The least square B-nucleolus for fuzzy cooperative games

Abstract

The least square B-nucleolus for fuzzy cooperative games is proposed based on the bi-excess of fuzzy coalitions. The proposed solutions consider not only the size of fuzzy coalitions, but also the blocking and constructive powers. The uniqueness of the least square B-prenucleolus for fuzzy cooperative games is proved detailedly. Some quadratic programming models for generating the least square B-nucleolus of complete and incomplete fuzzy cooperative games are presented, respectively. The least square B-prenucleolus is extended to the multiplicative setting, and the logarithmic least square B-prenucleolus for multiplicative fuzzy cooperative games is derived.

Keywords

Least square nucleolus cooperative game quadratic programming fuzzy coalition

1 Introduction

Cooperative game is a useful tool to deal with profit allocation problems. The existing research works focus on finding effective and rational solution of cooperative games. Core [1] and Shapley value [2] are two widely used solution concepts for cooperative games. Generally, the core contains a set of imputations. Therefore, in the practical profit allocation problems, we need to select a proper imputation in the core. However, it is difficult to establish the rule for selecting the optimal imputation in the core. It can not even guarantee that the core is non-empty. Although Shapley value is a single point solution, it is not always in the non-empty core. To overcome above problems, Schmeidler [3] proposed the nucleolus based on the lexicographic order of excess of coalitions at imputation. The focus of the nucleolus is to definite an effective excess of coalition at imputation. Schmeidler pointed out that the nucleolus is a single point solution and always belongs to the non-empty core. As an important solution concept for cooperative games, the nucleolus has been widely studied over the last decades [4 –7]. To consider both the blocking and constructive powers of coalitions, an excess containing both sides is introduced by Sudholter [8], and then the modified nucleolus is presented. Tarashnina [9] simplified the structure of the modified nucleolus, and explained what ratio the blocking and constructive powers are considered. To distinguish the cardinality of coalitions, a weighted excess is developed by Wallmeier [10], the f-nucleolus is proposed synchronously. Since the concept of f-nucleolus appearance, many weighted excesses have been presented in the literature [11 –13]. Puerto and Perea [14] proposed an effective method for calculating the nucleolus by solving a linear programming model. Faigle et al. [15] developed a new algorithm for computing the nucleolus of cooperative games. Fromen [16] compared the proposed algorithm for finding the nucleolus with two existing algorithms, and pointed out that nucleolus can be solved by a finite sequence of linear programs. Fragnelli et al. [17] proved that well-posedness of a lexicographic minimum over a compact set can be guaranteed under suitablehypotheses, and the nucleolus is well-posed.

As mentioned above, the nucleolus usually generates by some iterative algorithms or programming models. However, the explicit expression of nucleolus is difficut to obtain. This greatly limits the application of nucleolus in practical cooperative games. Difference from the lexicographic order based nucleolus, the least square nucleolus is given by Ruiz et al. [18] to minimize the variance of the resulting excesses of the coalitions. A quadratic programming model is established to derive the proposed solution. Furthermore, Ruiz proved that the least square nucleolus is a single point solution. The explicit expression of least square nucleolus can be obtained by solving a system of equations. The least square nucleolus is important and has received more and more attention. Dragan [19] pointed out that the weighted least square value is the related Shapley value of cooperative games. Molina and Tejada [20] proved that the least square nucleolus is a general nucleolus. Ruiz et al. [21] introduced the family of least square values, and clarified that the additive efficient normalization of any semivalue belong to the proposed family. Laruelle and Merlin [22] proposed different linear solutions forTU-games by least square values, and pointed out that least square values induce a ranking of players. Ruiz et al. [23] introduced some new properties of the least square family of values. The stability of these least square values from a dynamic point of view is studied. Dragan [24] proved that the least square values of TU-games can be expressed in terms of averages of values of characteristic function. However, the constructive and blocking powers of coalitions are not considered together by any of the above-mentioned least square nucleolus. In many pratical situations, some players may partly participate in a coalition. The cooperative game with fuzzy coalitions is a suitable concept to deal with this phenomenon [25 –27]. Nevertheless, the least square nucleolus is also seldom discussed in cooperative games with fuzzy coalitions. Therefore, it is very necessary to develop the least square nucleolus for fuzzy cooperative games based on the constructive and blocking powers. In this paper, a new solution concept called least square B-nucleolus is proposed based on the bi-excess of fuzzy coalitions. The uniqueness of the least square B-prenucleolus for fuzzy cooperative games is proved in detail. The least square B-nucleolus can be obtained by calculating some quadratic programming models. Furthermore, we extend the least square B-prenucleolus to the multiplicative setting, and the logarithmic least square B-prenucleolus for multiplicative fuzzy cooperative game is given.

The rest of this paper is structured as follows.Section 2 introduces some basic definitions of cooperative games and the least square nucleolus. In Section 3, we establish some quadratic programming models to derive the least square B-nucleolus and least square B-prenucleolus. Section 4 presents the least square B-nucleolus for incomplete games. The extension of the least square B-prenucleolus to multiplicative setting is discussed in Section 5. In Section 6, two numerical examples are provided to illustrate the validity of the proposed solutions. The paper is concluded inSection 7.

2 Preliminaries

In the following, some basic concepts on cooperative games are introduced in detail.

Let N = {1, 2, …, n} be the player set. We denote the crisp cooperative game by (N, v). For any coalition S ∈ P (N) ∖ {∅} and payoff vector $\hat{x}$ , the excess of S with respect to $\hat{x}$ is defined by $e (S, v, \hat{x}) = v (S) - \sum_{i \in S} x_{i}$ (1)

The value $e (S, v, \hat{x})$ can be interpreted as a scale of the dissatisfaction of coalition S when payoff vector $\hat{x}$ is selected. Ruiz et al. [18] proposed the least square nucleolus by solving a quadratic programming model (M-1), which selects the imputation to minimize the variance of the resulting excesses of the coalitions, shown as follows:

(M-1): $\begin{matrix} min \underset{S \neq \emptyset}{\sum_{S \subseteq N}} {[e (S, v, \hat{x}) - \bar{E} (v, \hat{x})]}^{2} \\ s . t . x_{i} \geq v ({i}), i = 1, 2, \dots, n . \\ \sum_{i = 1}^{n} x_{i} = v (N) . \end{matrix}$ where $\bar{E} (v, \hat{x})$ is the average excess at payoff vector $\hat{x}$ , and can be expressed as $\bar{E} (v, \hat{x}) = \frac{1}{2^{n} - 1} \underset{S \neq \emptyset}{\sum_{S \subseteq N}} e (S, v, \hat{x})$ (2)

If condition x_i ≥ v ({i}) (i = 1, 2, …, n .) is not required in (M-1), then the solution of the corresponding model is called the least square prenucleolus of cooperative game (N, v).

