The least square pre-nucleolus for interval cooperative games based on anti-symmetric interval excess values

Abstract

Considering both cardinal characteristics and double powers, the anti-symmetric interval excess value is defined. The least square pre-nucleolus for interval cooperative games is presented by making a single-objective programming model. We obtain the analytic expression of least square pre-nucleolus using Lagrange multiplier method, and construct an effective quadratic programming model to derive the least square pre-nucleolus of incomplete interval cooperative games. In addition, the application of least square pre-nucleolus in land pollution control is provided to show the validity of the proposed solution concepts.

Keywords

Least square pre-nucleolus interval excess value cooperative game incomplete payoff

1 Introduction

With the rapid development of social production and industry economy, it is increasingly prominent to seek cooperation in a competitive environment. As an effective tool of dealing with income distribution and cost allocation, cooperative game has potential applications in many fields, such as resource optimization [1 –3], environmental governance [4, 5], adaptive control [6 –8] and so on. The key content of cooperative game is to find reasonable and effective solution concepts, in which core [9] and Shapley value [10] are widely used. Since the core often contains infinite imputations, further selection of the most effective imputation is needed in practical application. It is not often guaranteed that the core is non-empty. Although Shapley value is unique and always exists, it does not necessarily belong to the non-empty core. Therefore, Shapley value is fair, but lack of stability as an imputation of cooperative game.

In order to overcome the shortcomings of the above solution concept, based on the lexicographic order of excess values, Schmeidler [11] proposed the nucleolus of cooperative games, which belonged to core, and was a single point solution. Since the concept of nucleolus was put forward, it has been analyzed and perfected by many scholars. Considering the cardinality of coalitions on the excess values, Wallmeier [12] developed the F-nucleolus based on weighted excess values. By defining the weighted excess value of the coalitions, Derks and Haller [13] put forward the weighted nucleoli of cooperative games. Faigle et al. [14] established a set of weighted functions of the coalitions, and gave the definition of CF-nucleolus. Moreover, they proved the degenerative compatibility between F-nucleolus and CF-nucleolus. By expanding the F-nucleolus, Kern and Paulusma [15] proposed the novel nucleolus of flow games. In view of both the blocking and constructive powers of coalitions, Sudholter [16] gave the concept of modified nucleolus by defining Bi-excess values. Tarashnina [17] clarified the ratio of blocking and constructive powers, and proposed the simplified modified nucleolus. It reduced the computational complexity of the nucleolus as the structure of the Bi-excess values was effectively simplified. By constructing the generalized excess value with cardinal characteristics and double powers, Lin and Zhang [18] introduced the G-nucleolus for cooperative game with fuzzy coalition. Zhang et al. [19] pointed out that the nucleolus of fuzzy cooperative games is well-defined following from the total order of fuzzy numbers. Tae [20] reduced the computation time of calculating the nucleolus of cooperative games. The relationship between nucleolus and non-empty core is discussed.

In the existing research on the nucleolus-like solution, it is often necessary to establish an optimization model or iterative algorithm to derive the nucleolus. However, Obtaining the intuitive analytic expression of nucleolus is always difficult. This disadvantage makes a limit on the application of nucleolus in practice. In order to facilitate the application of nucleolus-like solution, Ruiz et al. [21] presented the least square nucleolus by minimizing the variance of excess values. The analytical expression of the least square nucleolus can be obtained by solving the quadratic programming model using Lagrange multiplier method. Furthermore, Ruiz et al. [22, 23] studied the family of least square nucleolus, and discussed the desirable properties of the solution concepts, in which the stability of the least square nucleolus was dynamically analyzed. Molina and Tejada [24] pointed out that the least square nucleolus was a general kind of nucleolus. Lin and Zhang [25] considered the cardinal characteristics and double powers in excess values, and gave the least square Bi-nucleolus for fuzzy cooperative games. Dragan [26, 27] proved that the weighted least square value was one kind of Shapley value, and can be expressed as the mean value of characteristic functions. Kong et al. [28] pointed out that the least square general pre-nucleolus minimized the variance of player complaints. Laruelle and Merlin [29] investigated different linear solutions based on the least square values, which induced the ranking of players. Li and Ye [30] proposed a direct simplified method of obtaining the interval-valued least square pre-nucleolus of interval-valued cooperative games. Li [31] developed a fast method of computing the least square interval-valued nucleoli of interval-valued cooperative games. Some useful properties were studied in detail. Zhao and Liu [32] constructed the interval-valued least square excess solution, and applied the proposed solution to the profit allocation of the road freight coalition. However, the cardinal characteristics, constructive and blocking powers of coalitions have not been considered together by any of the above-mentioned cooperative games with interval payoffs. So it is necessary to propose the least square pre-nucleolus for interval cooperative games based on the new definition of excess values to overcome this shortage. In this paper, we define the anti-symmetric interval excess value considering both the cardinal characteristics and double powers. The least square pre-nucleolus for interval cooperative games can be obtained by making a single-objective programming model. According to Lagrange multiplier method, the analytic expression of least square pre-nucleolus is derived. And we construct a novel quadratic programming model of generating the least square pre-nucleolus of interval cooperative games with incomplete payoffs. To illustrate the validity and practicality of the least square pre-nucleolus, we make the land pollution control as a typical application case.

