On the core,nucleolus and bargaining sets of cooperative games with fuzzy payoffs

1. Introduction

A model called cooperative game with fuzzy payoffs (cooperative fuzzy game in short) presented by Mallozzi et al. [6] in cooperative game theory has been developed when the worth of any coalition is given by means of a fuzzy number. It can be used to analyze and resolve cooperative situations under fuzzy environments. For example, let us consider a bankruptcy problem where an estate has to be divided among some creditors. In the classical model, any creditor has an individual claim on a certain amount of estate, and the total claim is weakly larger than the estate. The individual claim is the best expectation of a creditor, while nothing is said about the minimum that a creditor is willing to accept. Now, any creditor can try to improve his situation and gives bounds for what he wants to accept. It becomes an interval bankruptcy problem introduced by Branzei et al. [12]. Actually, if creditor claims with their minimum and best expectation bounds, it is natural to assume that the creditor prefers the amounts in the interval consisting of their minimum claim and the best expectation claim according to an increasing utility function. In addition, the result of asset evaluation may be varied for different assessment methods. Accordingly, the estate and the claims of creditors become fuzzy numbers. This is a fuzzy bankruptcy problem. Obviously, the classical bankruptcy problem and the interval bankruptcy problem are the special cases of the fuzzy bankruptcy problem. How to allocate the fuzzy estate among the creditors following their fuzzy claims for such fuzzy bankruptcy problem, which involves with the concepts of core and bargaining sets for the cooperative fuzzy games. Our interests will focus on the core and bargaining sets, in which any coalition prefers to the allocations for the cooperative fuzzy games.

The core of a cooperative fuzzy game consists of all payoff vectors which distribute the worth of the grand coalition, under the condition that the players in each coalition receive at least the amount that they can obtain by cooperating. So then, it is guaranteed that the grand coalition is stable that no coalition has an incentive to split off it. However, it is empty in some cases. This leads us to considering other solution concepts. The bargaining sets of cooperative fuzzy game are more closely tied to bargaining process since they take account of the possible threats and counterthreats made by coalitions. They are non-empty for all cooperative fuzzy games with a non-empty imputation set. Under specific situations, the bargaining sets coincide with the core of the cooperative fuzzy game.

Based on a partial order relation defined by the α-level sets of fuzzy numbers, the F -core proposed by Mallozzi et al. [6] is a set-valued solution that distributes the worth of the grand coalition and guarantees that the players of any coalition gain at least what they could obtain by themselves. In other words, the F -core collects feasible outcomes which cannot be improved by any coalition. Depended on the partial order relation of fuzzy numbers, Mallozzi et al. [6] defined the F -balancedness as a condition for a cooperative fuzzy game. This condition requires that it is not advantageous for the worth of the grand coalition to divide the grand coalition into the sets of any balanced collection over the grand coalition, in case that the worths of the involved sets are weighted according to the corresponding weights. Supposing that the F -core is not empty, then the cooperative fuzzy game is F -balanced, but the opposite is not true. However, the F -core is empty in some cases, three sufficient conditions for non-emptiness of the F -core were given by Wang and Zhang [7].

As mentioned earlier, even if the F -core provides a viewpoint in decision making processes where the vagueness over the worth of coalitions is given by means of fuzzy numbers, one of the main drawback of the F -core is that it contains only few payoff vectors, what’s worse, it is empty in many cases just because of the partial order relation, that means we may hardly select some reasonable allocations to keep the grand coalition stable. Another question is that players hesitate on whether to depart from the grand coalition or not, seeing that they are not sure their payoff from this allocation is bigger than the worth gained on their own. The root cause of the questions lies in the ranking criterion of fuzzy numbers. In this current paper, we intend to define a total order relation to settle the issues based on the expected values of fuzzy numbers which are the centers of the expected values of interval random sets generated by these fuzzy numbers.

In order to better define the core and the bargaining sets for the cooperative fuzzy game, a total order relation is proposed in this paper by the expected values of fuzzy numbers, in which any two fuzzy numbers with the same expected values are defined as an indifference relation. In other words, the size relation of any two fuzzy numbers coincides with the size relation of their corresponding expected values. It is no longer a puzzle for us to choose a bigger fuzzy number between any two.

