Coalitional game with fuzzy payoffs and credibilistic nucleolus

Abstract

In this paper, the coalitional game with fuzzy payoffs is considered. Based on the credibility theory, two new solution concepts of fuzzy coalitional game are proposed, one of which is the expected nucleolus and the other is the α-optimistic nucleolus. Furthermore, the non-emptiness and uniqueness characterizations of credibilistic nucleolus are proved, and the relationship between credibilistic nucleolus and credibilistic core is presented. Two examples are given to demonstrate the solution process.

Keywords

Credibilistic coalitional game credibilistic nucleolus fuzzy payoff credibility measure

1 Introduction

Game theory was established in 1944, when von Neumann and Morgenstern published the seminal work, The Theory of Games and Economic Behavior [29]. From then on, game theory has been used extensively in many fields, such as economics, politics, sociology and military, etc. Considering whether the players can make restrictive agreements, game theory can be categorized into cooperative game theory and non-cooperative game theory. The former is also called the coalitional game.

A coalitional game is distinguished from a non-cooperative game primarily by its focus on what groups of players can achieve rather than on what individual players can do, and by the fact that it does not consider the details of how groups of players function internally. Generally, the coalitional game uses characteristic function to analyze the properties of the coalition and allocates transferable payoffs (i.e., the outcomes of the coalition) to players of the coalition based on reasonability and stability. Many researchers introduced solution concepts to coalitional games from different perspectives [2 , 27]. A solution concept for coalitional games assigns to each game a set of outcomes. Each solution concept captures the consequences of a natural line of reasoning for the participants in a game. It defines a set of arrangements that are stable in some sense. The stability requirement is in general the outcome that is immune to deviations of a certain sort by groups of players. There are mainly two categories of solution concepts: one is domination, the other is estimation [20].

“Domination” requires that no set of players are able to break away and take a joint action that makes all of them to be better off. The requirement that the solution concepts belong to “domination” guarantees the dominate property of allocation. On the other hand, “estimation” gives a reasonable and unique allocation for coalitional game based on axiomatic systems. Shapley value and Banzhaf index of power are both “estimation” solution concepts. Different from estimation concepts, there are many domination concepts, such as core, bargaining set, kernel and nucleolus. Contrary to other domination concepts, nucleoles, which was introduced by Schemeidler [26], is the smallest concept, that means it is the subset of any other dominate concept sets, and it has the properties of existence and uniqueness.

Whatever concepts, the payoff of the game is necessary. But in many real-world circumstances, there are inadequate, even no historical records of statistical data available for probabilistic methods to predictthe payoff functions [9]. As Harsanyi [8] pointed out, the player may lack full information about other player’s (or even their own) payoffs. In such kind of situation, fuzzy set theory, which models the imprecise and vague payoffs as fuzzy variables [31], offers an appropriate and powerful tool to deal with the incomplete information. The fuzzy game was introduced by Butnaria [3] and Aubin [1]. From then on, the fuzzy strategic game [7, 17], fuzzy extensive game and fuzzy cooperative game [18 , 24] have been discussed. Based on credibility theory [11], a spectrum of credibilistic games have been developed by Gao and his coworkers [4–6 , 10], and fuzzy games have also been studied by many authors [22 , 23,28, 30].

In this paper, we focus on coalitional game with fuzzy payoffs, and propose two new solution concepts, i.e., expected and optimistic, of credibilistic nucleolus for the fuzzy coalitional game. When the players’ goals are to maximize the expected value of their fuzzy objectives in the decision process, the expected value criterion will be used. The expected nucleolus of fuzzy coalitional game indicates the expected value of nucleolus and gives the judgement basis for decision makers. On the other hand, we present the α-optimistic nucleolus for players when they are risk averse and want to avoid considerable losses. Under a predetermined confidence level α, the α-optimistic nucleolus can help players to maximize their optimistic profits.

The rest of the paper is organized as follows.Section 2 presents some preliminary results oncredibility theory and coalitional game. Then in Section 3, we give the definitions of the expected and optimistic nucleolus for the coalitional game with transferable fuzzy payoffs, respectively. Meanwhile, we testify the characteristics of non-emptiness and uniqueness of credibilistic nucleolus, and then analyze the relationship between credibilistic core and nucleolus. Section 4 presents two examples to demonstrate the solution process and illustrate the usefulness of the theory developed in the paper. Finally, Section 5 draws the conclusion of the paper and makes some discussions.