The rest of this paper is structured as follows. Section 2 introduces some corresponding concepts and notations of cooperative games. In Section 3, we establish some programming model to calculate the least square pre-nucleolus of interval cooperative games. Section 4 studies the least square nucleolus of interval cooperative games with incomplete payoffs. In Section 5, land pollution control is applied to present the validity of the proposed solutions. The comparison and analysis are studied in Section 6. The paper is concluded in Section 7.

2 Preliminaries

Some corresponding concepts and notations of cooperative games [17, 32] are reviewed as follows to facilitate the next discussion.

Let N = {1, 2, ⋯ , n} be the collection of players. For any subset S of N, S is called a coalition. Denoting Δ (N) = {S|S ⊆ N}, Δ (N) is the set of all coalitions on N. A cooperative game on grand coalition N is a function v : Δ (N) → R⁺ with v (ø) =0, R⁺ = {r|r ≥ 0}. ∀S ⊆ N, the non-negative value v (S) represents the profit of coalition S. For the sake of simplicity, a cooperative game on N can be denoted by (N, v).

Definition 2.1. For all S, T ∈ Δ (N), S∩ T = ø, if $v (S \cup T) \geq v (S) + v (T),$ (2.1) then the cooperative game (N, v) is super-additive.

Definition 2.2. Let $\hat{x} = (x_{1}, x_{2}, \dots, x_{n})$ be a n-dimensional payoff vector, the component x_i (i = 1, 2, …, n) is a crisp number. If $x_{i} \geq v ({i}),$ (2.2) $\sum_{i = 1}^{n} x_{i} = v (N),$ (2.3) then payoff vector $\hat{x}$ is called an imputation of crisp cooperative game (N, v). If condition (2.2) is not required, then $\hat{x}$ is called a pre-imputation of crisp cooperative game (N, v). The set of all pre-imputations of (N, v) is denoted by PI (N, v). The notation I (N, v) indicates the set of all imputations of crisp cooperative game (N, v).

Because of the uncertainty of the game environment and the fuzziness of human thinking, the payoffs of the coalitions often do not appear with the accurate real number. In this context, the coalition value is given in the form of some fuzzy variables, such as interval number, triangle fuzzy number, intuitionistic fuzzy number and so on. The expression of interval payoffs is widely used because of its intuitiveness and strong expressiveness. Therefore, it is of great theoretical and practical significance to study the cooperative game with interval payoffs.

Definition 2.3. Let H be the set of real numbers. For all a_L, a_R ∈ H and a_L ≤ a_R, $\tilde{a} = [a_{L}, a_{R}]$ is called an interval number. If a_L = a_R holds, then interval number $\tilde{a}$ is reduced to a real number a_L (or a_R).

An interval cooperative game on grand coalition N is a function $\tilde{v} : Δ (N) \to {IN}^{+}$ with $\tilde{v} (ø) = 0$ , IN⁺ is the set of all non-negative interval numbers. ∀S ⊆ N, the interval number $\tilde{v} (S) = [v_{L} (S), v_{R} (S)]$ represents the profit interval of coalition S. For simplicity, the interval cooperative game is denoted by $(N, \tilde{v})$ .

Definition 2.4. For all S, T ∈ Δ (N), S∩ T = ø, if $v_{L} (S \cup T) \geq v_{L} (S) + v_{L} (T),$ (2.4) $v_{R} (S \cup T) \geq v_{R} (S) + v_{R} (T),$ (2.5) then the interval cooperative game $(N, \tilde{v})$ is said to be super-additive.

Definition 2.5. Let $\hat{x} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})$ be a n-dimensional interval payoff vector, the component ${\tilde{x}}_{i} = [x_{Li}, x_{Ri}]$ is an interval number. If $x_{Li} \geq v_{L} ({i}),$ (2.6) $x_{Ri} \geq v_{R} ({i}),$ (2.7) $\sum_{i = 1}^{n} x_{Li} = v_{L} (N),$ (2.8) $\sum_{i = 1}^{n} x_{Ri} = v_{R} (N),$ (2.9) then interval payoff vector $\hat{x}$ is called an imputation of interval cooperative game $(N, \tilde{v})$ . If conditions (2.6) and (2.7) are not required, then $\hat{x}$ is called a pre-imputation of interval cooperative game $(N, \tilde{v})$ . Denoting $\tilde{X} (S) = [X_{L} (S), X_{R} (S)]$ , where X_L (S) = ∑_i∈Sx_Li and X_R (S) = ∑_i∈Sx_Ri. For the convenience of the following description, the set of all pre-imputations of $(N, \tilde{v})$ is denoted by $PI (N, \tilde{v})$ . The notation $I (N, \tilde{v})$ indicates the set of all imputations of interval cooperative game $(N, \tilde{v})$ .

3 Least square pre-nucleolus for interval cooperative games

It is well known that the least square pre-nucleolus is a solution concept based on the excess values of coalitions. Inspired by references [17, 25], in order to take the influence of coalition cardinality, blocking power and constructive power of coalitions into account, the definition of interval excess value of coalition is given as follows.