Players may be not fully participate in a coalition, the cooperative game with fuzzy coalitions is given to deal with this situation [25 –27]. A fuzzy subset in N is called a fuzzy coalition, which can be denoted by a n-dimensional vector $\tilde{S} = (S_{1}, S_{2}, \dots, S_{n})$ with S_i ∈ [0, 1], i = 1, 2, …, n. The number S_i indicates the participation level of player i in fuzzy coalition $\tilde{S}$ . For any two fuzzy coalitions $\tilde{S}$ and $\tilde{T}$ , $\tilde{S} \subseteq \tilde{T}$ if and only if S_i = 0 or S_i = T_i (i = 1, 2, …, n). We denote the set of all subset of grand fuzzy coalition $\tilde{N}$ by $F (\tilde{N})$ . Accordingly, we have $F (\tilde{N}) = {\tilde{S} | \tilde{S} \subseteq \tilde{N}}$ . In the fuzzy setting, for any crisp coalition R(R ⊆ N), we denote R by e^R = (R₁, R₂, …, R_n) satisfying R_i = 1 when i ∈ R and R_i = 0 when i ∈ N ∖ R, i = 1, 2, …, n. ∀ $\tilde{S}, \tilde{T} \in F (\tilde{N})$ , the union and intersection of $\tilde{S}, \tilde{T}$ are defined by the following maximum and minimumoperators [26]: $(\tilde{S} \lor \tilde{T})_{i} = max {S_{i}, T_{i}}, \forall i \in N .$ (3) $(\tilde{S} \land \tilde{T})_{i} = min {S_{i}, T_{i}}, \forall i \in N .$ (4)

A cooperative game with grand fuzzy coalition $\tilde{N}$ is a function $\tilde{v} : F (\tilde{N}) \to R^{+}$ with $\tilde{v} (\emptyset) = 0$ . The worth of fuzzy coalition $\tilde{S}$ is represented by $\tilde{v} (\tilde{S})$ . The cooperative game with fuzzy coalitions is also called fuzzy cooperative game for short. The notation $(\tilde{N}, \tilde{v})$ is used to indicate the fuzzy cooperative game. For any two fuzzy coalitions $\tilde{S}$ and $\tilde{T}$ ( $\tilde{S} \land \tilde{T} = \emptyset$ ), a fuzzy cooperative game $(\tilde{N}, \tilde{v})$ is said to be superadditive if and only if it satisfies [26]: $\tilde{v} (\tilde{S} \lor \tilde{T}) \geq \tilde{v} (\tilde{S}) + \tilde{v} (\tilde{T})$ (5)

Yu and Zhang [26] proposed the imputation for a fuzzy cooperative game $(\tilde{N}, \tilde{v})$ . An imputation for $(N, \tilde{v})$ is a payoff vector $\hat{x} = (x_{1}, x_{2}, \dots, x_{n})$ satisfying $x_{i} = 0, \forall i \notin supp \tilde{N} .$ (6) $x_{i} \geq \tilde{v} (N_{i} e^{{i}}), \forall i \in supp \tilde{N} .$ (7) $\sum_{i \in supp \tilde{N}} x_{i} = \tilde{v} (\tilde{N}) .$ (8)where $supp \tilde{N} = {j | N_{j} > 0}$ is the support of fuzzy coalition $\tilde{N}$ . If condition (7) is not required, then payoff vector $\hat{x}$ is said to be a preimputation of fuzzy cooperative game $(\tilde{N}, \tilde{v})$ . The set of all imputation and preimputation of fuzzy cooperative game $(\tilde{N}, \tilde{v})$ are denoted by $I (\tilde{N}, \tilde{v})$ and $I_{0} (\tilde{N}, \tilde{v})$ , respectively.

3 Least square B-nucleolus and its uniqueness

The least square nucleolus is proposed based on the excess of the coalitions. Thus, the definition of excess is very important in the model. To distinguish thecardinality of coalitions, Wallmeier [10] proposed the formula of f-excess of coalition S at imputation $\hat{x}$ , shown as follows: $δ (S, v, \hat{x}) = f (| S |) (v (S) - \sum_{i \in S} x_{i})$ (9)where f : T → R⁺ be a nonincreasing function, T = {|S||S ⊆ N}, and |S| denotes the cardinality of coalition S. However, “the blocking power” of a coalition is not taken into account in Equation (9). To improve this situation, the bi-excess of fuzzy coalition $\tilde{S}$ at $\hat{x}$ is proposed to considering both the constructive power and blocking power of fuzzy coalitions as follows:

$\begin{matrix} σ (\tilde{S}, \tilde{v}, \hat{x}) & = & f (| \tilde{S} |) e (\tilde{S}, \tilde{v}, \hat{x}) \\ - f (| \tilde{N} ∖ \tilde{S} |) e (\tilde{N} ∖ \tilde{S}, \tilde{v}, \hat{x}) \end{matrix}$ (10)where $| \tilde{S} | = \sum_{i \in supp \tilde{S}} S_{i}$ denotes the cardinality of fuzzy coalition $\tilde{S}$ , $\forall \tilde{S} \subseteq \tilde{N}$ . Moreover, Equation (10) can be further written as $\begin{matrix} σ (\tilde{S}, \tilde{v}, \hat{x}) = f (\sum_{i \in supp \tilde{S}} S_{i}) e (\tilde{S}, \tilde{v}, \hat{x}) \\ - f (\sum_{i \in supp \tilde{N} ∖ \tilde{S}} (\tilde{N} ∖ \tilde{S})_{i}) e (\tilde{N} ∖ \tilde{S}, \tilde{v}, \hat{x}) \end{matrix}$ (11)

Since $e (\tilde{S}, \tilde{v}, \hat{x}) = \tilde{v} (\tilde{S}) - \sum_{i \in supp \tilde{S}} x_{i}$ , we get $\begin{matrix} σ (\tilde{S}, \tilde{v}, \hat{x}) = f (| \tilde{S} |) [\tilde{v} (\tilde{S}) - \sum_{j \in supp \tilde{S}} x_{j}] \\ - f (| \tilde{N} ∖ \tilde{S} |) [\tilde{v} (\tilde{N} ∖ \tilde{S}) - \sum_{j \in supp \tilde{N} ∖ \tilde{S}} x_{j}] \end{matrix}$ (12)