2 Preliminaries

In this section, we review some basic results on credibility theory and the concepts of nucleolus of coalitional game.

2.1 Credibility theory

In 1965, Zadeh [31] presented the concept of fuzzy set, and set up the possibility theory with the axiomatic foundation. The concept of credibility measure was proposed by Liu and Liu [14] in 2002. Later, the credibility theory was founded by Liu [11] and refined by Liu [13]. Credibility theory has an axiomatic foundation similar to that of probability theory, and can deal with complicated fuzzy phenomenon. Hence, it has been widely applied in the areas of fuzzy optimization and decision-making theory [12].

Let ξ be a fuzzy variable with membership function μ, and B be a set of real numbers. The credibility measure Cr is defined as follows:

$Cr {ξ \in B} = \frac{1}{2} (sup_{x \in B} μ (x) + 1 - sup_{x \in B^{c}} μ (x)) .$ (1) The formula (1) is also called the credibility inversion theorem.

Definition 2.1. (Liu and Liu [14]). Let ξ be a fuzzy variable. The expected value of ξ is defined as

$E [ξ] = \int_{0}^{\infty} Cr {ξ \geq r} dr - \int_{- \infty}^{0} Cr {ξ \leq r} dr,$ (2)

provided that at least one of the two integrals is finite.

Definition 2.2. (Liu and Liu [16]). Let ξ be a fuzzy variable, and α ∈ (0, 1]. Then $ξ_{sup} (α) = sup {r ∣ Cr {ξ \geq r} \geq α}$ (3)

is called the critical value at confidence level α.

Definition 2.3. (Liu and Gao [15]). The fuzzy variables ξ₁, ξ₂, …, ξ_m are said to be independent if $Cr {⋂_{i = 1}^{m} {ξ_{i} \in B_{i}}} = min_{1 \leq i \leq m} Cr {ξ_{i} \in B_{i}}$ (4) for any sets B₁, B₂, …, B_m of real numbers.

The properties of fuzzy variables are given as follows:

Linear property: Let ξ and η be independent fuzzy variables. Then for any real numbers a, b and α ∈ (0, 1], we have E [aξ + bη] = aE [ξ] + bE [η] and (aξ + bη) _sup (α) = aξ_sup (α) + bη_sup (α). Equivalently, both the expected value operator and the critical value operator have linear property under the assumption of independence.

Fuzzy ranking criteria: Let r be a real number. Based on the credibility theory, we may have the following ranking criteria:

Expected value criterion: ξ < η if and only if E [ξ] < E [η];

Critical value criterion: ξ < η if and only if η_sup (α) < ξ_sup (α) for some predetermined confidence level α ∈ (0, 1].

2.2 Coalitional game and the nucleolus

A coalitional game model mainly analyzes the actions taken by groups of players, especially focuses on the allocation of outcome in coalition members. The solution concept of the coalitional game is just as induction, which gives reasonable allocation of the outcome among players who are in one group. There are many solution concepts, and they can be categorized into two classes. One is domination, the other is estimation [20]. Nucleolus introduced by Schemeidler [26] belongs to the former. Of course, there are many other concepts, such as core, ε-core, stable set, bargaining set, kernel, etc., which also belong to domination solution. But the set of nucleolus is the subset of all other solution sets. Therefore, nucleolus is also called the smallest solution concept.

In order to analyze the solution concept, we first give the definition of coalitional game with transferable payoffs.

Definition 2.4. (Osborne and Rubinstein [25]). A coalitional game with transferable payoff 〈N, v〉 consists of:

a finite set of N (the set of the players) ; and

a function v, called characteristic function, which associates a real number v (S), the worth of S, with every nonempty subset S of N.

An important property of v is superadditivity.

Definition 2.5. In a coalitional game G = 〈N, v〉, if there is v (S) + v (T) ≤ v (S ∪ T) for all S ⊂ N, T ⊂ N and S∩ T = ∅, then the characteristic function v is called with property of superadditivity.

This definition means that, the payoff of a coalition must be more than the sum of the payoff that each player could receive if he or she does not join the coalition. The property of superadditivity presents the value of coalition.

Definition 2.6. In a coalitional game G = 〈N, v〉, where N = {1, 2, …, n}, let x = (x₁, x₂, …, x_n) if it satisfies the following: x_i ≥ v ({i}) for i = 1, …, n and $\sum_{i = 1}^{n} x_{i} = v (N)$ . Then x is called an allocation (or imputation) of N. Denote the set of all allocations as I (N, v).