Definition 3.1. Let $(N, \tilde{v})$ be an interval cooperative game on grand coalition N. $\hat{x}$ is an pre-imputation of interval cooperative game $(N, \tilde{v})$ . ∀S ⊆ N, the left and right excess values of coalition S at pre-imputation $\hat{x}$ are defined as follows:

$\begin{matrix} η_{L} (S, \tilde{v}, \hat{x}) = \frac{1}{{| S |}^{θ}} (v_{L} (S) - X_{L} (S)) \\ - \frac{1}{{| N ∖ S |}^{θ}} (v_{L} (N ∖ S) - X_{L} (N ∖ S)) \end{matrix}$ (3.1)

$\begin{matrix} η_{R} (S, \tilde{v}, \hat{x}) = \frac{1}{{| S |}^{θ}} (v_{R} (S) - X_{R} (S)) \\ - \frac{1}{{| N ∖ S |}^{θ}} (v_{R} (N ∖ S) - X_{R} (N ∖ S)) \end{matrix}$ (3.2) where |S| denotes the cardinality of coalition S. The parameter θ can be determined according to the risk preference of players, θ ≥ 0. To maintain logical consistency, | ø |^-1 = 1 is defined in this study.

According to the structural characteristics of Equations (3.1) and (3.2), the left and right excess values are also called the anti-symmetric interval excess values. In order to minimize the change in the left and right excess values, the following two-objective programming model (M-1) can be constructed by extending the programming model in [25] to interval payoffs setting.

(M-1): $\begin{matrix} min (\sum_{S \subseteq N, S \neq ø} {(η_{L} (S, \tilde{v}, \hat{x}) - {\bar{η}}_{L} (S, \tilde{v}, \hat{x}))}^{2}, \\ \sum_{S \subseteq N, S \neq ø} {(η_{R} (S, \tilde{v}, \hat{x}) - {\bar{η}}_{R} (S, \tilde{v}, \hat{x}))}^{2}) \end{matrix}$ $s . t . {\begin{matrix} \sum_{i = 1}^{n} x_{Li} = v_{L} (N), \\ \sum_{i = 1}^{n} x_{Ri} = v_{R} (N) . \end{matrix}$ where ${\bar{η}}_{L} (S, \tilde{v}, \hat{x}) = \frac{1}{2^{n} - 1} \sum_{S \subseteq N, S \neq ø} η_{L} (S, \tilde{v}, \hat{x})$ (3.3) ${\bar{η}}_{R} (S, \tilde{v}, \hat{x}) = \frac{1}{2^{n} - 1} \sum_{S \subseteq N, S \neq ø} η_{R} (S, \tilde{v}, \hat{x}) .$ (3.4)

Obviously, ${\bar{η}}_{L} (S, \tilde{v}, \hat{x})$ and ${\bar{η}}_{R} (S, \tilde{v}, \hat{x})$ are the arithmetic average of left and right excess values of non-empty coalition S in Δ (N), respectively. By introducing the risk preference coefficient λ of players, the above two-objective programming model (M-1) can be transformed into the following single-objective programming model (M-2):

(M-2): $\begin{matrix} min ((1 - λ) \sum_{S \subseteq N, S \neq ø} {(η_{L} (S, \tilde{v}, \hat{x}) - {\bar{η}}_{L} (S, \tilde{v}, \hat{x}))}^{2} \\ + λ \sum_{S \subseteq N, S \neq ø} {(η_{R} (S, \tilde{v}, \hat{x}) - {\bar{η}}_{R} (S, \tilde{v}, \hat{x}))}^{2}) \end{matrix}$ $s . t . {\begin{matrix} \sum_{i = 1}^{n} x_{Li} = v_{L} (N), \\ \sum_{i = 1}^{n} x_{Ri} = v_{R} (N) . \end{matrix}$

The coefficient λ reflects players’ attitude to risk. For example, when the coefficients are λ = 0, λ = 1 and λ = 0.5, it corresponds to the risk preference types of players: risk aversion, risk pursuit and risk neutral.

Assuming that players have no additional preference for risk, in the following, λ = 0.5 is always selected. Since multiplication of objective function by non-zero constant does not affect the optimal solution of the model, the model (M-2) can be simplified to model (M-3) as follows:

(M-3): $\begin{matrix} min (\sum_{S \subseteq N, S \neq ø} {(η_{L} (S, \tilde{v}, \hat{x}) - {\bar{η}}_{L} (S, \tilde{v}, \hat{x}))}^{2} \\ + \sum_{S \subseteq N, S \neq ø} {(η_{R} (S, \tilde{v}, \hat{x}) - {\bar{η}}_{R} (S, \tilde{v}, \hat{x}))}^{2}) \end{matrix}$ $s . t . {\begin{matrix} \sum_{i = 1}^{n} x_{Li} = v_{L} (N), \\ \sum_{i = 1}^{n} x_{Ri} = v_{R} (N) . \end{matrix}$

Definition 3.2. Let $(N, \tilde{v})$ be an interval cooperative game on grand coalition N. The solutions of single-objective programming model (M-3) is the least square pre-nucleolus for interval cooperative game $(N, \tilde{v})$ .

Because ${\bar{η}}_{L} (S, \tilde{v}, \hat{x})$ and ${\bar{η}}_{R} (S, \tilde{v}, \hat{x})$ are the arithmetic average of 2ⁿ - 1 excess values, it greatly increases the computational complexity of the objective function in model (M-3). The following theorem shows how to reduce the computational complexity of the objective function in the model (M-3).

Theorem 3.1. The single-objective programming model (M-3) is equivalent to the model (M-4) as follows.

