (Null player axiom). If player i is null for b ∈ BGN in (S, T ), then,

�′i (b)(S, T ) = 0

Axiom b4 (Intra-coalition symmetry). If b ∈ BGN and a permutation � is defined on N , such that �S = S and �T = T ,then it holds that, for all i ∈ N , �′

�i (�b)(�S, �T ) = �′i (b)(S, T ) where �b(�S, �T ) = b(S, T ) and �S = {�i : i ∈ S}.

Axiom b5 (Inter-coalition symmetry). Let i ∈ S and j ∈ T , and bi , b j be two bi-cooperative games such that for all(S′, T ′) ∈ Q(S,T )(N \ {i, j}),

bi (S′ ∪ i, T ′) − bi (S′, T ′) = b j (S′, T ′) − b j (S′, T ′ ∪ j)

bi (S′ ∪ i, T ′ ∪ j) − bi (S′, T ′ ∪ j) = b j (S′ ∪ i, T ′) − b j (S′ ∪ i, T ′ ∪ j)

Then,

�′i (bi )(S, T ) = −�′

j (b j )(S, T ) (2.3)

Axiom b6 (Monotonicity). Given b, b′ ∈ BGN such that ∃i ∈ N with

b′(S′, T ′) = b(S′, T ′) (2.4)

50 S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58

b′(S′ ∪ i, T ′) ≥ b(S′ ∪ i, T ′) (2.5)

b′(S′, T ′ ∪ i) ≥ b(S′, T ′ ∪ i) (2.6)

for all (S′, T ′) ∈ Q(S,T )(N \ i), then �′i (b

′)(S, T ) ≥ �′i (b)(S, T ).

Note that the value defined above is specified not only by the game but also by the bi-coalition (S, T ).

Theorem 2.5. Let �′ be a value on BGN for (S, T ) ∈ Q(N ). The value �′ satisfies Axioms (b1)–(b6) if and only if ithas for all i ,

�′i (b)(S, T ) =

∑K⊆(S∪T )\i

k!(s + t − k − 1)!

(s + t)![V (K ∪ i) − V (K )] (2.7)

where for K ⊆ S ∪ T , V (K ) := b(S ∩ K , T ∩ K ).

The following result as a corollary to Theorem 2.5 is also important for our findings:

Result 2.6. We have,

∀i ∈ N \ (S ∪ T ), �′i (b)(S, T ) = 0 (2.8)

∀i ∈ S, with i positively monotone, �′i (b)(S, T ) ≥ 0 (2.9)

∀i ∈ T, with i negatively monotone, �′i (b)(S, T ) ≤ 0 (2.10)

3. Bi-cooperative game with fuzzy bi-coalitions and Shapley value

In this section, we define a bi-cooperative game with fuzzy bi-coalitions and a corresponding Shapley function as asolution concept. Let us now define a fuzzy bi-coalition as follows:

Definition 3.1. Let N = {1, 2, . . . , n} be as usual the players’ set. A fuzzy bi-coalition is an expression A of Ngiven by

A ={〈i, �N

A (i), �NA (i)〉|i ∈ N , min

i∈N(�N

A (i), �NA (i)) = 0

}

where �NA : N → [0, 1], �N

A : N → [0, 1] represent, respectively, the membership functions over N of the fuzzy setsof positive and negative contributors of A.

Remark 3.2. Admission of the minimum condition in the above definition is justified by the fact that the twooptions (positive and negative contributions) being mutually exclusive, one cannot choose a little bit of both optionssimultaneously.

Thus, it is apparent from Definition 3.1 that a fuzzy bi-coalition A of N is fully characterized by the functions �NA

and �NA . As N is fixed here, now onwards we omit N from �N

A and �NA and simply use �A and �A . We call i a positive

contributor in A if �A(i) > 0 and a negative contributor if �A(i) > 0. Let FB(N ) denote the set of all fuzzy bi-coalitionson N . Moreover, every crisp bi-coalition can be considered as a fuzzy bi-coalition with memberships either 0 or 1.Thus with an abuse of notations, we write Q(N ) ⊆ FB(N ).

For comparing two A, B ∈ FB(N ), we adopt the following operations and relations in accordance with their crispcounterparts given in [17]:

A � B ⇔ �A(i) ≤ �B(i) and �A(i) ≤ �B(i) ∀i ∈ N

A = B ⇔ �A(i) = �B(i) and �A(i) = �B(i) ∀i ∈ N

For any A ∈ FB(N ), we denote by FB(A), the set of all fuzzy bi-coalitions B such that B � A.