Let x (S) = ∑_i∈Sx_i, where $(x_{i})_{i \in N} \in ℝ^{N}$ is the payoff profile for any S. A vector (x_i) _i∈N is an S-feasible payoff vector if x (S) ≤ v (S), which means the sum of allocations among coalition S cannot beyond the value of S.

Definition 2.7. (Gillies [7]). The core of the coalitional game with transferable payoff 〈N, v〉 is the set of feasible payoff profiles (x_i) _i∈N for which there is no coalition S and S-feasible payoff vector (y_i) _i∈N for which y_i > x_i, for all i ∈ S, denotedas C (v).

For any S-feasible payoff vector (x_i) _i∈N, there is x (S) ≤ v (S), hence according to Definition 2.7, we know that x ∈ C (v) if and only if x (S) ≥ y (S) for any S ⊆ N and any S-feasible payoff y (S). For any coalition S, e (S, x) = v (S) - x (S) is called the excess of coalition S based on allocation x. We also give the definitions of objection and counter objection as follows.

A pair (S, y) consisting of a coalition S and an allocation y is an objection to the allocation x if e (S, x) > e (S, y) (i.e., y (S) > x (S)).

A coalition S′ is a counter objection to the objection (S, y) if e (S′, y) > e (S′, x) (i.e., x (S′) > y (S′) and e (S′, y) ≥ e (S, x)).

The nucleolus of a coalitional game with transferable payoff is the set of all allocations x with the property that for every objection (S, y) to x, there is a counter objection to (S, y). The definition of the nucleolus is presented in the followingDefinition 2.8.

Definition 2.8. (Schemeidler [26]). In the n-player coalitional game with transferable payoff G = 〈N, v〉, nucleolus is a subset of I (N, v), denoted as N (v), which can be expressed as N (v) = {x ∣ x ∈ I (N, v), for all y ∈ I (N, v), y ≠ x, there is θ (x) <_Lθ (y)}, where θ (x) = [e (S₁, x), e (S₂, x) , …, e (S_2ⁿ, x)] and e (S_i, x) ≥ e (S_i+1, x), i = 1, 2, …, 2ⁿ- 1 and S_i ⊆ N, i = 1, 2, …, 2ⁿ. θ (x) <_Lθ (y) means that according to the lexicographical order, θ (x) is less than θ (y), i.e., there exists k₀ such that θ_k (x) < θ_k (y) for all k < k₀ and θ_{k
₀} (x) = θ_{k
₀} (y).

Definition 2.8 tells us that an allocation x is a nucleolus of a coalitional game with transferable payoffs if and only if the vector θ (x) is lexicographically minimal.

Three properties of nucleolus were given by Schemeidler [26]:

In the n-player coalitional game with transferable payoff G = 〈N, v〉, if C (v)≠ ∅, then N (v) ⊆ C (v).

In the n-player coalitional game with transferable payoff G = 〈N, v〉, N (v)≠ ∅.

In the n-player coalitional game with transferable payoff G = 〈N, v〉, |N (v) |=1, i.e., there is only one element in N (v).

3 Credibilistic nucleolus of coalitional game with fuzzy transferable payoffs

In this section, we discuss the coalitional game with fuzzy payoffs. Using the credibility theory introduced in Section 2, we propose two new solution concepts of credibilistic nucleolus of coalitional game with fuzzy transferable payoffs, one of which is the expected nucleolus, the other is the α-optimistic nucleolus. Moreover, we prove that the properties of nucleolus mentioned above are also satisfied under these two new definitions.

Definition 3.1. A coalitional game with fuzzy transferable payoff $〈 N, \tilde{v} 〉$ consists of:

a finite set N (the set of the players) ; and

a fuzzy variable $\tilde{v} (S)$ which is the fuzzy payoff of coalition S, where S is any nonempty subset of N.

As the traditional definition of transferable payoffs, the fuzzy payoff $\tilde{v} (S)$ is the total payoff that is available for the division among the members of the coalition S. Meanwhile, $〈 N, \tilde{v} 〉$ iscohesive, i.e., $\tilde{v} (N) \geq \sum_{k = 1}^{K} \tilde{v} (S_{k})$ for each partition {S₁, …, S_K}.

3.1 Expected nucleolus of fuzzy coalitional game

In this subsection, we assume that the players of the coalition S consider the expected values of fuzzy payoffs as the income of this coalitionalmodel.