(M-4): $min (\sum_{S \subseteq N, S \neq ø} η_{L}^{2} (S, \tilde{v}, \hat{x}) + \sum_{S \subseteq N, S \neq ø} η_{R}^{2} (S, \tilde{v}, \hat{x}))$ $s . t . {\begin{matrix} \sum_{i = 1}^{n} x_{Li} = v_{L} (N), \\ \sum_{i = 1}^{n} x_{Ri} = v_{R} (N) . \end{matrix}$

Proof: Since N ∖ (N ∖ S) = S, from Equations (3.1) and (3.2), we have

$\begin{matrix} η_{L} (N ∖ S, \tilde{v}, \hat{x}) & = \frac{1}{{| N ∖ S |}^{θ}} (v_{L} (N ∖ S) - X_{L} (N ∖ S)) \\ - \frac{1}{{| S |}^{θ}} (v_{L} (S) - X_{L} (S)) \end{matrix}$ (3.5)

$\begin{matrix} η_{R} (N ∖ S, \tilde{v}, \hat{x}) & = \frac{1}{{| N ∖ S |}^{θ}} (v_{R} (N ∖ S) - X_{R} (N ∖ S)) \\ - \frac{1}{{| S |}^{θ}} (v_{R} (S) - X_{R} (S)) \end{matrix}$ (3.6)

For any coalition S ⊆ N, S≠ ø and payoff vector $\hat{x}$ , by Equations (3.1) and (3.5), we get $\begin{matrix} η_{L} (S, \tilde{v}, \hat{x}) + η_{L} (N ∖ S, \tilde{v}, \hat{x}) \\ = \frac{1}{{| S |}^{θ}} (v_{L} (S) - X_{L} (S)) \\ - \frac{1}{{| N ∖ S |}^{θ}} (v_{L} (N ∖ S) - X_{L} (N ∖ S)) \\ + \frac{1}{{| N ∖ S |}^{θ}} (v_{L} (N ∖ S) - X_{L} (N ∖ S)) \\ - \frac{1}{{| S |}^{θ}} (v_{L} (S) - X_{L} (S)) = 0 \end{matrix}$

In a similar way, by Equations (3.2) and (3.6), it follows that $\begin{matrix} η_{R} (S, \tilde{v}, \hat{x}) + η_{R} (N ∖ S, \tilde{v}, \hat{x}) \\ = \frac{1}{{| S |}^{θ}} (v_{R} (S) - X_{R} (S)) \\ - \frac{1}{{| N ∖ S |}^{θ}} (v_{R} (N ∖ S) - X_{R} (N ∖ S)) \\ + \frac{1}{{| N ∖ S |}^{θ}} (v_{R} (N ∖ S) - X_{R} (N ∖ S)) \\ - \frac{1}{{| S |}^{θ}} (v_{R} (S) - X_{R} (S)) = 0 \end{matrix}$ ∀S ⊆ N and S≠ ø, N ∖ S ⊂ N holds. By the relativity of sets and their complements, we have $\sum_{S \subseteq N, S \neq ø, N} η_{L} (S, \tilde{v}, \hat{x}) = 0$ and ∑_{S⊆N,S≠ø,N}η_R $(S, \tilde{v}, \hat{x}) = 0$ .

Considering v_L (N) = X_L (N) and v_R (N) = X_R (N), from Equations (3.1) and (3.2), we have $\begin{matrix} η_{L} (N, \tilde{v}, \hat{x}) = \frac{1}{{| N |}^{θ}} (v_{L} (N) - X_{L} (N)) \\ - \frac{1}{{| ø |}^{θ}} (v_{L} (ø) - X_{L} (ø)) = 0, \end{matrix}$ $\begin{matrix} η_{R} (N, \tilde{v}, \hat{x}) = \frac{1}{{| N |}^{θ}} (v_{R} (N) - X_{R} (N)) \\ - \frac{1}{{| ø |}^{θ}} (v_{R} (ø) - X_{R} (ø)) = 0 . \end{matrix}$

To sum up, we get ${\bar{η}}_{L} (S, \tilde{v}, \hat{x}) = \frac{1}{2^{n} - 1} \sum_{S \subseteq N, S \neq ø} η_{L} (S, \tilde{v}, \hat{x}), = 0$ ${\bar{η}}_{R} (S, \tilde{v}, \hat{x}) = \frac{1}{2^{n} - 1} \sum_{S \subseteq N, S \neq ø} η_{R} (S, \tilde{v}, \hat{x}) = 0 .$

Therefore, the model (M-3) is equivalent to the model (M-4). The Theorem 3.1 is proved. □

Since v_L (N) = X_L (S) + X_L (N ∖ S) and v_R (N) = X_R (S) + X_R (N ∖ S), by Equations (3.1) and (3.2) we have $\begin{matrix} η_{L} (S, \tilde{v}, \hat{x}) = \frac{1}{{| S |}^{θ}} (v_{L} (S) - X_{L} (S)) \\ - \frac{1}{{| N ∖ S |}^{θ}} (v_{L} (N ∖ S) - X_{L} (N ∖ S)) \end{matrix}$ $\begin{matrix} = \frac{1}{{| S |}^{θ}} (v_{L} (S) - X_{L} (S)) \\ - \frac{1}{{| N ∖ S |}^{θ}} (v_{L} (N ∖ S) - X_{L} (N ∖ S)) \end{matrix}$ $\begin{matrix} + \frac{1}{{| N ∖ S |}^{θ}} X_{L} (N) \\ - \frac{1}{{| N ∖ S |}^{θ}} (X_{L} (S) + X_{L} (N ∖ S)), \end{matrix}$ $\begin{matrix} η_{R} (S, \tilde{v}, \hat{x}) = \frac{1}{{| S |}^{θ}} (v_{R} (S) - X_{R} (S)) \\ - \frac{1}{{| N ∖ S |}^{θ}} (v_{R} (N ∖ S) - X_{R} (N ∖ S)) \end{matrix}$ $\begin{matrix} = \frac{1}{{| S |}^{θ}} (v_{R} (S) - X_{R} (S)) \\ - \frac{1}{{| N ∖ S |}^{θ}} (v_{R} (N ∖ S) - X_{R} (N ∖ S)) \end{matrix}$ $\begin{matrix} + \frac{1}{{| N ∖ S |}^{θ}} X_{R} (N) \\ - \frac{1}{{| N ∖ S |}^{θ}} (X_{R} (S) + X_{R} (N ∖ S)) . \end{matrix}$ which can be simplified as $η_{L} (S, \tilde{v}, \hat{x}) = \frac{1}{{| S |}^{θ}} v_{L} (S) + \frac{1}{{| N ∖ S |}^{θ}} (v_{L} (N) - v_{L} (N ∖ S))$ $- (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) X_{L} (S)$ (3.7) $η_{R} (S, \tilde{v}, \hat{x}) = \frac{1}{{| S |}^{θ}} v_{R} (S) + \frac{1}{{| N ∖ S |}^{θ}} (v_{R} (N) - v_{R} (N ∖ S))$ $- (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) X_{R} (S)$ (3.8)