The indifference fuzzy core defined in our paper contains all payoff vectors, by which the players of any coalition receive at least they could gain by themselves in view of the total order relation of fuzzy numbers. In this case, the above-mentioned drawback of the F -core is entirely avoidable. Simultaneously, the players are determined to join the grand coalition so long as they receive from the allocation no less than the worth obtained by themselves. Moreover, we come to a conclusion that the indifference fuzzy core is non-empty for convex cooperative fuzzy game. However, the indifference fuzzy core may be empty. A necessary and sufficient condition for non-emptiness of the indifference fuzzy core called balancedness, is introduced, which differs from the F -balancedness defined by the partial order relation of fuzzy numbers being a necessary but not sufficient condition for non-emptiness of the F -core.

However, every player pursues the maximum payoff with the premise of the payoff vector in the indifference fuzzy core. So then it is hard to say which of payoff vectors is better in the indifference fuzzy core. Here the indifference fuzzy nucleolus is presented by minimizing the excess of coalitions in the lexicographic order over the non-empty compact convex indifference fuzzy imputation set based on the total order relation. We obtain that there exists at least one fuzzy payoff vector in the indifference fuzzy nucleolus. It is also proved that the indifference fuzzy nucleolus is a subset of the indifference fuzzy core when the indifference fuzzy core is non-empty.

Nevertheless, the indifference fuzzy core of the cooperative fuzzy game may be empty, which makes us to taking into account other solution concepts. Additionally, the concepts which have been introduced in the previous paragraphs neglect the bargaining process that may actually take place during a play of a game. The major reason of bargaining is that if every player will demand the maximum he can get in the coalition, no agreement will be reached. In this paper, we assume that all the players can bargain together, and settle at a unanimous outcome which is based on the threats and counterthreats that they possess. The set of all the unanimous outcomes for cooperative fuzzy games, called indifference fuzzy bargaining set, is defined and some of its properties are discussed.

In view of the aforementioned threats and counterthreats by coalitions, we introduce the Aumann-Maschler (or Mas-Colell) indifference fuzzy bargaining set which no player (or no coalition) has a justified objection at an indifference fuzzy imputation against any other player (or coalition). Only objections without counter-objections are considered as justified. Consequently, the indifference fuzzy core is a subset of the indifference fuzzy bargaining sets since blocking an allocation becomes more difficult. Moreover, it is proved that for convex or superadditive cooperative fuzzy games the indifference fuzzy bargaining sets coincide with the indifference fuzzy core. We further acquire a conclusion that the Aumann-Maschler indifference fuzzy bargaining set of superadditive cooperative fuzzy games is a subset of the Mas-Colell indifference fuzzy bargaining set.

The rest of this paper is arranged as follows. In Section 2, we recall some basic concepts of fuzzy numbers. Section 3 introduces the indifference fuzzy core and the balanced condition of the cooperative fuzzy games. The indifference fuzzy nucleolus is defined in Section 4. Section 5 proposes two kinds of indifference fuzzy bargaining sets, which coincide with the indifference fuzzy core for convex cooperative fuzzy games. The last section concludes with a brief summary.

2. Preliminaries

3. Indifference fuzzy core

A classical game introduced by Gillies [2] is a pair game where N = {1, 2, ⋯ , n} is the set of players and v : 2 ^N  → R is a map, assigning any coalition S ∈ 2 ^N to a real number with v (∅) =0. For any coalition S ⊆ N, v (S) can be interpreted as the worth that the players of the coalition S can obtain when they cooperate. A coalition S is said to be trivial if S = N, ∅. The number of players in any coalition S ⊆ N is denoted by |S|. The family of all classical games with player set N is denoted by G ^N . A classical game (N, v) ∈ G ^N is convex if and only if v (S ∪ T) + v (S ∩ T) ≥ v (S) + v (T) for any S, T ∈ 2 ^N . Denote the set of all convex classical games by C ^N . A classical game (N, v) ∈ G ^N is said to be balanced if for any map λ : 2 ^N  → R₊ such that ∑_{S⊆Natopi∈S}λ (S) =1 for any i ∈ N, we have v (N) ≥ ∑_S⊆Nλ (S) v (S). For simplicity, we write v (i) instead of v ({i}) for any i ∈ N. A solution of game ∈ G ^N is a function Φ : G ^N  → R ⁿ assigning to any game (N, v) a set of n-dimensional payoff vectors. The imputation set I(N, v) of a classical game (N, v) is the set I ( N , υ ) = { ( x 1 , ⋯ , x n ) ∈ R n | ∑ i ∈ N x i = υ ( N ) and x i ≥ υ ( i ) , ∀ i ∈ N } , and the core C(N, υ) of a classical game (N, υ) is the set C ( N , υ ) = { ( x 1 , ⋯ , x n ) ∈ R n | ∑ i ∈ N x i = υ ( N ) and ∑ i ∈ S x i ≥ υ ( S ) , ∀ S ⊆ N } .