For any coalition S, let $E [e (S, x)] = E [\tilde{v} (S)] - x (S)$ , which is called the expected excess value of the coalition S. Then $\tilde{θ} (x) = ({\tilde{θ}}_{1} (x), {\tilde{θ}}_{2} (x), \dots, {\tilde{θ}}_{2^{n}} (x))$ , where ${\tilde{θ}}_{i} (x) = E [e (S_{i}, x)]$ for i = 1, 2, …, 2ⁿ and ${\tilde{θ}}_{i} (x) \geq {\tilde{θ}}_{i + 1} (x)$ for i = 1, 2, …, 2ⁿ- 1.

Definition 3.2. The expected nucleolus of the n-player coalitional game with fuzzy transferable payoffs, $E N (\tilde{v})$ , is defined as $\begin{matrix} E N (\tilde{v}) & = {x ∣ x \in I (N, \tilde{v}), for all y \in I (N, \tilde{v}) \\ with y \neq x, there holds \tilde{θ} (x) <_{L} \tilde{θ} (y)}, \end{matrix}$

where $\tilde{θ} (x) <_{L} \tilde{θ} (y)$ means that $\tilde{θ} (x)$ is less than $\tilde{θ} (y)$ based on the lexicographical order, i.e., there exists k₀ such that ${\tilde{θ}}_{k} (x) = {\tilde{θ}}_{k} (y)$ for k < k₀ and ${\tilde{θ}}_{k_{0}} (x) < {\tilde{θ}}_{k_{0}} (y)$ .

Theorem 3.1. For any n-player coalitional games with transferable fuzzy payoffs, the expected nucleolus is nonempty and only has one element.

Proof. First, we prove the property of non-emptiness. Let N = {1, 2, …, n} and C be the set of all subsets of N, i.e., C = {S₁, S₂, …, S_2ⁿ}. Denote ${\tilde{θ}}_{1} (x) = max_{S \in C} E [e (S, x)] = E [\tilde{v} (S_{1})] - x (S_{1})$ . It is obvious that ${\tilde{θ}}_{1} (x)$ is continuous, then $x_{1} = arg min_{x \in ℝ^{n}} {\tilde{θ}}_{1} (x)$ is nonempty and compact. Similarly to ${\tilde{θ}}_{1} (x)$ and x₁, we have

$\begin{matrix} {\tilde{θ}}_{2} (x) & = max_{S \in C \ {S_{1}}} E [e (S, x)] \\ = E [\tilde{v} (S_{2})] - x (S_{2}), \\ x_{2} = arg min_{x \in X_{1}} {\tilde{θ}}_{2} (x), \\ ⋮ \\ {\tilde{θ}}_{k} (x) & = max_{S \in C \ {S_{1} \cup \dots \cup S_{k - 1}}} E [e (S, x)] \\ = E [\tilde{v} (S_{k})] - x (S_{k}), \\ x_{k} = arg min_{x \in X_{k - 1}} {\tilde{θ}}_{k} (x) . \end{matrix}$

As n is a finite number and $\begin{matrix} {\tilde{θ}}_{2^{n}} (x) & = max_{S \in C \ {S_{1} \cup \dots \cup S_{2^{n} - 1}}} E [e (S, x)] \\ = E [\tilde{v} (S_{2^{n}})] - x (S_{2^{n}}), \end{matrix}$ according to Definition 3.1, we know that $\begin{matrix} x_{2^{n}} = arg min_{x \in X_{2^{n} - 1}} {\tilde{θ}}_{2^{n}} (x) \end{matrix}$ is the expected nucleolus of the fuzzy coalitional game $〈 N, \tilde{v} 〉$ .

Then, we prove the uniqueness of expected nucleolus. Let B₁ (x) , B₂ (x) , …, B_K (x) be the subsets of N = {1, 2, …, n}, satisfying the following condition: For any S_i ∈ B_k (x), then S_j ∈ B_k (x) if and only if there holds ${\tilde{θ}}_{i} (x) = {\tilde{θ}}_{j} (x)$ . This means that the expected excess values of all the elements in the set B_k (x) are equal to each other. Denote e_k (x) to be the value of B_k (x), and |B_k (x) | to be the number of the elements of B_k (x). If we suppose that the expected excess values are listed in decreasing order, then there holds e₁ (x) > e₂ (x) > ⋯ > e_K (x).