Since ∑_j∈øx_Lj = ∑_j∈øx_Rj = 0, and $u_{L} (ø) = \frac{1}{{| ø |}^{θ}} v_{L} (ø) + \frac{1}{{| N |}^{θ}} (v_{L} (N) - v_{L} (N)) = 0,$ $u_{R} (ø) = \frac{1}{{| ø |}^{θ}} v_{R} (ø) + \frac{1}{{| N |}^{θ}} (v_{R} (N) - v_{R} (N)) = 0 .$

By Equations (3.7) and (3.8), the model (M-4) can be further written as follows:

(M-5): $\begin{matrix} min (\sum_{S \subseteq N} (u_{L} (S) - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N \ S |}^{θ}}) \sum_{j \in S} x_{Lj})^{2} \\ + \sum_{S \subseteq N} (u_{R} (S) - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N \ S |}^{θ}}) \sum_{j \in S} x_{Rj})^{2}) \end{matrix}$ $s . t . {\begin{matrix} \sum_{i = 1}^{n} x_{Li} = v_{L} (N), \\ \sum_{i = 1}^{n} x_{Ri} = v_{R} (N) . \end{matrix}$ where $u_{L} (S) = \frac{1}{{| S |}^{θ}} v_{L} (S) + \frac{1}{{| N ∖ S |}^{θ}} (v_{L} (N) - v_{L} (N ∖ S)),$ (3.9) $u_{R} (S) = \frac{1}{{| S |}^{θ}} v_{R} (S) + \frac{1}{{| N ∖ S |}^{θ}} (v_{R} (N) - v_{R} (N ∖ S)) .$ (3.10)

Theorem 3.2. For any interval cooperative game $(N, \tilde{v})$ , the pre-nucleolus of $(N, \tilde{v})$ is the unique solution of model (M-5), and can be expressed by an interval vector $\hat{x} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})$ . The component ${\tilde{x}}_{i} = [x_{Li}, x_{Ri}]$ of $\hat{x}$ are derived as $x_{Li} = \frac{1}{| N |} v_{L} (N) + \frac{\underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S) - \frac{1}{| N |} \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S)}{\sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}}, i \in N;$ (3.11) $x_{Ri} = \frac{1}{| N |} v_{R} (N) + \frac{\underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S) - \frac{1}{| N |} \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S)}{\sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}}, i \in N .$ (3.12) where | * | is the cardinality of the coalition *, and $C_{| N | - 2}^{| S | - 1}$ is a combinatorial number.

Proof. From the above deduction process and the equivalence between models (M-4) and (M-5), it is clear that the unique solution of model (M-5) is the pre-nucleolus defined in Definition 3.2. In the following, we will prove that the pre-nucleolus of $(N, \tilde{v})$ has unique expression, which has been shown in Equations (3.11) and (3.12). In order to obtain the conditional extremum of model (M-5), the Lagrangian function of single-objective programming model (M-5) is given as $\begin{matrix} L (\hat{x}, α, β) = \sum_{S \subseteq N} (u_{L} (S) - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) \sum_{j \in S} x_{Lj})^{2} \\ + \sum_{S \subseteq N} (u_{R} (S) - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) \sum_{j \in S} x_{Rj})^{2} \\ + α (\sum_{j \in N} x_{Lj} - v_{L} (N)) + β (\sum_{j \in N} x_{Rj} - v_{R} (N)) . \end{matrix}$

For any i ∈ N, the following partial derivatives of Lagrangian function L with respect to variables x_Li and x_Ri are obtained, respectively.