An interval game introduced by AlparslanGök et al. [15] is a pair interval (N, v) where N = {1, 2, ⋯ , n} is the set of players, and ν : 2 N → 𝕀 R is the characteristic function such that ν (∅) =0. Denote the set of all interval games with player set N by 𝕀 G N . An interval game ( N , v ) ∈ 𝕀 G N is said to be I -balanced if for any map λ : 2 ^N  → R₊ such that ∑_{S⊆Natopi∈S}λ (S) =1 for any i ∈ N, we have ν (N) succapprox∑_S⊆Nλ (S) ν (S). The interval imputation set I(N, v) and the interval core C(N, v) of an interval game (N, v), are defined by the partial order relation succapprox of intervals as follows: I ( N , v ) = { ( J 1 , ⋯ , J n ) ∈ 𝕀 R n ∣ ∑ i ∈ N J i = ν ( N ) and J i ∈ ν ( i ) , ∀ i ∈ N } ,

C ( N , v ) = { ( J 1 , ⋯ , J n ) ∈ 𝕀 R n ∣ ∑ i ∈ N J i = ν ( N ) and ∑ i ∈ S J i ∈ ν ( S ) , ∀ S ⊆ N } , where ∑ i ∈ S J i = [ ∑ i ∈ S J i _ , ∑ i ∈ S J i ¯ ] .

Han et al. [17] discovered the total order relation ≿ of intervals, which allowed to define alternative imputation set and interval core of an interval game (N, v). The indifference interval imputation setI^*(N, v) is I * ( N , v ) = { ( J 1 , ⋯ , J n ) ∈ 𝕀 R n ∣ ∑ i ∈ N J i ∼ ν ( N ) and J i ≿ ν ( i ) , ∀ i ∈ N } , and the indifference interval core C^*(N, v) is C * ( N , v ) = { ( J 1 , ⋯ , J n ) ∈ 𝕀 R n ∣ ∑ i ∈ N J i ∼ ν ( N ) and ∑ i ∈ S J i ≿ ν ( S ) , ∀ S ⊆ N } , where ∑ i ∈ S J i = [ ∑ i ∈ S J i _ , ∑ i ∈ S J i ¯ ] .

A cooperative fuzzy game proposed by Mallozzi et al. [6] is a pair ( N , v ˜ ) , where N = {1, 2, ⋯ , n} is the set of players and v ˜ : 2 N → F ( R ) is a mapping which assigns any coalition S ∈ 2 ^N to a fuzzy number with v ˜ ( ∅ ) = 0 . For any coalition S ⊆ N, v ˜ ( S ) denotes the worth that coalition S achieves with its members cooperative altogether. The class of all cooperative fuzzy games with player set N is denoted by F G N . A cooperative fuzzy game ( N , v ˜ ) ∈ F G N is said to be F -balanced if for any map λ : 2 ^N  → R₊ such that ∑_{S⊆Natopi∈S}λ (S) =1 for any i ∈ N, we have v ˜ ( N ) ≽ ∑ ˜ S ⊆ N λ ( S ) v ˜ ( S ) . For simplicity, write v ˜ ( i ) instead of v ˜ ( { i } ) for any i ∈ N, use the standard notation x ˜ ( S ) : = ∑ ˜ i ∈ S x ˜ i for any ∅ ≠ S ⊆ N and x ˜ ( ∅ ) = 0 . A solution of ( N , v ˜ ) ∈ F G N is a function Ψ : F G N → F R n assigning to any cooperative fuzzy game ( N , v ˜ ) a set of n-dimensional payoff vectors with fuzzy numbers. The fuzzy core ( F -core for short) of a cooperative fuzzy game ( N , v ˜ ) , is defined by Mallozzi et al. [6] by the partial order relation ≽ of fuzzy numbers as follows: C F ( N , v ˜ ) = { ( x ˜ 1 , ⋯ , x ˜ n ) ∈ F ( R ) n ∣ x ˜ ( N ) = v ˜ ( N ) and x ˜ ( S ) ≽ v ˜ ( S ) , ∀ S ⊆ N } .