It follows that $σ (\tilde{S}, \tilde{v}, \hat{x}) = P (\tilde{S}) - Q (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j}$ (13)where $P (\tilde{S}) = f (| \tilde{S} |) \tilde{v} (\tilde{S}) + f (| \tilde{N} ∖ \tilde{S} |) [\tilde{v} (\tilde{N}) - \tilde{v} (\tilde{N} ∖ \tilde{S})]$ (14) $Q (\tilde{S}) = f (| \tilde{S} |) + f (| \tilde{N} ∖ \tilde{S} |)$ (15)

To minimize the variance of the excesses of fuzzy coalitions, two quadratic programming models are developed to select the payoff vector asfollows:

(M-2): $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[P (\tilde{S}) - Q (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j} - E (\tilde{v}, \hat{x})]}^{2} \\ s . t . x_{i} = 0, \forall i \notin supp \tilde{N} . \\ x_{i} \geq \tilde{v} (N_{i} e^{{i}}), \forall i \in supp \tilde{N} . \\ \sum_{i \in supp \tilde{N}} x_{i} = \tilde{v} (\tilde{N}) \end{matrix}$

(M-3): $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[P (\tilde{S}) - Q (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j} - E (\tilde{v}, \hat{x})]}^{2} \\ s . t . x_{i} = 0, \forall i \notin supp \tilde{N} . \\ \sum_{i \in supp \tilde{N}} x_{i} = \tilde{v} (\tilde{N}) . \end{matrix}$ where $E (\tilde{v}, \hat{x})$ is the arithmetic average excess at payoff vector $\hat{x}$ , and can be expressed as $E (\tilde{v}, \hat{x}) = \frac{1}{2^{n} - 1} \underset{S \neq \emptyset}{\sum_{S \subseteq N}} σ (\tilde{S}, \tilde{v}, \hat{x})$ (16)

Definition 1. The solutions of quadratic programming models (M-2) and (M-3) are the least square B-nucleolus and the least square B-prenucleolus for fuzzy cooperative game $(\tilde{N}, \tilde{v})$ , respectively.

Theorem 1. The quadratic programming models (M-2) and (M-3) are equivalent to the following models (M-4) and (M-5), respectively.

(M-4): $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[P (\tilde{S}) - Q (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j}]}^{2} \\ s . t . x_{i} = 0, \forall i \notin supp \tilde{N} . \\ x_{i} \geq \tilde{v} (N_{i} e^{{i}}), \forall i \in supp \tilde{N} . \\ \sum_{i \in supp \tilde{N}} x_{i} = \tilde{v} (\tilde{N}) \end{matrix}$

(M-5): $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[P (\tilde{S}) - Q (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j}]}^{2} \\ s . t . x_{i} = 0, \forall i \notin supp \tilde{N} . \\ \sum_{i \in supp \tilde{N}} x_{i} = \tilde{v} (\tilde{N}) . \end{matrix}$

Proof: Let $\hat{x} = (x_{1}, x_{2}, \dots, x_{n})$ be a payoff vector of fuzzy cooperative game $(\tilde{N}, \tilde{v})$ . For any $\tilde{S} \in F (\tilde{N}) ∖ \emptyset$ , by Equation (10), we have $\begin{matrix} σ (\tilde{S}, \tilde{v}, \hat{x}) + σ (\tilde{N} ∖ \tilde{S}, \tilde{v}, \hat{x}) \\ = f (| \tilde{S} |) e (\tilde{S}, \tilde{v}, \hat{x}) - f (| \tilde{N} ∖ \tilde{S} |) e (\tilde{N} ∖ \tilde{S}, \tilde{v}, \hat{x}) \\ + f (| \tilde{N} ∖ \tilde{S} |) e (\tilde{N} ∖ \tilde{S}, \tilde{v}, \hat{x}) \\ - f (| \tilde{N} ∖ (\tilde{N} ∖ \tilde{S}) |) e (\tilde{N} ∖ (\tilde{N} ∖ \tilde{S}), \tilde{v}, \hat{x}) \\ = f (| \tilde{S} |) e (\tilde{S}, \tilde{v}, \hat{x}) - f (| \tilde{N} ∖ \tilde{S} |) e (\tilde{N} ∖ \tilde{S}, \tilde{v}, \hat{x}) \\ + f (| \tilde{N} ∖ \tilde{S} |) e (\tilde{N} ∖ \tilde{S}, \tilde{v}, \hat{x}) - f (| \tilde{S} |) e (\tilde{S}, \tilde{v}, \hat{x}) = 0 \end{matrix}$ Since fuzzy coalition $\tilde{S}$ is the complementary set of $\tilde{N} ∖ \tilde{S}$ , we have $\underset{S \neq \emptyset, \tilde{N}}{\sum_{\tilde{S} \subseteq \tilde{N}}} σ (\tilde{S}, \tilde{v}, \hat{x}) = 0 .$ In addition, by Equations (8) and (12) we have $\begin{matrix} σ (\tilde{N}, \tilde{v}, \hat{x}) = f (| \tilde{N} |) [\tilde{v} (\tilde{N}) \\ - \sum_{j \in supp \tilde{N}} x_{j}] - f (| \emptyset |) [\tilde{v} (\emptyset) - \sum_{j \in supp \emptyset} x_{j}] = 0 \end{matrix}$ In sum, it holds that $E (\tilde{v}, \hat{x}) = \frac{1}{2^{n} - 1} \underset{S \neq \emptyset}{\sum_{S \subseteq N}} σ (\tilde{S}, \tilde{v}, \hat{x}) = 0 .$ Accordingly, the models (M-2) and (M-3) are equivalent to the models (M-4) and (M-5), respectively. This completes the proof of Theorem 1. □

Theorem 2. For any nonincreasing function $f : {| supp \tilde{S} | | \tilde{S} \subseteq \tilde{N}} \to R^{+}$ and fuzzy cooperative game $(\tilde{N}, \tilde{v})$ , the (M-5) has a unique solution $\hat{x} = (x_{1}, x_{2}, \dots, x_{n})$ , which can be expressed by ${\begin{matrix} x_{i} = \frac{\underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) - \frac{1}{| supp \tilde{N} |} \sum_{i \in supp \tilde{N}} \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S})}{\sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} | - 1} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 1}} \\ + \frac{\tilde{v} (\tilde{N})}{| supp \tilde{N} |}, i \in supp \tilde{N}; (17) \\ x_{i} = 0, i \notin supp \tilde{N} . \end{matrix}$ (17) where $| supp \tilde{S} |$ and $| supp \tilde{N} |$ is the cardinality of the crisp coalitions $supp \tilde{S}$ and $supp \tilde{N}$ , respectively.