S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58 51

The union and intersection of two fuzzy bi-coalitions A and B are obtained using the maximum and minimumoperators “∨” and “∧” respectively. Formally we have:

A ∪ B = {〈i, �A(i) ∨ �B(i), �A(i) ∨ �B(i)〉|i ∈ N } (3.1)

A ∩ B = {〈i, �A(i) ∧ �B(i), �A(i) ∧ �B(i)〉|i ∈ N } (3.2)

The Support of a fuzzy bi-coalition A, denoted by Supp(A) is given by

Supp(A) = ({i ∈ N |�A(i) > 0}, {i ∈ N |�A(i) > 0}) (3.3)

Definition 3.3. The null fuzzy bi-coalition ∅B is given by

∅B = {〈i, �∅B(i), �∅B (i)〉|i ∈ N }

where �∅B(i) = 0, and �∅B (i) = 0 ∀i ∈ N .

Let us define a bi-cooperative game with fuzzy bi-coalitions as follows:

Definition 3.4. A bi-cooperative game with fuzzy bi-coalitions is a function w : FB(N ) → R with w(∅B) = 0. Wecall the value w(A), the worth of A due to the fuzzy or partial contributions by the members of N .

The worth w(A) for every A ∈ FB(N ) may be interpreted as the gain (whenever w(A) > 0) or loss (wheneverw(A) < 0) that A can receive when the players can participate in it in two distinct capacities. Now onwards, weshall call a “bi-cooperative game with fuzzy bi-coalitions” a “fuzzy bi-cooperative game” in short. Let us denote byGFB(N ) the class of all fuzzy bi-cooperative games. It is apparent from the above definition that the class BGN , ofcrisp bi-cooperative games is a subclass of the class GFB(N ) of fuzzy bi-cooperative games.

Definition 3.5. Let w ∈ GFB(N ). Player i is called positively monotone in fuzzy sense with respect to w if for everyA ∈ FB(N ) with �A(i) = 0 = �A(i), and all I ∈ FB(N ) such that �I (i) > 0 and �I ( j) = 0 = �I ( j) for j � i ,we have w(A ∪ I ) ≥ w(A). Similarly, player i is negatively monotone in fuzzy sense with respect to w if for everyA ∈ FB(N ) with �A(i) = 0 = �A(i), and all I ∈ FB(N ) such that �I (i) > 0 and �I ( j) = 0 = �I ( j) for j � i , we havew(A ∪ I ) ≤ w(A).

The game w ∈ GFB(N ) is monotone in fuzzy sense if every player is both positively and negatively monotone infuzzy sense.

Preparatory to the definition of the Shapley axioms for a fuzzy bi-cooperative game, we describe the following:

Definition 3.6. If A ∈ FB(N ), and w ∈ GFB(N ), the player i ∈ N is said to be null for w in A if w(B ∪ I ) = w(B)for all B ∈ FB(A) with �B(i) = �B(i) = 0 and all I ∈ FB(N ) such that �I ( j) = �I ( j) = 0 when j � i , where ∪ isdefined in relation (3.1).

Definition 3.7. Let A ∈ FB(N ), for any permutation � on N , define the fuzzy bi-coalition �A by

��A(i) = �A(�−1i) (3.4)

��A(i) = �A(�−1i) (3.5)

Then �A is called a permutation of the fuzzy bi-coalition A.

52 S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58

Now we define the Shapley function for bi-cooperative games with fuzzy bi-coalitions as follows:

Definition 3.8. A function � : GFB(N ) → (Rn)FB (N ) is said to be a Shapley function on GFB(N ) if it satisfies thefollowing six axioms:

Axiom f1 (Efficiency). If w ∈ GFB(N ) and A ∈ FB(N ), then∑i∈N

�i (w)(A) = w(A)

Axiom f2 (Linearity). For �, � ∈ R and w, w′ ∈ GFB(N ) we must have

�(�w + �w′) = ��(w) + ��(w′)

Axiom f3 (Null player axiom). If player i ∈ N is a null player for w ∈ GFB(N ), in A ∈ FB(N ), then, �i (w)(A) = 0.