The F -core contains few payoff vectors or even no one just because of the partial order relation. Even worse, players are unaware of whether to take part in the grand coalition or not, since they are not sure their payoff from this allocation is greater than the worth gained by themselves. The key of such problems is the total order relation of fuzzy numbers. Here, we propose an indifference fuzzy core of a cooperative fuzzy game based on the total order relation ⊵ of fuzzy numbers. In a cooperative fuzzy game ( N , v ˜ ) , we say a payoff vector x ˜ ∈ F ( R ) n is indifferent efficient, if x ˜ ( N ) ≈ v ˜ ( N ) ; individually rational, if x ˜ i ⊵ v ˜ ( i ) and coalitionally rational, if x ˜ ( S ) ⊵ v ˜ ( S ) . A payoff vector x ˜ is called an indifference fuzzy imputation if it is indifferent efficient and individually rational. I ( N , v ˜ ) denotes the indifference fuzzyimputation set.

Definition 2. An indifference fuzzy core C ( N , v ˜ ) of a cooperative fuzzy game ( N , v ˜ ) ∈ F G N is definedas C ( N , v ˜ ) = { ( x ˜ 1 , ⋯ , x ˜ n ) ∈ F ( R ) n ∣ x ˜ ( N ) ≈ v ˜ ( N ) and x ˜ ( S ) ⊵ v ˜ ( S ) , ∀ S ⊆ N } .

Remark 2. (i) Obviously, it is true that C F ( N , v ˜ ) ⊆ C ( N , v ˜ ) . (ii) In addition, C ( N , v ˜ )  = Cgame when ( N , v ˜ ) ∈ F G N degenerates into the classical game (N, v) ∈ G ^N . C ( N , v ˜ )  = C^*(N, v) if ( N , v ˜ ) ∈ F G N becomes the interval game ( N , v ) ∈ 𝕀 G N .

Example 2. Suppose that the fuzzy numbers in this example are trapezoidal. Let the cooperative fuzzy game ( N , v ˜ ) ∈ F G N with N = {1, 2, 3} and v ˜ ( i ) = ( 0 , 0 , 0 , 0 ) , i = 1, 2, 3, v ˜ ( 12 ) = ( 1 , 2 , 2 , 3 ) , v ˜ ( 23 ) = ( 1 , 1 , 2 , 2 ) , v ˜ ( 13 ) = ( 4 , 5 , 6 , 7 ) , v ˜ ( 123 ) = ( 4 , 5 , 7 , 8 ) . We may check the fuzzy payoff vector ( x ˜ 1 , x ˜ 2 , x ˜ 3 ) ∈ C ( N , v ˜ ) , where x ˜ 1 = ( 0 , 1 , 3 , 6 ) , x ˜ 2 = ( 0 , 0 , 1 , 1 ) and x ˜ 3 = ( 2 , 3 , 3 , 4 ) . In fact, ( E ( x ˜ 1 ) , E ( x ˜ 2 ) , E ( x ˜ 3 ) ) = ( 5 2 , 1 2 , 3 ) , E ( x ˜ 1 + ˜ x ˜ 2 + ˜ x ˜ 3 ) = 6 = E ( v ˜ ( 123 ) ) , E ( x ˜ i ) > 0 = E ( v ˜ ( i ) ) , E ( x ˜ 1 + ˜ x ˜ 2 ) = 3 > E ( v ˜ ( 12 ) ) = 2 , E ( x ˜ 1 + ˜ x ˜ 3 ) = 11 2 = E ( v ˜ ( 13 ) ) , E ( x ˜ 2 + ˜ x ˜ 3 ) = 7 2 > E ( v ˜ ( 23 ) ) = 3 2 , which are equivalent to x ˜ 1 + ˜ x ˜ 2 + ˜ x ˜ 3 ≈ v ˜ ( 123 ) , x ˜ i ≈ v ˜ ( i ) , x ˜ 1 + ˜ x ˜ 2 ≈ v ˜ ( 12 ) , x ˜ 1 + ˜ x ˜ 3 ≈ v ˜ ( 13 ) and x ˜ 2 + ˜ x ˜ 3 ≈ v ˜ ( 23 ) . However, ( x ˜ 1 + ˜ x ˜ 2 ) [ α ] = [ α , 7 - 3 α ] , v ˜ ( 12 ) [ α ] = [ α + 1 , 3 - α ] . It is obvious that x ˜ 1 + ˜ x ˜ 2 and v ˜ ( 12 ) are non-comparable by the partial order relation ≽ of fuzzy numbers, which means ( x ˜ 1 , x ˜ 2 , x ˜ 3 ) ∉ C F ( N , v ˜ ) .