Suppose that the allocations x and y belong to $E N (\tilde{v})$ . If there exists at least one coalition S^*, which satisfies E [e (S^*, x)] ≠ E [e (S^*, y)], i.e., ${\tilde{θ}}_{*} (x) \neq {\tilde{θ}}_{*} (y)$ , then we consider another coalition $z = \frac{1}{2} (x + y)$ . Because of ${\tilde{θ}}_{*} (x) \neq {\tilde{θ}}_{*} (y)$ , there exists a smallest k^* such that B_{k
^*} (x) ≠ B_{k
^*} (y). There are two cases: one is B_{k
^*} (x)∩ B_{k
^*} (y) ≠ ∅, then denote B_{k
^*} (z) = B_{k
^*} (x) ∩ B_{k
^*} (y); the other one is B_{k
^*} (x)∩ B_{k
^*} (y) = ∅, then e_{k
^*} (z) < e_{k
^*} (x) = e_{k
^*} (y). In both cases, the lexicographical order of $\tilde{θ} (z)$ is smaller than that of $\tilde{θ} (x)$ . So the hypothesis does not hold. Therefore, for any coalition S_i, there holds ${\tilde{θ}}_{i} (x) = E [e (S_{i}, x)] = {\tilde{θ}}_{i} (y) = E [e (S_{i}, y)], i = 1, \dots, 2^{n}$ . Especially, for any $i \in N$ , there holds E [e ({i} , x)] = E [e ({i} , y)]. This means that thevectors x and y are equal. The proof of uniqueness is complete.□

The relationship between the expected nucleolus $E N (\tilde{v})$ and the expected core $E C (\tilde{v})$ of fuzzy coalitional game can be expressed in the following Theorem 3.2. The readers may refer to [23] for more details on the expected core.

Theorem 3.2. For any n-player fuzzy coalitional game $〈 N, \tilde{v} 〉$ , if the set of expected core is nonempty, then there holds $E N (\tilde{v}) \subseteq E C (\tilde{v})$ .

Proof. Suppose that there exists an allocation xsuch that x ∈ $E N (\tilde{v})$ but x ∉ $E C (\tilde{v})$ . Because of x ∉ $E C (\tilde{v})$ , there is a coalition S such that x (S) < $E [\tilde{v} (S)]$ , then the expected excess value E [e (S, x)] = $E [\tilde{v} (S)]$ - x (S) >0. Let ${\tilde{θ}}_{i} (x)$ = E [e (S_i, x)], i = 1, 2, …, 2ⁿ, $\tilde{θ} (x)$ = $({\tilde{θ}}_{1} (x)$ , ${\tilde{θ}}_{2} (x)$ , …, ${\tilde{θ}}_{2^{n}} (x))$ , where ${\tilde{θ}}_{1} (x)$ is the maximum value of all ${\tilde{θ}}_{i} (x)$ , i = 1, 2, …, 2ⁿ. If $E C (\tilde{v})$ ≠ ∅, there exists an allocation y ∈ $E C (\tilde{v})$ . According to the properties of allocation, for any S_i, i = 1, 2,…, 2ⁿ, there is E [e (S_i, y)] = $E [\tilde{v} (S_{i})]$ - y ≤ 0. Therefore, $\tilde{θ} (y)$ = $({\tilde{θ}}_{1} (y)$ , ${\tilde{θ}}_{2} (y)$ , …, ${\tilde{θ}}_{2^{n}} (y))$ , ${\tilde{θ}}_{i} (y)$ ≤ 0 for i = 1, 2, …, 2ⁿ, where ${\tilde{θ}}_{1} (y)$ is the maximum value. Then we have $\tilde{θ} (x) = ({\tilde{θ}}_{1} (x), {\tilde{θ}}_{2} (x), \dots, {\tilde{θ}}_{2^{n}} (x)) > 0, {\tilde{θ}}_{1} (x) > 0$ for $x \in E N (\tilde{v})$ , and $\tilde{θ} (y) = ({\tilde{θ}}_{1} (y), {\tilde{θ}}_{2} (y), \dots, {\tilde{θ}}_{2^{n}} (y)) \leq 0, {\tilde{θ}}_{1} (y) \leq 0 .$ for $y \in E C (\tilde{v})$ , which contradict to the fact that $\tilde{θ} (x) <_{L} \tilde{θ} (y)$ . So the hypothesis is not true. The proof is complete. □

3.2 Optimistic nucleolus of fuzzy coalitional game

In this subsection, similar to Subsection 3.1, we give the definition of α-optimistic nucleolus of fuzzy coalitional game, then analyze the non-emptiness and uniqueness properties of the optimistic nucleolus under a predetermined confidence level α. Meanwhile, the relationship between α-optimistic nucleolus and α-optimistic core (refer to [10]) is also analyzed.