Proof. The Lagrangian of quadratic programming model (M-5) is $\begin{matrix} L (\hat{x}, λ) = \sum_{\tilde{S} \subseteq \tilde{N}} {[P (\tilde{S}) - Q (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j}]}^{2} \\ + λ [\sum_{i \in supp \tilde{N}} x_{i} - \tilde{v} (\tilde{N})] \end{matrix}$ For any $i \in supp \tilde{N}$ , it holds

$\begin{matrix} \frac{\partial L}{\partial x_{i}} = - 2 \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} [P (\tilde{S}) - Q (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j}] \\ Q (\tilde{S}) + λ = 0 \end{matrix}$ (18)The Equation (18) can be further written as $\begin{matrix} \frac{λ}{2} = \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) - \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} [Q^{2} (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j}] \\ = \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) - \sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} |} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 1}^{| supp \tilde{S} | - 1} x_{i} \\ - \sum_{| supp \tilde{S} | = 2}^{| supp \tilde{N} |} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 2} \sum_{j \in supp \tilde{N} ∖ {i}} x_{j} \end{matrix}$ Since $\sum_{j \in supp \tilde{N} ∖ {i}} x_{j} = \sum_{j \in supp \tilde{N}} x_{j} - x_{i} = \tilde{v} (\tilde{N}) - x_{i},$ It follows that $\begin{matrix} \frac{λ}{2} = \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) - \sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} |} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 1}^{| supp \tilde{S} | - 1} x_{i} \\ - \sum_{| supp \tilde{S} | = 2}^{| supp \tilde{N} |} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 2} [\tilde{v} (\tilde{N}) - x_{i}] \\ = \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) - \sum_{| supp \tilde{S} | = 2}^{| supp \tilde{N} |} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 2} \tilde{v} (\tilde{N}) \\ - [\sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} |} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 1}^{| supp \tilde{S} | - 1} \\ - \sum_{| supp \tilde{S} | = 2}^{| supp \tilde{N} |} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 2}] x_{i} \end{matrix}$ (19)

Taking the summation over all $i \in supp \tilde{N}$ in Equation (19), we get $\begin{matrix} \frac{1}{2} | supp \tilde{N} | λ = \sum_{i \in supp \tilde{N}} \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) \\ - | supp \tilde{N} | \sum_{| supp \tilde{S} | = 2}^{| supp \tilde{N} |} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 2} \tilde{v} (\tilde{N}) \\ - [\sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} | - 1} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 1}] \sum_{i \in supp \tilde{N}} x_{i} \end{matrix}$ (20)then Equation (20) can be equivalently written as $\begin{matrix} \frac{λ}{2} = \frac{1}{| supp \tilde{N} |} \sum_{i \in supp \tilde{N}} \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) \\ - \sum_{| supp \tilde{S} | = 2}^{| supp \tilde{N} |} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 2} \tilde{v} (\tilde{N}) \\ - \frac{\tilde{v} (\tilde{N})}{| supp \tilde{N} |} \sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} | - 1} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 1} \end{matrix}$ (21)

The difference of Equations (19) and (21) is taken to eliminate λ, we have $\begin{matrix} \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) - x_{i} \sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} | - 1} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 1} \\ - \frac{1}{| supp \tilde{N} |} \sum_{i \in supp \tilde{N}} \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) \\ + \frac{\tilde{v} (\tilde{N})}{| supp \tilde{N} |} \sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} | - 1} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 1} = 0 \end{matrix}$ which can be further written as $\begin{matrix} x_{i} = \\ \frac{\underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S}) - \frac{1}{| supp \tilde{N} |} \sum_{i \in supp \tilde{N}} \underset{i \in supp \tilde{S}}{\sum_{\tilde{S} \subseteq \tilde{N}}} P (\tilde{S}) Q (\tilde{S})}{\sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} | - 1} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 1}} \\ + \frac{\tilde{v} (\tilde{N})}{| supp \tilde{N} |} . \end{matrix}$

Moreover, $\forall i \notin supp \tilde{N}$ , x_i = 0 holds. In sum, the Theorem 2 is proved. □

It is clear that the B-prenucleolus always exists, and has the unique expression as shown in Equation (20).

4 Least square B-nucleolus for incomplete games

Due to the uncertainty and complexity in real life, the payoffs information may be partly unknown in a cooperative game. Masuya and Inuiguchi [28] introduced crisp cooperative games with incomplete payoff information, the incomplete payoffs are evaluated by the corresponding interval based on superadditivity. Motivated by their idea, the definition of fuzzy cooperative games with incomplete payoff information is given as follows:

Definition 2. Let $(\tilde{N}, \tilde{v})$ be a fuzzy cooperative game. The set of all fuzzy coalitions whose payoffs are known, is denoted by Ω. If it satisfies

(1) $Ω \subseteq F (\tilde{N})$ and $Ω \neq F (\tilde{N})$ ;

(2) ∅, N ∈ Ω;

(3) N_ie^{i} ∈ Ω, i = 1, 2, …, n.

then $(\tilde{N}, Ω, \tilde{v})$ is called an incomplete fuzzy cooperative game.

Definition 3. For any fuzzy coalition $\tilde{S} \in Ω$ , ∃ natural number s and fuzzy coalitions ${\tilde{T}}_{i} \in Ω$ (i = 1, 2, …, s) such that $\lor_{i = 1}^{s} {\tilde{T}}_{i} \subseteq \tilde{S}$ and ${\tilde{T}}_{j} \land {\tilde{T}}_{k} = \emptyset$ , j, k ∈ {1, 2, …, s}. If $\tilde{v} (\tilde{S}) \geq \sum_{i = 1}^{s} \tilde{v} ({\tilde{T}}_{i})$ holds for each choice of the ${\tilde{T}}_{i}$ , then incomplete fuzzy cooperative game $(\tilde{N}, Ω, \tilde{v})$ is superadditive.