Axiom f4 (Intra-coalition symmetry). For any w ∈ GFB(N ), and a permutation �, defined on N , it holds for all i ∈ N ,

�i (w)(A) = ��i (�w)(�A) (3.6)

where �w ∈ GFB(N ) is defined by �w(�A) = w(A), with �A defined as in Definition 3.7.

Axiom f5 (Inter-coalition symmetry). Given A ∈ FB(N ) and i, j ∈ N , if wi and w j are two bi-cooperative gameswith fuzzy bi-coalitions such that for every B ∈ FB(A) with i, j /∈ Supp(B) (i.e. �B(i) = 0 = �B( j) and �B(i) = 0 =�B( j)), and every pair of I, J ∈ FB(N ), such that �I (i) = �J ( j) > 0 or �J ( j) = �I (i) > 0 and �I (k) = �J (k) = 0 =�I (k) = �J (k) ∀k ∈ N \ {i, j}, it holds that

wi (B ∪ I ) − wi (B) = w j (B) − w j (B ∪ J )

wi (B ∪ I ∪ J ) − wi (B ∪ J ) = w j (B ∪ I ) − w j (B ∪ I ∪ J )

then, �i (w)(A) = −� j (w)(A).

Axiom f6 (Monotonicity). Let w and w′ be two bi-cooperative games with fuzzy bi-coalitions and A ∈ FB(N ). Letfurther that there exists an i ∈ N such that for every I ∈ FB(N ) with �I (i) > 0 or �I (i) > 0, �I ( j) = �I ( j) = 0, ∀ j�i ,and for all B ∈ FB(A) such that �B(i) = �B(i) = 0, it holds that

w′(B) = w(B)

w′(B ∪ I ) ≥ w(B ∪ I )

Then, �i (w′)(A) ≥ �i (w)(A).

Remark 3.9. It is easy to see that if � satisfies Axioms f1–f6 then �|BGN (restriction of � to the class of crisp bi-cooperative games) satisfies Axioms b1–b6. Thus we can recover the crisp value from its fuzzy counterpart underrestriction of its domain. As a matter of fact all the above axioms are obtained intuitively from their crisp analoguesby generalizing the idea of participation of players. For example, Axiom f5 (an analogue to Axiom b5), says that whencontribution of a player i to a game wi , is exactly opposite of that of player j , to a game w j (i and j having equal ratesof participations) then i’s payoff will be exactly opposite to the one for j . Similarly, Axiom f6 implies that if i providessome positive contribution to B ∈ FB(A), and the added value for w′ is greater than that for w or if i provides somenegative contribution to B ∈ FB(A), and the negative added value for w′ is lesser than that for w in absolute value,then its payoff due to w′ cannot be lesser than the one due to w. This establishes a well deserved link between the crispand fuzzy frameworks pertaining to bi-cooperative games. Furthermore, the definition above can be adopted for anyclass of bi-cooperative games with fuzzy bi-coalitions. In the next section, we propose a new family of bi-cooperativegames with fuzzy bi-coalitions and discuss the corresponding properties.

S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58 53

4. Fuzzy bi-cooperative games in Choquet Integral form

Here, we define the class of games in Choquet Integral form and study their properties. Note that any gamewe consider here, associates a bi-cooperative game as its crisp counterpart. We begin with the followingdefinitions:

Definition 4.1. For A ∈ FB(N ) and � ∈ [0, 1], the expression ([A]�, [A]�) is called the �-level set of A where

[A]� = {i ∈ N |�A(i) ≥ �}[A]� = {i ∈ N |�A(i) ≥ �}

Remark 4.2. It is evident from Definition 4.1 that

(a) Every �-level set ([A]�, [A]�), (� ∈ [0, 1]) of a fuzzy bi-coalition A, is indeed a crisp bi-coalition (associated withA) where all players in positive or negative roles have participations level not below �. Thus for h ∈ (0, 1], thesymbol Q([A]h ,[A]h )(N ) will represent as usual the set:

Q([A]h ,[A]h )(N ) = {(S′, T ′) ∈ Q(N ), S′ ⊆ [A]h, T ′ ⊆ [A]h} (4.1)

(b) Every (S′, T ′) ∈ Q([A]h ,[A]h )(N ), h ∈ (0, 1] has a representation ([B]h, [B]h) for some B ∈ FB(A) given by�B(i) = �A(i) when i ∈ S′ with �B(i) = 0 otherwise and �B(i) = �A(i) when i ∈ T ′ with �B(i) = 0 otherwise.Moreover, for every A ∈ FB(N ), ([A]h, [A]h) ([A]k, [A]k), whenever h, k ∈ (0, 1] and h ≥ k.