$\begin{matrix} \frac{\partial L}{\partial x_{Li}} = - 2 \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) (u_{L} (S) \\ - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) \sum_{j \in S} x_{Lj}) + α \end{matrix}$ (3.13)

$\begin{matrix} \frac{\partial L}{\partial x_{Ri}} = - 2 \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) (u_{R} (S) \\ - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) \sum_{j \in S} x_{Rj}) + β \end{matrix}$ (3.14) Setting $\frac{\partial L}{\partial x_{Li}} = 0$ and $\frac{\partial L}{\partial x_{Ri}} = 0$ , we have

$\begin{matrix} α = 2 \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) (u_{L} (S) \\ - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) \sum_{j \in S} x_{Lj}), \end{matrix}$ (3.15)

$\begin{matrix} β = 2 \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) (u_{R} (S) \\ - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) \sum_{j \in S} x_{Rj}) . \end{matrix}$ (3.16)

From Equations (3.15) and (3.16), we have $\begin{matrix} \frac{α}{2} = \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S) \\ - \underset{i \in S}{\sum_{S \subseteq N}} ((\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} \sum_{j \in S} x_{Lj}) \end{matrix}$ $\begin{matrix} = \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S) \\ - \sum_{| S | = 1}^{| N |} C_{| N | - 1}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} x_{Li} \end{matrix}$ $\begin{matrix} - \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} \sum_{j \in N ∖ {i}} x_{Lj}, \end{matrix}$ $\begin{matrix} \frac{β}{2} = \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S) \\ - \underset{i \in S}{\sum_{S \subseteq N}} ((\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} \sum_{j \in S} x_{Rj}) \end{matrix}$ $\begin{matrix} = \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S) \\ - \sum_{| S | = 1}^{| N |} C_{| N | - 1}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} x_{Ri} \end{matrix}$ $\begin{matrix} - \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} \sum_{j \in N ∖ {i}} x_{Rj} . \end{matrix}$

Considering of ∑_j∈N∖{i}x_Lj = v_L (N) - x_Li and ∑_j∈N∖{i}x_Rj = v_R (N) - x_Ri, we have $\begin{matrix} \frac{α}{2} = \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S) \\ - \sum_{| S | = 1}^{| N |} C_{| N | - 1}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} x_{Li} \end{matrix}$ $- \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} (v_{L} (N) - x_{Li})$ $\begin{matrix} = \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S) \\ - \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} v_{L} (N) \end{matrix}$

$\begin{matrix} - (\sum_{| S | = 1}^{| N |} C_{| N | - 1}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} \\ - \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}) x_{Li}, \end{matrix}$ (3.17) $\begin{matrix} \frac{β}{2} = \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S) \\ - \sum_{| S | = 1}^{| N |} C_{| N | - 1}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} x_{Ri} \end{matrix}$ $\begin{matrix} - \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} (v_{R} (N) - x_{Ri}) \end{matrix}$ $\begin{matrix} = \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S) \\ - \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} v_{R} (N) \end{matrix}$

The summation over all i ∈ N in Equations (3.17) and (3.18) are respectively calculated as follows: $\begin{matrix} \frac{1}{2} | N | α = \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S) \\ - | N | \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} v_{L} (N) \end{matrix}$

$- (\sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}) \sum_{i \in N} x_{Li},$ (3.19) $\begin{matrix} \frac{1}{2} | N | β = \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S) \\ - | N | \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} v_{R} (N) \end{matrix}$ $- (\sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}) \sum_{i \in N} x_{Ri} .$ (3.20)

Dividing the two sides of Equations (3.19) and (3.20) by ∥N∥ at the same time, we have $\begin{matrix} \frac{α}{2} = \frac{1}{| N |} \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S) \\ - \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} v_{L} (N) \end{matrix}$ $- \frac{v_{L} (N)}{| N |} \sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2},$ (3.21) $\begin{matrix} \frac{β}{2} = \frac{1}{| N |} \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S) \\ - \sum_{| S | = 2}^{| N |} C_{| N | - 2}^{| S | - 2} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2} v_{R} (N) \end{matrix}$ $- \frac{v_{R} (N)}{| N |} \sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2},$ (3.22)

Observing the difference between the Equations (3.17) and (3.21), and eliminating the parameter α accordingly: $x_{Li} \sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}$ $= \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S)$ $\begin{matrix} - \frac{1}{| N |} \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S) \\ + \frac{v_{L} (N)}{| N |} \sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}, \end{matrix}$ which can be equivalently expressed as $x_{Li} = \frac{1}{| N |} v_{L} (N) + \frac{\underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (S) - \frac{1}{| N |} \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{L} (\tilde{S})}{\sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}} .$

Similarly, the parameter β can be eliminated according to the difference between the Equations (3.18) and (3.22), it follows that $x_{Ri} \sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}$ $= \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S)$ $\begin{matrix} - \frac{1}{| N |} \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S) \\ + \frac{v_{R} (N)}{| N |} \sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}, \end{matrix}$ which can be equivalently expressed as $x_{Ri} = \frac{1}{| N |} v_{R} (N) + \frac{\underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (S) - \frac{1}{| N |} \sum_{i \in N} \underset{i \in S}{\sum_{S \subseteq N}} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) u_{R} (\tilde{S})}{\sum_{| S | = 1}^{| N | - 1} C_{| N | - 2}^{| S | - 1} (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}})^{2}} .$

This complete the proof of Theorem 3.2. □

4 Least square nucleolus for interval cooperative games with incomplete payoffs

Payoff of coalitions are partly unknown due to the coexistence of competition and cooperation among coalitions. Masuya [33] studied the cooperative games with unknown worth of some coalitions. The Shapley value and its axioms are discussed in detail.

Yu et al. [34] proposed the proportional Owen value for partially defined cooperative games with coalition structure. The proportional Owen value is derived by two steps. Salamanca [35] compared the value-like solution concepts in a partially defined NTU cooperative games. The difference between these solution concepts is analyzed by payoff strategic possibilities. The crisp cooperative game with incomplete payoff information is defined by Masuya and Inuiguchi [36] as follows:

Definition 4.1. Let (N, v) be a crisp cooperative game. Λ denoted the set of all crisp coalitions with known payoff information, which satisfies

ø, N∈ Λ ;

Λ ⊆ Δ (N) and Λ≠ Δ (N) ;

∀ i = 1, 2, ⋯ , n, {i} ∈ Ω .

then (N, Λ, v) is called an incomplete cooperative game.