Masuya and Inuiguchi [28] proposed the upper and lower game of incomplete crisp cooperative game based on superadditivity. The boundary of unknown payoffs in incomplete crisp cooperative game are properly evaluated. For any superadditive incomplete fuzzy cooperative game $(\tilde{N}, Ω, \tilde{v})$ , the following corresponding upper and lower fuzzy games are given, which is similar to the formulae proposed by Masuya and Inuiguchi [28].

Definition 4. Let $(\tilde{N}, Ω, \tilde{v})$ be a incomplete fuzzy cooperative game. For any $\tilde{S} \in F (\tilde{N})$ , if ${\tilde{v}}^{-} (\tilde{S}) = \underset{j, k \in {1, 2, \dots, s}}{\underset{\lor_{i = 1}^{s} T_{i} \subseteq \tilde{S}, {\tilde{T}}_{j} \land {\tilde{T}}_{k} = \emptyset}{max_{{\tilde{T}}_{i} \in Ω, i = 1, 2, \dots, s}}} \sum_{i = 1}^{s} \tilde{v} ({\tilde{T}}_{i})$ (22) ${\tilde{v}}^{+} (\tilde{S}) = min_{\tilde{T} \in Ω, \tilde{T} \supseteq \tilde{S}} [\tilde{v} (\tilde{T}) - {\tilde{v}}^{-} (\tilde{T} ∖ \tilde{S})]$ (23)then $(\tilde{N}, {\tilde{v}}^{+})$ and $(\tilde{N}, {\tilde{v}}^{-})$ are called the upper and lower fuzzy cooperative games with respect to $(\tilde{N}, Ω, \tilde{v})$ , respectively.

Remark 1. From Equations (22) and (23), $(\tilde{N}, {\tilde{v}}^{+})$ and $(\tilde{N}, {\tilde{v}}^{-})$ are complete fuzzy cooperative games. It is clear that ${\tilde{v}}^{-} (\tilde{S}) = \tilde{v} (\tilde{S})$ in case of a superadditive game, for all $\tilde{S} \in Ω$ .

To minimize the variance of the resulting excesses of coalitions, the imputation of incomplete fuzzy cooperative game $(\tilde{N}, Ω, \tilde{v})$ is selected by the following quadratic programming model (M-6).

(M-6): $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[P (\tilde{S}) - Q (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j}]}^{2} \\ s . t . x_{i} = 0, \forall i \notin supp \tilde{N}; \\ \sum_{i \in supp \tilde{N}} x_{i} = \tilde{v} (\tilde{N}); \end{matrix}$ $\begin{matrix} x_{i} \geq \tilde{v} (N_{i} e^{{i}}), \forall i \in supp \tilde{N}; \\ \tilde{v} (\tilde{S} \lor \tilde{T}) \geq \tilde{v} (\tilde{S}) + \tilde{v} (\tilde{T}), \\ \tilde{S}, \tilde{T} \in F (\tilde{N}), \tilde{S} \land \tilde{T} = \emptyset, \\ | supp ({\tilde{S}, \tilde{T}, \tilde{S} \lor \tilde{T}} \land (F (\tilde{N}) ∖ Ω)) | > 0 . \end{matrix}$

Since $(\tilde{N}, Ω, \tilde{v})$ is superadditive, by Equation (13), the model (M-6) can be equivalently expressed to model (M-7).

(M-7): $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[P (\tilde{S}) - Q (\tilde{S}) \sum_{j \in supp \tilde{S}} x_{j}]}^{2} \\ s . t . x_{i} = 0, \forall i \notin supp \tilde{N}; \\ \sum_{i \in supp \tilde{N}} x_{i} = \tilde{v} (\tilde{N}); \\ x_{i} \geq \tilde{v} (N_{i} e^{{i}}), \forall i \in supp \tilde{N}; \\ {\tilde{v}}^{-} (\tilde{S}) \leq \tilde{v} (\tilde{S}) \leq {\tilde{v}}^{+} (\tilde{S}), \forall \tilde{S} \in F (\tilde{N}) ∖ Ω; \\ \tilde{v} (\tilde{S} \lor \tilde{T}) \geq \tilde{v} (\tilde{S}) + \tilde{v} (\tilde{T}), \\ \tilde{S}, \tilde{T} \in F (\tilde{N}), \tilde{S} \land \tilde{T} = \emptyset, \\ | supp ({\tilde{S}, \tilde{T}, \tilde{S} \lor \tilde{T}} \land (F (\tilde{N}) ∖ Ω)) | \geq 2 . \end{matrix}$

We denote $\bar{Ω} = F (\tilde{N}) ∖ Ω = {{\tilde{H}}_{1}, {\tilde{H}}_{2}, \dots, {\tilde{H}}_{t}}$ , i.e. set $\bar{Ω}$ is all the fuzzy coalitions in $\tilde{N}$ with unknown payoffs. For any ${\tilde{H}}_{i} \in \bar{Ω}$ , $\tilde{v} ({\tilde{H}}_{i})$ is denoted by θ_i, i = 1, 2, …, t. Accordingly, the optimal solution of quadratic programming model (M-7) can be denoted by ${\hat{η}}^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*}, θ_{1}^{*}, θ_{2}^{*}, \dots, θ_{t}^{*})$ . Logically, ${\hat{x}}^{*} = (x_{1}^{*}, x_{2}^{*}, \dots, x_{n}^{*})$ is called the least squareB-nucleolus for incomplete fuzzy cooperative game $(\tilde{N}, Ω, \tilde{v})$ .

Remark 2. It is apparent that least square B-nucleolus ${\hat{x}}^{*}$ always exists. From the first three constraints of model (M-7), we have ${\hat{x}}^{*} \in I (\tilde{N}, \tilde{v})$ . Therefore, the least square B-nucleolus ${\hat{x}}^{*}$ is an effective imputation for incomplete fuzzy cooperative game $(\tilde{N}, Ω, \tilde{v})$ .

5 Extension of the least square B-prenucleolus to multiplicative setting

Recently, Ortmann [29] proposed a new positive cooperative game model in the multiplicative setting. In the multiplicative model, all players are using the same utility scale, and the multiplicative Shapley value is given based on the proposed utility scale. Brockman and Wright [30] pointed out that the advantage of the multiplicative model is the simplicity in dealing with banking problems, medical science and insurance problems without a major sacrifice in accuracy. In the multiplicative setting, a positive cooperative game with player set N is denoted by (N, v_M), with v_M : P (N) → R⁺ and v_M (∅) =1. Ortmann [29] defined the power of a coalition as a multiple of the power of the empty set. Moreover, for any effective payoff vector $\hat{y} = (y_{1}, y_{2}, \dots, y_{n})$ , the multiplicative efficiency is introduced as $\prod_{i \in N} y_{i} = v_{M} (N)$ .