Definition 3.3. The optimistic nucleolus of the fuzzy coalitional game $〈 N, \tilde{v} 〉$ under a given confidence level α is denoted as $ξ_{sup}^{α} [N (\tilde{v})]$ , and $\begin{matrix} ξ_{sup}^{α} [N (\tilde{v})] & = {x ∣ x \in I (N, \tilde{v}), for any y \in I (N, \tilde{v}), \\ ξ_{sup}^{α} [θ (x)] < ξ_{sup}^{α} [θ (y)]}, \end{matrix}$ where $\begin{matrix} ξ_{sup}^{α} [θ (x)] & = (ξ_{sup}^{α} [θ_{1} (x)], ξ_{sup}^{α} [θ_{2} (x)], \dots, \\ ξ_{sup}^{α} [θ_{2^{n}} (x)]), \end{matrix}$ and $ξ_{sup}^{α} [θ (x)] < ξ_{sup}^{α} [θ (y)]$ means that there exists k₀ such that $ξ_{sup}^{α} [θ_{k} (x)] = ξ_{sup}^{α} [θ_{k} (y)]$ for all k < k₀, and $ξ_{sup}^{α} [θ_{k_{0}} (x)] < ξ_{sup}^{α} [θ_{k_{0}} (y)]$ . Equivalently, $ξ_{sup}^{α} [θ_{k} (x)]$ is less than $ξ_{sup}^{α} [θ_{k} (y)]$ based on the lexicographical order. For any coalitions S_i, $S_{i} \subseteq N$ , i = 1, 2, …, 2ⁿ, there hold $\begin{matrix} ξ_{sup}^{α} [θ_{i} (x)] & = ξ_{sup}^{α} [e (S_{i}, x)] \\ = ξ_{sup}^{α} [\tilde{v} (S_{i}, x)] - x (S_{i}), \\ ξ_{sup}^{α} [\tilde{v} (S_{i})] & = sup {r ∣ Cr {\tilde{v} (S_{i}) \geq r} \geq α} . \end{matrix}$

Theorem 3.3. For any n-player coalitional game with transferable fuzzy payoffs, the α-optimistic nucleolus is nonempty and only has one element.

Theorem 3.4. For any n-player fuzzy coalitional game $〈 N, \tilde{v} 〉$ , if the set of α-optimistic core $ξ_{sup}^{α} (x)$ is nonempty, then there holds $ξ_{sup}^{α} [θ (x)] \subseteq ξ_{sup}^{α} (x)$ for any given confidence level α ∈ [0, 1]. That means the set of optimistic nucleolus is the subset of the optimistic core at any confidence level α.

The proofs of Theorems 3.3 and 3.4 are similar to those of Theorems 3.1 and 3.2. So we omit them here.

4 Two examples

In this section, we give two examples to explain how to calculate the expected nucleolus and the α-optimistic nucleolus of fuzzy coalitional game.

Based on Theorem 3.2, we know that the expected nucleolus is the subset of the expected core. Therefore, following the relationship, we can express the expected core firstly and then calculate the expected nucleolus.

Example 4.1. Suppose that there are three players in the game, i.e., N = {1, 2, 3}. Excluding Players 2 and 3, they alone cannot get any pay. The payoff of three players and other payoffs of coalitions that include one or two players are triangular fuzzy numbers. The fuzzy payoffs of the coalitions are expressed as follows: $\begin{matrix} \tilde{v} (S_{1}) = \tilde{v} ({1, 2, 3}) = (5, 9, 17), \\ \tilde{v} (S_{2}) = \tilde{v} ({1, 2}) = (1.5, 4.3, 9.9), \\ \tilde{v} (S_{3}) = \tilde{v} ({1, 3}) = (2, 6, 14), \\ \tilde{v} (S_{4}) = \tilde{v} ({2, 3}) = (1, 5, 13), \\ \tilde{v} (S_{5}) = \tilde{v} ({1}) = (1.5, 3.5, 7.5), \\ \tilde{v} (S_{6}) = \tilde{v} ({2}) = 0, \\ \tilde{v} (S_{7}) = \tilde{v} ({3}) = 0 . \end{matrix}$