Based on the super-additivity of cooperative games, Masuya and Inuiguchi [36] developed the following upper and lower cooperative game of (N, Λ, v). The unknown payoffs can be effective evaluated by upper and lower bounds.

Definition 4.2. Let (N, Λ, v) be an incomplete crisp cooperative game. ∀ R ∈ Δ (N), if $v^{-} (R) = \underset{j, k \in {1, 2, \dots, r}}{\underset{\cup_{i = 1}^{r} S_{i} \subseteq R, S_{j} \land S_{k} = ø}{max_{S_{i} \in Λ, i = 1, 2, \dots, r}}} \sum_{i = 1}^{r} v (S_{i})$ (4.1) $v^{+} (R) = min_{S \in Λ, S \supseteq R} (v (S) - v^{-} (S ∖ R))$ (4.2) then (N, v^-) and (N, v⁺) are respectively called the lower and upper cooperative games of incomplete crisp cooperative game (N, Λ, v).

In order to minimize the change in the excess values, by adding constraints to the model (M-5), the programming model (M-6) is constructed as follows:

(M-6): $\begin{matrix} min (\sum_{S \subseteq N} (u_{L} (S) - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) \sum_{j \in S} x_{Lj})^{2} \\ + \sum_{S \subseteq N} (u_{R} (S) - (\frac{1}{{| S |}^{θ}} + \frac{1}{{| N ∖ S |}^{θ}}) \sum_{j \in S} x_{Rj})^{2}) \end{matrix}$ $s . t . {\begin{matrix} \sum_{i = 1}^{n} x_{Li} = v_{L} (N), \\ \sum_{i = 1}^{n} x_{Ri} = v_{R} (N), \\ v_{L}^{-} (S) \leq v_{L} (S) \leq v_{L}^{+} (S), \forall S \in Δ (N) ∖ Λ, \\ v_{R}^{-} (S) \leq v_{R} (S) \leq v_{R}^{+} (S), \forall S \in Δ (N) ∖ Λ, \\ v_{L} (S_{1} \cup S_{2}) \geq v_{L} (S_{1}) + v_{L} (S_{2}), \\ v_{R} (S_{1} \cup S_{2}) \geq v_{R} (S_{1}) + v_{R} (S_{2}), \\ S_{1} \cap S_{2} = ø, S_{1}, S_{2}, S_{1} \cup S_{2} \in Δ (N), \\ | (Δ (N) ∖ Λ) \cap {S_{1}, S_{2}, S_{1} \cup S_{2}} | > 1 . \end{matrix}$ where $(N, v_{L}^{-})$ and $(N, v_{L}^{+})$ are the lower and upper cooperative games of incomplete crisp cooperative game (N, Λ, v_L), $(N, v_{R}^{-})$ and $(N, v_{R}^{+})$ are the lower and upper cooperative games of incomplete crisp cooperative game (N, Λ, v_R). The constraint | (Δ (N) ∖ Λ) ∩ {S₁, S₂, S₁ ∪ S₂} | > 1 ensures that there are at least two unknown payoffs in the super-additive constraints, which effectively avoids the recurrence of redundant constraints and greatly reduces the computational complexity.

By solving the model (M-6), the optimal values of x_L1, x_L2, ⋯ , x_Ln, x_R1, x_R2, ⋯ , x_Rn are derived. Accordingly, $\hat{x} = ([x_{L 1}, x_{R 1}]$ , [x_L2, x_R2] , ⋯ , [x_Ln,x_Rn]) is called the least square pre-nucleolus of incomplete interval cooperative game $\tilde{v}$ .

5 Application of least square pre-nucleolus in land pollution control

Due to the influence of industrial sewage discharge, pesticide abuse and heavy metal product, land pollution has become a major environmental problem which needs to be solved urgently. There are about 3 million potential pollution sites in the European Economic Area and the Western Balkans. More than 1300 polluted sites in the United States are listed on the Superfund National Priorities List. China classifies 19% of its agricultural land as contaminated land. Land pollution is becoming more and more serious in the world.

Suppose that there are five chemical enterprises in Fuzhou City, Fujian Province in China that discharge sewage and industrial waste improperly. It has led to serious pollution of the land around these factories. Land pollution control and remediation mainly follow the “polluter pays” principle, and punish the polluters who refuse to control. In order to improve the environmental ecology and food safety in the region, the local government demanded the five chemical enterprises concerned to pay the corresponding pollution control fees, and control the pollution of surrounding land jointly. As five chemical enterprises actively cooperate on land pollution control, the cost saving is estimated to be between 600,000 and 700,000 RMB. In order to take the fairness and efficiency of all parties involved in pollution control into account, the cost-saving reallocation scheme should consider the differences of pollution control capacity and efficiency among five chemical enterprises. For the sake of convenience, five chemical enterprises (P₁, P₂, ⋯ , P₅) are numbered as players 1–5 in turn. That is, we have the grand coalition N = {1, 2, ⋯ , 5}.