If some players not fully participate in a coalition under multiplicative setting, we denote the multiplicative fuzzy cooperative game by $(\tilde{N}, {\tilde{v}}_{M})$ . $\tilde{N}$ is the grand fuzzy coalition, ${\tilde{v}}_{M} : F (\tilde{N}) \to {r | r > 0}$ is the characteristic function with ${\tilde{v}}_{M} (\emptyset) = 1$ . In the following, the formula of multiplicative bi-excess of fuzzy coalition $\tilde{S}$ at payoff vector $\hat{y}$ is $β (\tilde{S}, {\tilde{v}}_{M}, \hat{y}) = \frac{[μ (\tilde{S}, {\tilde{v}}_{M}, \hat{y})]^{f (| \tilde{S} |)}}{[μ (\tilde{N} ∖ \tilde{S}, {\tilde{v}}_{M}, \hat{y})]^{f (| \tilde{N} ∖ \tilde{S} |)}}$ (24)where $μ (\tilde{S}, {\tilde{v}}_{M}, \hat{y}) = \frac{{\tilde{v}}_{M} (\tilde{S})}{\prod_{i \in supp \tilde{S}} y_{i}}$ , $f : {| \tilde{S} | | \tilde{S} \subseteq \tilde{N}} \to R^{+}$ be a nonincreasing function. For any $\tilde{S} \subseteq \tilde{N}$ , $| \tilde{S} | = \sum_{i \in supp \tilde{S}} S_{i}$ denotes the cardinality of fuzzy coalition $\tilde{S}$ . Furthermore, the Equation (24) can be wrriten as $\begin{matrix} ln β (\tilde{S}, {\tilde{v}}_{M}, \hat{y}) = f (| \tilde{S} |) [ln {\tilde{v}}_{M} (\tilde{S}) - \sum_{i \in supp \tilde{S}} ln y_{i}] \\ - f (| \tilde{N} ∖ \tilde{S} |) [ln {\tilde{v}}_{M} (\tilde{N} ∖ \tilde{S}) - \sum_{i \in supp \tilde{N} ∖ \tilde{S}} ln y_{i}] \end{matrix}$ (25)

Since $\prod_{i \in supp N} y_{i} = {\tilde{v}}_{M} (\tilde{N})$ , we get $\sum_{i \in supp \tilde{N} ∖ \tilde{S}} ln y_{i} = \frac{{\tilde{v}}_{M} (\tilde{N})}{\sum_{i \in supp \tilde{S}} ln y_{i}}$ (26)

By Equation (26), we have $\begin{matrix} ln β (\tilde{S}, {\tilde{v}}_{M}, \hat{y}) = f (| \tilde{S} |) [ln {\tilde{v}}_{M} (\tilde{S}) - \sum_{i \in supp \tilde{S}} ln y_{i}] \\ - f (| \tilde{N} ∖ \tilde{S} |) [ln {\tilde{v}}_{M} (\tilde{N} ∖ \tilde{S}) - ln {\tilde{v}}_{M} (\tilde{N}) + \sum_{i \in supp \tilde{S}} ln y_{i}] \end{matrix}$

It follows that $ln β (\tilde{S}, {\tilde{v}}_{M}, \hat{y}) = α (\tilde{S}) - Q (\tilde{S}) \sum_{i \in supp \tilde{S}} ln y_{i}$ (27)where $Q (\tilde{S})$ is defined in Equation (15), and

$\begin{matrix} α (\tilde{S}) = f (| \tilde{S} |) ln {\tilde{v}}_{M} (\tilde{S}) + f (| \tilde{N} ∖ \tilde{S} |) \\ \times [ln {\tilde{v}}_{M} (\tilde{N}) - ln {\tilde{v}}_{M} (\tilde{N} ∖ \tilde{S})] \end{matrix}$ (28)

To minimize the variance of the excesses of fuzzy coalitions, a logarithmic least square model is developed to select the payoff vector as

(M-8): $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[ln \frac{β (\tilde{S}, {\tilde{v}}_{M}, \hat{y})}{\bar{E} ({\tilde{v}}_{M}, \hat{y})}]}^{2} \\ s . t . y_{i} = 1, \forall i \notin supp \tilde{N} . \\ \prod_{i \in supp \tilde{N}} y_{i} = {\tilde{v}}_{M} (\tilde{N}) . \end{matrix}$ where $\bar{E} ({\tilde{v}}_{M}, \hat{y})$ is the geometric average excess at payoff vector $\hat{y}$ , and can be expressed as $\bar{E} ({\tilde{v}}_{M}, \hat{y}) = {[\underset{S \neq \emptyset}{\prod_{S \subseteq N}} β (\tilde{S}, {\tilde{v}}_{M}, \hat{y})]}^{\frac{1}{2^{n} - 1}}$ (29)

By Equation (27), the model (M-8) can be further wrriten by

(M-9): $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[α (\tilde{S}) - Q (\tilde{S}) \sum_{i \in supp \tilde{S}} ln y_{i} - ln \bar{E} ({\tilde{v}}_{M}, \hat{y})]}^{2} \\ s . t . y_{i} = 1, \forall i \notin supp \tilde{N} . \\ \sum_{i \in supp \tilde{N}} ln y_{i} = ln [{\tilde{v}}_{M} (\tilde{N})] . \end{matrix}$

Definition 5. The solution of logarithmic least square model (M-8) is the logarithmic least square B-prenucleolus for multiplicative fuzzy cooperative game $(\tilde{N}, {\tilde{v}}_{M})$ .

Theorem 3. The logarithmic least square model(M-9) is equivalent to the following optimal programming model (M-10).