This is a three-player majority game with fuzzy payoffs. For k = 1, 2, …, 5, the membership functions of fuzzy payoffs are defined below: $\begin{matrix} μ (\tilde{v} (S_{k})) = {\begin{matrix} 0, & \tilde{v} (S_{k}) < a_{k}, \\ \frac{\tilde{v} (S_{k}) - a_{k}}{b_{k} - a_{k}}, & a_{k} \leq \tilde{v} (S_{k}) \leq b_{k}, \\ \frac{\tilde{v} (S_{k}) - c_{k}}{b_{k} - c_{k}}, & b_{k} \leq \tilde{v} (S_{k}) \leq c_{k}, \\ 0, & \tilde{v} (S_{k}) > c_{k}, \end{matrix} \end{matrix}$ (5) where (a_k, b_k, c_k) stands for the fuzzy payoff vector of the coalition S_k, k = 1, 2, …, 5.

We assume that the players all adopt the expected criterion. Then the expected payoffs of the coalitions are given as follows: $\begin{matrix} E [\tilde{v} (S_{1})] = 10, E [\tilde{v} (S_{2})] = 5, \\ E [\tilde{v} (S_{3})] = 7, E [\tilde{v} (S_{4})] = 6, \\ E [\tilde{v} (S_{5})] = 4, E [\tilde{v} (S_{6})] = E [\tilde{v} (S_{7})] = 0 . \end{matrix}$

Based on the properties of the expected core, the necessary and sufficient condition of $x = (x_{1}, x_{2}, x_{3}) \in E C (\tilde{v})$ is (see [10]) $\begin{matrix} {\begin{matrix} x_{1} \geq 4, x_{2} \geq 0, x_{3} \geq 0, \\ x_{1} + x_{2} \geq 5, \\ x_{1} + x_{3} \geq 7, \\ x_{2} + x_{3} \geq 6, \\ x_{1} + x_{2} + x_{3} = 10 . \end{matrix} \end{matrix}$ (6)

Then, we can get the solution of (6) as $E C (\tilde{v}) = {(4, 6 - x, x) : 3 \leq x \leq 5}$ . It is obvious that the expected core is nonempty. According to Theorem 3.3, there is $E N (\tilde{v}) \subseteq E C (\tilde{v})$ . Because of $\begin{matrix} E [e (S, x)] = E [\tilde{v} (S)] - \sum_{i \in S} x_{i}, x \in E C (\tilde{v}), \end{matrix}$ we have $\begin{matrix} E [e (S_{1}, x)] = 0, \\ E [e (S_{2}, x)] = x - 5, \\ E [e (S_{3}, x)] = 3 - x, \\ E [e (S_{4}, x)] = E [e (S_{5}, x)] = 0, \\ E [e (S_{6}, x)] = x - 6, E [e (S_{7}, x)] = - x . \end{matrix}$

In the proof of Theorem 3.2, we have given the method of finding the expected nucleolus. According to Definition 3.2, we denote $\begin{matrix} \tilde{θ} (x) & = min max_{1 \leq i \leq 7} E [e (S_{i}, x)] \\ = min max_{3 \leq x \leq 5} {5 - x, x - 3} . \end{matrix}$

Then x = 4 satisfies the equality. The expected nucleolus of this fuzzy coalitional game is $E N (\tilde{v}) = {(4, 6 - x, x) : x = 4} = {(4, 2, 4)}$ . This result shows that the set of expected nucleolus is nonempty, and it has only one element.□

For the sake of simplicity, we give another example to express the α-optimistic nucleolus of coalitional game with fuzzy payoffs.

Example 4.2. Suppose that there are four players in the game, N = {1, 2, 3, 4}, and the players all want to get the optimal profit under a confidence level α. The payoff of every coalition is a triangular fuzzy number. The payoff of four players is denoted by (200, 300, 500), the payoff of any coalition that includes three players in this game is denoted by (150, 225, 375), and the payoff of any coalition with two players is denoted by (100, 150, 250). Meanwhile, we assume that every player alone cannot get any payoff. Then the membership functions of fuzzy payoffs are defined as follows: $\begin{matrix} μ (\tilde{v} (4)) = {\begin{matrix} 0, & \tilde{v} (4) < 200, \\ \frac{\tilde{v} (4) - 200}{100}, & 200 \leq \tilde{v} (4) \leq 300, \\ \frac{\tilde{v} (4) - 500}{- 200}, & 300 \leq \tilde{v} (4) \leq 500, \\ 0, & \tilde{v} (4) > 500, \end{matrix} \end{matrix}$ (7)