In what follows next, for ready reference, we shall mention here the definition and some pertinent properties of asigned Choquet integral which is a generalization of what Choquet [7] proposed in 1955. The rationale behind definingone is to relax the monotonicity of a regular Choquet integral so that the belonging fuzzy measures are not necessarilymonotone and may even take negative values. If no ambiguity arises, we call a signed Choquet integral simply a Choquetintegral. For details we recommend [22,23,26–30].

Definition 4.3. Let v be an arbitrary set function. For any x ∈ Rn , we define the signed Choquet integral∫

x dv,analogously to the regular Choquet integral given in [12–15] as

∫ ∞

0[v({ j : x j ≥ t}) − v(∅)] dt +

∫ 0

−∞[v( j : x j ≥ t) − v({1, · · · n})] dt (4.2)

It follows that we can compute the Choquet integral of x with respect to v in following three steps:

(i) Take a permutation � on {1, . . . , n} that is compatible with x , i.e. x�(1) ≥ · · · ≥ x�(n).(ii) Define ��( j) := v({�(1), . . . , �( j)}) − v({�(1), . . . , �( j − 1)}) ∀ j .

(iii) Then∫

x dv = ∑nj=1 � j x j is the signed Choquet integral of x with respect to v. We denote it by Cv(x).

The numbers � j are called decision weights. In general, they can well be negative. They are all nonnegative if andonly if v is monotonic. Let x, x ′ ∈ Rn . We say that x, x ′ are comonotonic if there exists a permutation � on {1, . . . , n}such that x�(1) ≤ x�(2) · · · ≤ x�(n) and x ′

�(1) ≤ x ′�(2) · · · ≤ x ′

�(n). The characteristic property of the signed Choquetintegral is comonotonic additivity, which means that Cv(x + y) = Cv(x) + Cv(y) whenever x and y are comonotonic.Comonotonic additivity leads to Cauchy equation. Moreover any such signed Choquet integral is positively homoge-neous i.e. Cv(x) = Cv(x), ∀ ≥ 0. The following result due to Schmeidler [23] characterizes a signed Choquetintegral:

Theorem 4.4. V : Rn → R is a signed Choquet integral if and only if it is continuous and satisfies comonotonicadditivity.

54 S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58

We define the following:

Definition 4.5. Given A ∈ FB(N ), let Q(A) = Q1(A) ∪ Q2(A) where Q1(A) = {�A(i)|�A(i) > 0, i ∈ N } andQ2(A) = {�A(i)|�A(i) > 0, i ∈ N } and let q(A) be the cardinality of Q(A). Let us write the elements of Q(A) in theincreasing order as h1 < · · · < hq(A). Then given b ∈ BGN , a game w ∈ GFB(N ) is said to be a fuzzy bi-cooperativegame in Choquet Integral form over FB(N ) if it holds that

w(A) =q(A)∑l=1

b([A]hl , [A]hl )(hl − hl−1) (4.3)

whence, h0 = 0 . We denote by GcFB(N ) the class of all bi-cooperative games with fuzzy bi-coalitions in Choquet

Integral form over BGN .

For A ∈ FB(N ), define x A ∈ [−1, 1]N , by

x Ai =

⎧⎪⎨⎪⎩

�A(i) if �A(i) > 0

−�A(i) if �A(i) > 0

0 else

then, the expression (4.3) for w(A) is exactly the Choquet integral Cb(x A) of x A with respect to b. Again, if we constructFS,T ∈ FB(N ) by

�FS,T(i) =

{1 if i ∈ S

0 else

�FS,T (i) ={

1 if i ∈ T

0 else

then we get

w(FS,T ) = Cb(1S, −1T , 0N\(S∪T )) = Cb(1S,T ) = b(S, T ) (4.4)

What follows, is simply an analogue of the “properly weighted with respect to bi-capacities (PWBC)” property givenby Grabisch et al. [14].

Furthermore, there is a one to one correspondence between the members of BGN and GcFB(N ). We call the crisp

game corresponding to the game with fuzzy bi-coalitions in Choquet Integral form, the associated crisp game and thegame with fuzzy bi-coalitions in Choquet Integral form corresponding to a crisp game, the associated fuzzy game.