(M-10): $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[α (\tilde{S}) - Q (\tilde{S}) \sum_{i \in supp \tilde{S}} ln y_{i}]}^{2} \\ s . t . y_{i} = 1, \forall i \notin supp \tilde{N} . \\ \sum_{i \in supp \tilde{N}} ln y_{i} = ln [{\tilde{v}}_{M} (\tilde{N})] . \end{matrix}$

Proof: Let $\hat{y} = (y_{1}, y_{2}, \dots, y_{n})$ be a payoff vector of multiplicative fuzzy cooperative game $(\tilde{N}, {\tilde{v}}_{M})$ . $\forall \tilde{S} \in F (\tilde{N}) ∖ \emptyset$ , by Equation (24), it follows that $\begin{matrix} β (\tilde{S}, {\tilde{v}}_{M}, \hat{y}) \cdot β (\tilde{N} ∖ \tilde{S}, {\tilde{v}}_{M}, \hat{y}) \\ = \frac{[μ (\tilde{S}, {\tilde{v}}_{M}, \hat{y})]^{f (| \tilde{S} |)}}{[μ (\tilde{N} ∖ \tilde{S}, {\tilde{v}}_{M}, \hat{y})]^{f (| \tilde{N} ∖ \tilde{S} |)}} \cdot \\ \frac{[μ (\tilde{N} ∖ \tilde{S}, {\tilde{v}}_{M}, \hat{y})]^{f (| \tilde{N} ∖ \tilde{S} |)}}{[μ (\tilde{N} ∖ (\tilde{N} ∖ \tilde{S}), {\tilde{v}}_{M}, \hat{y})]^{f (| \tilde{N} ∖ (\tilde{N} ∖ \tilde{S}) |)}} = 1 \end{matrix}$

Since fuzzy coalition $\tilde{S}$ is the complementary set of $\tilde{N} ∖ \tilde{S}$ , we have $\underset{S \neq \emptyset}{\prod_{S \subseteq N}} β (\tilde{S}, {\tilde{v}}_{M}, \hat{y}) = 1$ .

Besides, it holds that $β (\tilde{N}, {\tilde{v}}_{M}, \hat{y}) = \frac{{[\frac{{\tilde{v}}_{M} (\tilde{N})}{\prod_{j \in supp \tilde{N}} y_{j}}]}^{f (| \tilde{N} |)}}{{[\frac{{\tilde{v}}_{M} (\emptyset)}{\prod_{j \in supp \emptyset} y_{j}}]}^{f (| \emptyset |)}} = 1 .$

To sum up, we have $\bar{E} ({\tilde{v}}_{M}, \hat{y}) = {[\underset{S \neq \emptyset}{\prod_{S \subseteq N}} β (\tilde{S}, {\tilde{v}}_{M}, \hat{y})]}^{\frac{1}{2^{n} - 1}} = 1,$ i.e. $ln \bar{E} ({\tilde{v}}_{M}, \hat{y}) = 0$ . Therefore, the model (M-9) is equivalent to the model (M-10). □

Theorem 4. For any nonincreasing function $f : {| supp \tilde{S} | | \tilde{S} \subseteq \tilde{N}} \to R^{+}$ and multiplicative fuzzy cooperative game $(\tilde{N}, {\tilde{v}}_{M})$ , the (M-10) exists a unique solution $\hat{y} = (y_{1}, y_{2}, \dots, y_{n})$ , which can be denoted by ${\begin{matrix} y_{i} = e^{ɛ_{i}}, i \in supp \tilde{N} . \\ y_{i} = 1, i \notin supp \tilde{N} . \end{matrix}$ (30)where

$\begin{matrix} ɛ_{i} = \\ \frac{\underset{S \neq \emptyset, \tilde{N}}{\sum_{\tilde{S} \subseteq \tilde{N}}} α (\tilde{S}) Q (\tilde{S}) - \frac{1}{| supp \tilde{N} |} \sum_{i \in supp \tilde{N}} \underset{S \neq \emptyset, \tilde{N}}{\sum_{\tilde{S} \subseteq \tilde{N}}} α (\tilde{S}) Q (\tilde{S})}{\sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} | - 1} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 1}} \\ + \frac{ln [{\tilde{v}}_{M} (\tilde{N})]}{| supp \tilde{N} |} \end{matrix}$ (31)

Proof. Let ln y_i = ɛ_i, i = 1, 2, …, n. The model(M-10) is transformed into the following model $\begin{matrix} min \sum_{\tilde{S} \subseteq \tilde{N}} {[α (\tilde{S}) - Q (\tilde{S}) \sum_{i \in supp \tilde{S}} ɛ_{i}]}^{2} \\ s . t . ɛ_{i} = 0, \forall i \notin supp \tilde{N} . \\ \sum_{i \in supp \tilde{N}} ɛ_{i} = ln [{\tilde{v}}_{M} (\tilde{N})] . \end{matrix}$

Similar to the proof of theorem 2 by using the method of Lagrange multipliers, the unique solution can be derived as follows: $\begin{matrix} ɛ_{i} = \\ \frac{\underset{S \neq \emptyset, \tilde{N}}{\sum_{\tilde{S} \subseteq \tilde{N}}} α (\tilde{S}) Q (\tilde{S}) - \frac{1}{| supp \tilde{N} |} \sum_{i \in supp \tilde{N}} \underset{S \neq \emptyset, \tilde{N}}{\sum_{\tilde{S} \subseteq \tilde{N}}} α (\tilde{S}) Q (\tilde{S})}{\sum_{| supp \tilde{S} | = 1}^{| supp \tilde{N} | - 1} Q^{2} (\tilde{S}) C_{| supp \tilde{N} | - 2}^{| supp \tilde{S} | - 1}} \\ + \frac{ln [{\tilde{v}}_{M} (\tilde{N})]}{| supp \tilde{N} |}, i \in supp \tilde{N} . \end{matrix}$

Thus, $\forall i \in supp \tilde{N}$ , we have y_i = e^{ɛ
_i}. Besides, for any $i \notin supp \tilde{N}$ , ɛ_i = 0 holds, i.e. y_i = e^{ɛ
_i} = 1. In sum, the Theorem 4 is proved. □

Note that the logarithmic least square B-prenucleolus always exists, and has the unique expression as shown in Equation (30).

6 Illustrative example

Example 1. Suppose the function f (t) ≡1 is selected here. N = {1, 2} is the player set, and we know participation levels N₁ = 0.4 and N₂ = 0.8, respectively. The payoffs of fuzzy coalitions are $\tilde{v} (N_{1} e^{{1}}) = 1$ , $\tilde{v} (N_{2} e^{{2}}) = 3$ , $\tilde{v} (N_{1} e^{{1}} \lor N_{2} e^{{2}}) = 6$ . By Equations (14) and (15), we have $\begin{matrix} P (N_{1} e^{{1}}) = 4, P (N_{2} e^{{2}}) = 8, \\ P (N_{1} e^{{1}} \lor N_{2} e^{{2}}) = 12 . \\ Q (N_{1} e^{{1}}) = Q (N_{2} e^{{2}}) = Q (N_{1} e^{{1}} \lor N_{2} e^{{2}}) = 2 . \end{matrix}$

Based on Equation (17), we have x₁ = 2.5 and x₂ = 3.5. Thus, the least square B-nucleolus of fuzzy cooperative game $(\tilde{N}, \tilde{v})$ is $\hat{x} = (0.25, 0.35)$ .