From Example 2, we see that for any ( N , v ˜ ) ∈ F G N , C ( N , v ˜ ) ⊆ C F ( N , v ˜ ) is not always true, that means there exist some payoff vectors in the indifference fuzzy core but not in the F -core defined by Mallozzi et al. [6]. In other words, players become determined to join in the grand coalition, since they know their payoffs from those allocations in the indifference fuzzy core are greater than the worths acquired on their own by the total order relation ⊵ of fuzzy numbers.

An expected game (N, v _E ) of ( N , v ˜ ) ∈ F G N defined v E ( S ) = E ( v ˜ ( S ) ) for any S ∈ 2 ^N , is a classical game.

In the foregoing example, we consider the expected game (N, v _E ). Obviously, the expected vector ( E ( x ˜ 1 ) , E ( x ˜ 2 ) , E ( x ˜ 3 ) ) = ( 5 2 , 1 2 , 3 ) ∈ C ( N , v E ) . Conversely, suppose that there is a fuzzy payoff vector ( y ˜ 1 , y ˜ 2 , y ˜ 3 ) such that ( E ( y ˜ 1 ) , E ( y ˜ 2 ) , E ( y ˜ 3 ) ) = ( 5 2 , 1 2 , 3 ) ∈ C ( N , v E ) . Then ( y ˜ 1 , y ˜ 2 , y ˜ 3 ) ∈ C ( N , v ˜ ) , such as y ˜ 1 = ( 1 , 1 , 4 , 4 ) , y ˜ 2 = ( 1 3 , 1 3 , 2 3 , 2 3 ) , y ˜ 3 = ( 1 , 3 , 3 , 5 ) . We find the fact that such conclusion holds for any ( N , v ˜ ) ∈ F G N .

Theorem 1. For any ( N , v ˜ ) ∈ F G N , C ( N , v ˜ ) ≠ ∅ if and only if C (N, v _E )≠ ∅. That’s to say C ( N , v ˜ ) ≠ ∅ if and only if (N, v _E ) is balanced.

Proof. For any ( N , v ˜ ) ∈ F G N , suppose that C ( N , v ˜ ) ≠ ∅. There exists a fuzzy payoff vector ( x ˜ 1 , ⋯ , x ˜ n ) such that x ˜ ( N ) ≈ v ˜ ( N ) and x ˜ ( S ) ⊵ v ˜ ( S ) , for any S ⊆ N . Additionally, there exists a real payoff vector (y₁, ⋯ , y _n ) satisfying ( E ( x ˜ 1 ) , ⋯ , E ( x ˜ n ) ) = ( y 1 , ⋯ , y n ) . So we have ∑ i ∈ N y i = ∑ i ∈ N E ( x ˜ i ) = E ( ∑ ˜ i ∈ N x ˜ i ) = E ( v ˜ ( N ) ) = v E ( N ) , and ∑ i ∈ S y i = ∑ i ∈ S E ( x ˜ i ) = E ( ∑ ˜ i ∈ S x ˜ i ) ≥ E ( v ˜ ( S ) ) = v E ( S ) , for any S ⊆ N. Hence, C (N, v _E )≠ ∅.

Conversely, if C (N, v _E )≠ ∅, there exists a real payoff vector (x₁, ⋯ , x _n ) such that ∑ i ∈ N x i = v E ( N ) and ∑ i ∈ S x i ≥ v E ( S ) , for any S ⊆ N . Meanwhile, there exists at least one payoff vector ( z ˜ 1 , ⋯ , z ˜ n ) meeting ( E ( z ˜ 1 ) , ⋯ , E ( z ˜ n ) ) = ( x 1 , ⋯ , x n ) . It holds that E ( ∑ ˜ i ∈ N z ˜ i ) = ∑ i ∈ N E ( z ˜ i ) = ∑ i ∈ N x i = v E ( N ) = E ( v ˜ ( N ) ) , and E ( ∑ ˜ i ∈ S z ˜ i ) = ∑ i ∈ S E ( z ˜ i ) = ∑ i ∈ S x i ≥ v E ( S ) = E ( v ˜ ( S ) ) . That is ∑ ˜ i ∈ N z ˜ i ≈ v ˜ ( N ) and ∑ ˜ i ∈ S z ˜ i ⊵ v ˜ ( S ) . Therefore, C ( N , v ˜ ) ≠ ∅. □

We say that ( N , v ˜ ) ∈ F G N is convex if v ˜ ( S ) + ˜ v ˜ ( T ) ⊴ v ˜ ( S ∪ T ) + ˜ v ˜ ( S ∩ T ) , for any S , T ∈ 2 N . The class of all convex cooperative fuzzy games denotes by C N .