Remark 4.6. Let w ∈ GcFB(N ). Given A ∈ FB(N ), consider a set {r1, . . . , rm} ⊇ Q(A), such that 0 ≤ r1 ≤ · · · ≤

rm ≤ 1. Then the following holds:

w(A) =m∑

l=1

b([A]rl , [A]rl )(rl − rl−1)

where r0 = 0.

The following theorem is important:

Theorem 4.7. Define a metric d on FB(N ) by

d(A, B) =∑i∈N

|�A(i) − �B(i)| + |�A(i) − �B(i)|2

(4.5)

Then w ∈ GcFB(N ) is continuous with respect to a fuzzy bi-coalition under the usual metric on R.

S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58 55

Proof. The proof follows directly from the continuity of the Choquet Integral with respect to its integrand given byTheorem 4.4. �

Theorem 4.8. Every w ∈ GcFB(N ) is monotone in fuzzy sense whenever b ∈ BGM (N ).

Proof. The result follows from the non-decreasing monotonicity property proposed as a characterization of Choquetintegrals by Grabich et al. [15]. �

4.1. A Shapley function on GcFB(N )

We now provide a Shapley function as a solution concept to the class GcFB(N ). Prior to that, we state and prove an

important result as follows:

Lemma 4.9. Given w ∈ GcFB(N ), a permutation � on N , a permutation �A of A ∈ FB(N ) and the bi-cooperative

game �b, define �w ∈ GcFB(N ) by

�w(�A) = w(A)

Then,

�w(�A) =q(�A)∑l=1

�b([�A]hl , [�A]hl )(hl − hl−1) (4.6)

Proof. The proof follows directly from the facts that �[A]� = [�A]� , �[A]� = [�A]� for any � ∈ (0, 1] whereQ(A) = Q(�A). �

Given b ∈ BGN , let �′(b)(S, T ) denote the value of b for (S, T ) ∈ Q(N ), due to Labreuche and Grabisch given byTheorem 2.5.

Define � : GcFB(N ) → (Rn)FB (N ) by

�i (w)(A) =q(A)∑l=1

[�′i (b)]([A]hl , [A]hl )(hl − hl−1) (4.7)

In the next theorem, we show that � is a Shapley function on GcFB(N ).

Theorem 4.10. The function � : GcFB(N ) → (Rn)FB (N ) given by Eq. (4.7) is a Shapley function on Gc

FB(N ).

Proof. We require to prove that � satisfies Axioms f1–f6 over GcFB(N ) and all fuzzy bi-coalitions in FB(N ). However,

by the properties of linearity, comonotonicity and positive homogeneity of Choquet integrals, it suffices to take for anymembers A ∈ FB(N ), elements of the form FS,T ∈ FB(N ) with (S, T ) ∈ FB(N ). By virtue of expressions (4.3) and(4.4), it is easy to see that when � satisfies Axioms f1–f6, �′ of (4.7) satisfies Axioms b1–b6 and their conclusions.Therefore, as a consequence, reverting back to � from �′, we obtain the conclusions of Axioms f1–f6 as well. �

Our next goal would be to axiomatize the Shapley value so obtained.

Theorem 4.11. If a function � : GcFB(N ) → (Rn)FB (N ) satisfies to axioms of efficiency, linearity, null player, intra-

coalition symmetry, inter-coalition symmetry and monotonicity (Axiom f1–f6), and �′ given by Eq. (2.7), representsthe value for every crisp game associated to the corresponding member of Gc

FB(N ), then � must be given by Eq. (4.7).

Proof. For (K , L) ∈ Q(N ), define, UK ,L ∈ BGN by

UK ,L (S′, T ′) ={

1 if S′ = K and T ′ = L

0 otherwise

56 S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58

Then every b ∈ BGN is uniquely represented as b = ∑(K ,L)∈Q(N ) b(K , L)UK ,L . For A ∈ FB(N ), we have from (4.3),

w(A) =q(A)∑l=1

⎧⎨⎩

∑(K ,L)∈Q(N )

b(K , L)UK ,L ([A]hl , [A]hl )

⎫⎬⎭ (hl − hl−1)