If the fuzzy cooperative game is considered in the multiplicative setting, we suppose $\tilde{v} (\tilde{S}) = {\tilde{v}}_{M} (\tilde{S})$ for any $\tilde{S} \in F (\tilde{N}) ∖ \emptyset$ , and $\tilde{v} (\emptyset) = 1$ . By Equation (30), we get $\begin{matrix} α (N_{1} e^{{1}}) = ln 2, α (N_{2} e^{{2}}) = 2 ln 3 + ln 2, \\ α (N_{1} e^{{1}} \lor N_{2} e^{{2}}) = 2 ln 3 + 2 ln 2 . \end{matrix}$

Using Equation (31), we have $ɛ_{1} = \frac{1}{2} ln 2 + \frac{1}{4} ln 3$ and $ɛ_{2} = \frac{1}{2} ln 2 + \frac{3}{4} ln 3$ . Thus, the logarithmic least square B-prenucleolus of multiplicative fuzzy cooperative game $(\tilde{N}, {\tilde{v}}_{M})$ is $\hat{y} = (\sqrt[4]{12}, \sqrt[4]{108})$ .

Example 2. Suppose the function $f (t) = {\begin{matrix} \frac{1}{t}, t > 0 \\ 5, t = 0 \end{matrix}$ , N = {1, 2, 3} is the player set, and we know participation levels N₁ = 0.5, N₂ = 0.5 and N₃ = 0.2. The payoffs of fuzzy coalitions are $\begin{matrix} \tilde{v} (N_{1} e^{{1}}) = 1, \tilde{v} (N_{2} e^{{2}}) = 2, \tilde{v} (N_{3} e^{{3}}) = 3, \\ \tilde{v} (N_{1} e^{{1}} \lor N_{2} e^{{2}}) = 5, \tilde{v} (N_{1} e^{{1}} \lor N_{3} e^{{3}}) = 6, \\ \tilde{v} (N_{1} e^{{1}} \lor N_{2} e^{{2}} \lor N_{3} e^{{3}}) = 12 . \end{matrix}$

The payoff of fuzzy coalition N₂e^{2} ∨ N₃e^{3} is unknown. Denoting $\tilde{v} (N_{2} e^{{2}} \lor N_{3} e^{{3}}) = θ$ , by Equations (14) and (15), we have $\begin{matrix} P (N_{1} e^{{1}}) = 19.148 - 1.429 θ, P (N_{2} e^{{2}}) = 12.574, \\ P (N_{3} e^{{3}}) = 22, P (N_{1} e^{{1}} \lor N_{2} e^{{2}}) = 50, \end{matrix}$ $\begin{matrix} P (N_{1} e^{{1}} \lor N_{3} e^{{3}}) = 28.574, \\ P (N_{2} e^{{2}} \lor N_{3} e^{{3}}) = 22 + 1.429 θ \\ P (N_{1} e^{{1}} \lor N_{2} e^{{2}} \lor N_{3} e^{{3}}) = 70 . \\ Q (N_{1} e^{{1}}) = Q (N_{2} e^{{2}}) = 3.429, \\ Q (N_{1} e^{{1}} \lor N_{3} e^{{3}}) = Q (N_{2} e^{{2}} \lor N_{3} e^{{3}}) = 3.429, \\ Q (N_{3} e^{{3}}) = Q (N_{1} e^{{1}} \lor N_{2} e^{{2}}) = 6, \\ Q (N_{1} e^{{1}} \lor N_{2} e^{{2}} \lor N_{3} e^{{3}}) = 5.833 . \end{matrix}$

Based on model (M-7), we establish the following quadratic programming: $\begin{matrix} min [93.54 x_{1}^{2} + 93.54 x_{2}^{2} + 93.54 x_{3}^{2} + 4.08 θ^{2} \\ - 1743.90 x_{1} - 1653.73 x_{2} - 1427.46 x_{3} + 8.15 θ \\ + 140.05 x_{1} x_{2} + 91.56 x_{1} x_{3} + 91.56 x_{2} x_{3} \\ + 9.8 θ (x_{1} - x_{2} - x_{3})] \\ s . t . x_{1} + x_{2} + x_{3} = 12, \\ 5 \leq θ \leq 11, \\ x_{1} \geq 1, x_{2} \geq 2, x_{3} \geq 3 . \end{matrix}$

By solving the above quadratic programming, we get $x_{1}^{*} = 3.87$ , $x_{2}^{*} = 4.03$ , $x_{3}^{*} = 4.10$ . Thus, the least square B-nucleolus of incomplete fuzzy cooperative game $\tilde{v}$ is ${\hat{x}}^{*} = (3.87, 4.03, 4.10)$ .

7 Conclusions

This paper proposes an effective solution concept which is called the least square B-nucleolus for fuzzy cooperative games. The least square B-nucleolus considers both two sides power and the cardinality of fuzzy coalitions. For the superadditive fuzzy cooperative games with incomplete payoff information, the least square B-nucleolus is also derived by solving a quadratic programming model. Moreover, the logarithmic least square model is developed for deriving the corresponding B-prenucleolus in the multiplicative setting. The main theoretical contributions of this paper are shown as follows:

The least square B-nucleolus is defined based on bi-excess. An explicit expression of least square B-nucleolus is proposed.

The existence and uniqueness of least square B-prenucleolus are proved.

We construct a quadratic programming model to generate least square B-nucleolus for fuzzy cooperative games with incomplete payoffinformation.

The logarithmic least square B-nucleolus is proposed based on multiplicative bi-excess. We show that logarithmic least square B-nucleolus can be expressed by an explicit formula.

It is worth to pointed out that not only some new solutions are proposed here, but also the field of potential application is extended. In further study, we will consider several applications of the least square B-nucleolus in actual fields such as risk aversion, environmental improvement and assets management.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 71371030, 71071018), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20111101110036), the Science and Technique Foundation of Fujian Education Office (No. JA13114) and the Humanities and Social Sciences Fund of the Ministry of Education (No. 14YJC630114).

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