The core of a convex classical game is non-empty given out by Shapley [5]. By Theorem 1, we know that the indifference fuzzy core is non-empty if the expected game is convex. Besides, the expected game is convex if and only if the cooperative fuzzy game is convex.

Theorem 2. C ( N , v ˜ ) ≠ ∅ for any ( N , v ˜ ) ∈ C N .

Proof. From Theorem 1, we need to prove C (N, v _E )≠ ∅. Due to ( N , v ˜ ) ∈ C N , it holds that for any S, T ∈ 2 ^N , v ˜ ( S ) + ˜ v ˜ ( T ) ⊴ v ˜ ( S ∪ T ) + ˜ v ˜ ( S ∩ T ) . As the properties of the expected value, it is true that for any S, T ⊆ N, v E ( S ) + v E ( T ) ≤ v E ( S ∪ T ) + v E ( S ∩ T ) , which means v _E convex. That is, C (N, v _E )≠ ∅. □

Definition 3. For any ( N , v ˜ ) ∈ F G N , we say that ( N , v ˜ ) is balanced if v ˜ ( N ) ⊵ ∑ ˜ S ⊆ N λ ( S ) v ˜ ( S ) for any map λ : 2 ^N  → R₊ with ∑_{S⊆Natopi∈S}λ (S) =1 for any i ∈ N.

Remark 3. The Definition is on line with F -balanced introduced by Mallozzi et al. [6], I -balanced defined by AlparslanGök et al. [15] and balanced presented by Shapley [4].

Example 3. Let ( N , v ˜ ) ∈ F G N with N = 1, 2, 3 and v ˜ ( i ) = ( 0 , 0 , 0 , 0 ) , i = 1 , 2 , 3 , v ˜ ( 12 ) = ( 8 , 8 , 9 , 9 ) , v ˜ ( 23 ) = ( 7 , 8 , 9 , 10 ) , v ˜ ( 13 ) = ( 4 , 6 , 10 , 14 ) , v ˜ ( 123 ) = (10, 12, 14, 16). These fuzzy numbers are trapezoidal. We check all the minimal balanced collections: 1 ⋅ v ˜ ( 1 ) + ˜ 1 ⋅ v ˜ ( 2 ) + ˜ 1 ⋅ v ˜ ( 3 ) = ( 0 , 0 , 0 , 0 ) ⊴ v ˜ ( 123 ) = ( 10 , 12 , 14 , 16 ) 1 ⋅ v ˜ ( 1 ) + ˜ 1 ⋅ v ˜ ( 23 ) = ( 7 , 8 , 9 , 10 ) ⊴ v ˜ ( 123 ) = ( 10 , 12 , 14 , 16 ) 1 ⋅ v ˜ ( 3 ) + ˜ 1 ⋅ v ˜ ( 12 ) = ( 8 , 8 , 9 , 9 ) ⊴ v ˜ ( 123 ) = ( 10 , 12 , 14 , 16 ) 1 ⋅ v ˜ ( 2 ) + ˜ 1 ⋅ v ˜ ( 13 ) = ( 4 , 6 , 10 , 14 ) ⊴ v ˜ ( 123 ) = ( 10 , 12 , 14 , 16 ) 1 2 ⋅ v ˜ ( 12 ) + ˜ 1 2 ⋅ v ˜ ( 23 ) + ˜ 1 2 ⋅ v ˜ ( 13 ) = ( 19 2 , 11 , 14 , 33 2 ) ⊴ v ˜ ( 123 ) = ( 10 , 12 , 14 , 16 ) .

So ( N , v ˜ ) is balanced. However, ( 1 2 · v ˜ ( 12 ) + ˜ 1 2 · v ˜ ( 23 ) + ˜ 1 2 · v ˜ ( 13 ) ) [ α ] = ( 19 2 , 11 , 14 , 33 2 ) [ α ] = [ 3 2 α + 19 2 , 33 2 - 5 2 α ] , v ˜ ( 123 ) [ α ] = [ 10 + 2 α , 16 - 2 α ] . Obviously, ( 19 2 , 11 , 14 , 33 2 ) and v ˜ ( 123 ) are non-comparable by the partial order relation ≽ of fuzzy numbers. That’s to say ( N , v ˜ ) not F -balanced.