Let �|BGN = �′ (Remark 3.9). Also from definition, we have w|Q(N ) = b. Thus if � satisfies Axiom f1–f6, then for

every A ∈ FB(N ), �′ will satisfy Axiom b1–b6 for ([A]h, [A]h) ∀h ∈ (0, 1]. Thus �′ defines the value given byTheorem 2.5 for ([A]h, [A]h) ∀h ∈ (0, 1]. By linearity,

�i (w)(A) =q(A)∑l=1

∑(K ,L)∈Q(N )

b(K , L)�i (UK ,L )([A]hl , [A]hl )(hl − hl−1) (4.8)

�i (w)(A) =q(A)∑l=1

⎧⎨⎩

∑(K ,L)∈Q(N )

b(K , L)�′i (UK ,L )([A]hl , [A]hl )

⎫⎬⎭ (hl − hl−1) (4.9)

�i (w)(A) =q(A)∑l=1

�′i (b)([A]hl , [A]hl )(hl − hl−1) (4.10)

This completes the proof. �

5. An illustrative example

Let us now make use of our model in explaining the mega dam issue discussed in the Introduction. Let us considera hypothetical model with three players 1, 2, and 3, of which 1 and 3 put fractions of their resources for running theproject and 2 provides her resource to stop it. One can interpret these players as the Government (Player 1), Groupof Environmentalists (Player 2), and the Agency in charge of dam construction and production of hydro-electricity(Player 3). Assume that in the process, we get a fuzzy bi-coalition A = {〈1, 0.3, 0〉, 〈2, 0, 0.5〉, 〈3, 0.4, 0〉}. Let theassociated bi-cooperative game b in this case, represent the strength of the supporting group over its opponents ina bi-coalition towards the decision to construct the mega dam. The values of b at different bi-coalitions are givenas follows: b(∅, ∅) = 0, b(∅, 2) = 0, b(1, ∅) = 4, b(3, ∅) = 2, b(3, 2) = 1, b({1, 3}, 2) = 6, b({1, 3}, ∅) = 11,b(1, 2) = 2. Table 1 gives the corresponding values of b at these bi-coalitions.

Using Eq. (4.3), we obtain w(A) = 1.9. Here, w represents physically, the expected strength of the decision to supportthe mega dam issue in regards to the resources provided by the constituent agencies. Consequently, the components ofthe Shapley value for w, with respect to a fuzzy bi-coalition would represent the individual contributions of the playersin attaining that strength (Table 1).

After computations, the components of the Shapley value are found to be :

�1(w)(A) ≈ 1.45

�2(w)(A) ≈ −0.68

�3(w)(A) ≈ 1.15

Table 1Computation of the values of the associated crisp game at different bi-coalitions.

�′i (b)(S, T ) ({1, 3}, {2}) ({3}, {2}) (∅, {2})

�′1(b)(S, T ) 4.83 0 0

�′2(b)(S, T ) −2.1 −0.5 0

�′3(b)(S, T ) 3.333 1.5 0

S. Borkotokey, P. Sarmah / Fuzzy Sets and Systems 198 (2012) 46 –58 57

This example shows that although membership of player 1 is less than that of 3, yet she receives more payoff. This isjustified because player 1’s marginal contribution in a positive role is more than that of player 3 in situations when,player 2 puts her negative contribution or remains indifferent.

6. Conclusion

In this study, we have developed a framework for a bi-cooperative game with fuzzy bi-coalitions. Any member of thisclass incorporates the contributions by the players in two mutually exclusive roles so that the set of players at each levelof participation can be divided into three disjoint groups: group of positive contributors, negative contributors and thegroup of indifferent players. A hypothetical yet realistic example is formulated to illustrate our findings. Here, we havemaintained a strict bi-polarity of the games, thus defined, however it seems to be possible to introduce a similar notionfor multi-choice games as well. In our next study, we propose to highlight on this aspect. Further, we have defined herethe class of fuzzy bi-cooperative games in Choquet Integral form. It has been further shown that the members of thisclass satisfy both continuity and monotonicity. A solution concept namely the Shapley function to this class has beenproposed. There may exist other fuzzy bi-coalitions whose payoff cannot be expressed by crisp bi-coalitional valuesand participation levels. In our future work, we plan to introduce more of such games and study their properties in thiscontext.

Acknowledgment

The authors acknowledge the encouraging and constructive comments and suggestions of the anonymous reviewersand also are grateful to Prof. R. Mesiar for his constant support and useful comments. The suggestions of Dr. S. Sarangiare also highly acknowledged.

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