$\begin{matrix} μ (\tilde{v} (3)) = {\begin{matrix} 0, & \tilde{v} (3) < 150, \\ \frac{\tilde{v} (3) - 150}{75}, & 150 \leq \tilde{v} (3) \leq 225, \\ \frac{\tilde{v} (3) - 375}{- 150}, & 225 \leq \tilde{v} (3) \leq 375, \\ 0, & \tilde{v} (3) > 375, \end{matrix} \end{matrix}$ (8)

$\begin{matrix} μ (\tilde{v} (2)) = {\begin{matrix} 0, & \tilde{v} (2) < 100, \\ \frac{\tilde{v} (2) - 100}{50}, & 100 \leq \tilde{v} (2) \leq 150, \\ \frac{\tilde{v} (2) - 250}{- 100}, & 150 \leq \tilde{v} (2) \leq 250, \\ 0, & \tilde{v} (2) > 250 . \end{matrix} \end{matrix}$ (9)

In what follows, we present the α-optimistic nucleolus under different confidence levels. Firstly, we have the following conclusion: Letting ξ = (a, b, c) be a fuzzy variable with a < b < c, then there holds

$\begin{matrix} ξ_{sup} (α) = {\begin{matrix} 2 α b + (1 - 2 α) c, & if α \leq 0.5, \\ (2 α - 1) a + (2 - 2 α) b, & if α > 0.5 . \end{matrix} \end{matrix}$ (10)

Take the confidence level α = 0.6 for example. We have $\begin{matrix} ξ_{sup}^{0.6} [\tilde{v} (4)] = 280, ξ_{sup}^{0.6} [\tilde{v} (3)] = 210, \\ ξ_{sup}^{0.6} [\tilde{v} (2)] = 140, ξ_{sup}^{0.6} [\tilde{v} (1)] = 0 . \end{matrix}$

It is obvious that, under the given confidence level α = 0.6, the allocation x = (70, 70, 70, 70) not only belongs to the set of α-optimistic core but also satisfies the properties of α-optimistic nucleolus. For other levels, the corresponding optimistic nucleolus profiles are given in Table 1.

It can be seen from Table 1 that the optimistic income of players are equal at the same α-level. Following with the increasing of α, the optimistic income of players are decreasing correspondingly. This phenomenon shows that the more risk, the more income. That is, if the player is more risk-affordable, he would choose a lower confidence level to get more income. Conversely, if the player is risk-averse, he would prefer not to afford the risk of coalition and hopes to keep lower but stabler payoff under high confidencelevel.

Meanwhile, from the value of profiles, we can see that under different confidence level, every element of $ξ_{sup}^{α}$ is equal to zero, i.e., for all coalitions S_i ⊆ N, there are $ξ_{sup}^{α} [\tilde{v} (S_{i})] = x (S_{i})$ , i = 1, …, 2⁴, where α increases from 0.1 to 1.0. This means that for any coalition S_i, there are no unsatisfactory (or the smallest unsatisfactory) on the allocation x which belongs to the optimistic nucleolus. This is just the essence properties of nucleolus.□

5 Conclusions

In this paper, we proposed two new definitions for coalitional game with fuzzy transferable payoffs. One is the expected nucleolus, the other is α-optimistic nucleolus. Furthermore, we discussed the properties of non-emptiness and uniqueness of these two definitions and analyzed the relationship between expected (α-optimistic) nucleolus and expected (α-optimistic) core. This relationship provides the method of calculating the credibilistic nucleolus, which can be seen from the examples.

Based on the analysis for the examples presented in Section 4, we see that the credibilistic nucleolus given in this paper has the same properties as the traditional one (the nucleolus of coalitional game with crisp payoffs). But the new one that we propose has more management information and is more applicable to the real world than the traditional one. This shows the value of the paper.

With the concepts and results proposed in this paper, stability analysis for the coalition with fuzzy payoffs can be discussed, and behavior study of individuals in the coalition with fuzzy payoffs can be deeply researched. Furthermore, we may also study other solution concepts with the application of credibilistic theory into coalitional game. A third research issue is to construct credibilistic kernel, credibilistic Nash bargaining solution, and credibilistic egalitarian solution, etc. These will be the directions of our future work.

Footnotes

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant No. 16XNB036). The authors are grateful to the anonymous referees for invaluable comments and suggestions which helped to improve the quality of this paper.

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