Furthermore, we consider the expected game (N, v _E ) of the game ( N , v ˜ ) mentioned in Example a and calculate v _E  (i) =0, i = 1, 2, 3, v _E  (12) = v _E  (13) = v _E  (23) =8.5, v _E  (123) =13. It can be verified that ( 13 3 , 13 3 , 13 3 ) ∈ C ( N , v E ) . By Theorem 1, it follows directly that C ( N , v ˜ ) ≠ ∅. We find out that the conclusion is true for any ( N , v ˜ ) ∈ F G N .

Theorem 3. For any ( N , v ˜ ) ∈ F G N , C ( N , v ˜ ) ≠ ∅ if and only if ( N , v ˜ ) is balanced.

Proof. Let ( N , v ˜ ) ∈ F G N with C ( N , v ˜ ) ≠ ∅ and λ : 2 ^N  → R₊ be a map with ∑_{S⊆Natopi∈S}λ (S) =1 for any i ∈ N. From Theorem 1, C ( N , v ˜ ) ≠ ∅ if and only if the expected game (N, v _E ) is balanced. That is v E ( N ) ≥ ∑ S ⊆ N λ ( S ) v E ( S ) , which is equivalent to v ˜ ( N ) ⊵ ∑ ˜ S ⊆ N λ ( S ) v ˜ ( S ) . That say, ( N , v ˜ ) is balanced. □

Remark 4. The balancedness is a necessary and sufficient condition for non-emptiness of the indifference fuzzy core, instead of the F -balancedness being a necessary condition for non-emptiness of the F -core.

4. Indifference fuzzy nucleolus

Given a cooperative fuzzy game ( N , v ˜ ) , for any x ˜ ∈ I ( N , v ˜ ) and any coalition S ∈ 2 ^N , the excess e ˜ ( S , x ˜ ) of S at x ˜ is defined to be e ˜ ( S , x ˜ ) = v ˜ ( S ) - ˜ x ˜ ( S ) . It can be checked easily that for any payoff vector x ˜ ∈ F ( R ) n , if x ˜ ∈ C ( N , v ˜ ) , then e ˜ ( S , x ˜ ) ⊴ 0 ˜ for any S ⊆ N, where if e ˜ ( S , x ˜ ) ≈ 0 ˜ , then E ( e ˜ ( S , x ˜ ) ) = 0 , i.e., E ( v ˜ ( S ) ) = E ( x ˜ ( S ) ) , which means v ˜ ( S ) ≈ x ˜ ( S ) .

The nucleolus of the classical games defined by Schmeidler [3] consists of payoff vectors which minimize the excesses of coalitions in the lexicographic order over the compact convex imputation set. We give an indifference fuzzy nucleolus of a cooperative fuzzy game under the total order relation mentioned in Definition 1. For any payoff vector x ˜ ∈ F ( R ) n , let θ ˜ ( x ˜ ) be a 2 ⁿ -tuple vector whose components are the excesses of coalitions arranged in non-increasing order, i.e., θ i ˜ ( x ˜ ) ⊵ θ j ˜ ( x ˜ ) , 1 ≤ i ≤ j ≤ 2 n .

Definition 4. An indifference fuzzy nucleolus N ( N , v ˜ ) of a cooperative fuzzy game ( N , v ˜ ) ∈ F G N is the set of payoff vectors x ˜ ∈ I ( N , v ˜ ) satisfying θ ˜ ( x ˜ ) ⊴ L θ ˜ ( y ˜ ) , for any y ˜ ∈ I ( N , v ˜ ) , where the lexicographic order ⊴ _L on F ( R ) 2 n is used to order the excesses, i.e., for any x ˜ , y ˜ ∈ F ( R ) n ,

θ ˜ ( x ˜ ) ⊲ L θ ˜ ( y ˜ ) : if there exists an integer 1 ≤ k ≤ 2 ⁿ such that θ i ˜ ( x ˜ ) ≈ θ i ˜ ( y ˜ ) for 1 ≤ i < k, whereas θ ˜ k ( x ˜ ) ⊲ θ ˜ k ( y ˜ ) .

Remark 5. The indifference fuzzy nucleolus N ( N , v ˜ ) is same as the nucleolus of the classical games if ( N , v ˜ )  ∈ G ^N . When ( N , v ˜ ) ∈ 𝕀 G N , the definition of the nucleolus for the interval games may be given as the Definition AB. From another perspective, Kong et al. [9] defined the general prenucleolus of cooperative games with fuzzy coalition.

Theorem 4. For any ( N , v ˜ ) ∈ F G N , N ( N , v ˜ ) ≠ ∅ .

Proof. Let X₀ = I ( N , v ˜ ) . Denote θ ˜ 1 ( x ˜ ) ≈ min x ˜ ∈ X 0 max S ∈ 2 N e ˜ ( S , x ˜ ) : = e ˜ ( S 1 , x ˜ ) , X 1 = { x ˜ ∈ X 0 ∣ max S ∈ 2 N e ˜ ( S , x ˜ ) ≈ θ ˜ 1 ( x ˜ ) } , θ ˜ 2 ( x ˜ ) ≈ min x ˜ ∈ X 1 max S ∈ 2 N ∖ S 1 e ˜ ( S , x ˜ ) : = e ˜ ( S 2 , x ˜ ) , X 2 = { x ˜ ∈ X 1 ∣ max S ∈ 2 N ∖ S 1 e ˜ ( S , x ˜ ) ≈ θ ˜ 2 ( x ˜ ) } , ⋯ θ ˜ i ( x ˜ ) ≈ min x ˜ ∈ X i - 1 max S ∈ 2 N ∖ { S 1 , S 2 , ⋯ , S i - 1 } e ˜ ( S , x ˜ ) : = e ˜ ( S i , x ˜ ) , X i = { x ˜ ∈ X i - 1 ∣ max S ∈ 2 N ∖ { S 1 , S 2 , ⋯ , S i - 1 } e ˜ ( S , x ˜ ) ≈ θ ˜ i ( x ˜ ) } . Furthermore, n is a finite number, θ ˜ 2 n ( x ˜ ) ≈ min x ˜ ∈ X 2 n - 1 max S ∈ 2 N ∖ { S 1 , S 2 , ⋯ , S 2 n - 1 } e ˜ ( S , x ˜ ) : = e ˜ ( S 2 n , x ˜ ) ,

X 2 n = { x ˜ ∈ X 2 n - 1 ∣ max S ∈ 2 N ∖ { S 1 , S 2 , ⋯ , S 2 n - 1 } e ˜ ( S , x ˜ ) ≈ θ ˜ 2 n ( x ˜ ) } . As a result, we obtain that X_{2 ⁿ} is the indifference fuzzy nucleolus of the cooperative fuzzy game ( N , v ˜ ) from Definition 4.

From Theorem 4, we get the following corollary immediately.

Corollary 5. For any ( N , v ˜ ) ∈ F G N , if x ˜ 1 , x ˜ 2 ∈ N ( N , v ˜ ) , then x ˜ 1 ≈ x ˜ 2 .

Remark 6. The indifference fuzzy nucleolus of a cooperative fuzzy game is non-empty but not only one element, which differs from the property of the nucleolus defined by Schmeidler [3] for a classical game.

Theorem 6. For any ( N , v ˜ ) ∈ F G N , if C ( N , v ˜ ) ≠ ∅, then N ( N , v ˜ ) ⊆ C ( N , v ˜ ) .

Proof. We suppose that there exists an indifference fuzzy imputation x ˜ * meeting x ˜ * ∈ N ( N , v ˜ ) and θ ˜ ( x ˜ * ) ≈ ( θ ˜ 1 ( x ˜ * ) , θ ˜ 2 ( x ˜ * ) , ⋯ , θ ˜ 2 n ( x ˜ * ) ) , where θ ˜ i ( x ˜ * ) : = e ˜ ( S i , x ˜ * ) , S i ∈ 2 N defined in the proof of Theorem DG. But x ˜ * ∉ C ( N , v ˜ ) , which means there exists a coalition S ∈ 2 ^N satisfying e ˜ ( S , x ˜ * ) ≈ 0 ˜ .

If C ( N , v ˜ ) ≠ ∅, then there exists an indifference fuzzy imputation y ˜ ∈ C ( N , v ˜ ) such that for any T _i  ∈ 2 ^N , e ˜ ( T i , y ˜ ) ⊴ 0 ˜ and θ ˜ ( y ˜ ) ≈ ( θ ˜ 1 ( y ˜ ) , θ ˜ 2 ( y ˜ ) , ⋯ , θ ˜ 2 n ( y ˜ ) ) , where θ ˜ i ( y ˜ ) : = e ˜ ( T i , y ˜ ) . It is concluded that e ˜ ( S , x ˜ * ) ⊴ θ ˜ 1 ( x ˜ * ) ⊴ θ ˜ 1 ( y ˜ ) ⊴ 0 ˜ which is contradicted with the hypothesis. Hence, x ˜ * ∈ C ( N , v ˜ ) . □

5. Indifference fuzzy bargaining sets