Case 1. The interval payoffs of all coalitions are known

Due to the complexity of co-operation in pollution control, cost-saving values cannot be accurately estimated, which commonly given in the form of interval numbers. Suppose that the payoffs of coalitions are obtained as follows: $\begin{matrix} \tilde{v} ({1}) = [8, 10], \tilde{v} ({2}) = [5, 7], \tilde{v} ({3}) = [9, 10], \\ \tilde{v} ({4}) = [6, 8], \tilde{v} ({5}) = [5, 6], \\ \tilde{v} ({1, 2}) = [15, 19], \tilde{v} ({1, 3}) = [21, 25], \\ \tilde{v} ({1, 4}) = [17, 20], \tilde{v} ({1, 5}) = [15, 18], \\ \tilde{v} ({2, 3}) = [18, 21], \tilde{v} ({2, 4}) = [14, 18], \\ \tilde{v} ({2, 5}) = [13, 16], \tilde{v} ({3, 4}) = [18, 20], \\ \tilde{v} ({3, 5}) = [17, 19], \tilde{v} ({4, 5}) = [14, 17], \\ \tilde{v} ({1, 2, 3}) = [30, 34], \tilde{v} ({1, 2, 4}) = [26, 30], \\ \tilde{v} ({1, 2, 5}) = [25, 28], \tilde{v} ({1, 3, 4}) = [33, 36], \\ \tilde{v} ({1, 3, 5}) = [31, 34], \end{matrix}$ $\begin{matrix} \tilde{v} ({1, 4, 5}) = [28, 32], \tilde{v} ({2, 3, 4}) = [29, 33], \\ \tilde{v} ({2, 3, 5}) = [28, 31], \\ \tilde{v} ({2, 4, 5}) = [25, 28], \tilde{v} ({3, 4, 5}) = [29, 33], \\ \tilde{v} ({1, 2, 3, 4}) = [45, 51], \tilde{v} ({1, 2, 3, 5}) = [48, 56], \\ \tilde{v} ({1, 2, 4, 5}) = [39, 45], \\ \tilde{v} ({1, 3, 4, 5}) = [47, 53], \tilde{v} ({2, 3, 4, 5}) = [42, 48], \\ \tilde{v} ({1, 2, 3, 4, 5}) = [60, 70] . \end{matrix}$

Suppose the parameter θ = 1 is selected according to the risk preference of players. By Equation (3.9), we have $\begin{matrix} u_{L} ({1}) = 12.5, u_{L} ({2}) = 8.25, u_{L} ({3}) = 14.25, \\ u_{L} ({4}) = 9, u_{L} ({5}) = 8.75, \\ u_{L} ({1, 2}) = 17.83, u_{L} ({1, 3}) = 22.17, \\ u_{L} ({1, 4}) = 19.17, u_{L} ({1, 5}) = 17.83, \\ u_{L} ({2, 3}) = 19.67, u_{L} ({2, 4}) = 16.67, \\ u_{L} ({2, 5}) = 15.5, u_{L} ({3, 4}) = 20.67, \\ u_{L} ({3, 5}) = 19.83, u_{L} ({4, 5}) = 17, \\ u_{L} ({1, 2, 3}) = 33, u_{L} ({1, 2, 4}) = 30.17, \\ u_{L} ({1, 2, 5}) = 29.33, u_{L} ({1, 3, 4}) = 34.5, \\ u_{L} ({1, 3, 5}) = 33.33, \\ u_{L} ({1, 4, 5}) = 30.33, u_{L} ({2, 3, 4}) = 32.17, \\ u_{L} ({2, 3, 5}) = 30.83, \\ u_{L} ({2, 4, 5}) = 27.83, u_{L} ({3, 4, 5}) = 32.17, \\ u_{L} ({1, 2, 3, 4}) = 66.25, \\ u_{L} ({1, 2, 3, 5}) = 66, u_{L} ({1, 2, 4, 5}) = 60.75, \\ u_{L} ({1, 3, 4, 5}) = 66.75, \\ u_{L} ({2, 3, 4, 5}) = 62.5, u_{L} ({1, 2, 3, 4, 5}) = 12 . \end{matrix}$

Similarly, from Equation (3.10), the following results can be derived: $\begin{matrix} u_{R} ({1}) = 15.5, u_{R} ({2}) = 11.25, u_{R} ({3}) = 16.25, \\ u_{R} ({4}) = 11.5, \\ u_{R} ({5}) = 10.75, u_{R} ({1, 2}) = 21.83, \\ u_{R} ({1, 3}) = 26.5, u_{R} ({1, 4}) = 23, \\ u_{R} ({1, 5}) = 21.33, u_{R} ({2, 3}) = 23.17, \\ u_{R} ({2, 4}) = 21, u_{R} ({2, 5}) = 19.33, \end{matrix}$ $\begin{matrix} u_{R} ({3, 4}) = 24, u_{R} ({3, 5}) = 22.83, \\ u_{R} ({4, 5}) = 20.5, u_{R} ({1, 2, 3}) = 37.83, \\ u_{R} ({1, 2, 4}) = 35.5, u_{R} ({1, 2, 5}) = 34.33, \\ u_{R} ({1, 3, 4}) = 39, u_{R} ({1, 3, 5}) = 37.33, \\ u_{R} ({1, 4, 5}) = 35.17, u_{R} ({2, 3, 4}) = 37, \\ u_{R} ({2, 3, 5}) = 35.33, \\ u_{R} ({2, 4, 5}) = 31.83, u_{R} ({3, 4, 5}) = 36.5, \\ u_{R} ({1, 2, 3, 4}) = 76.75, \\ u_{R} ({1, 2, 3, 5}) = 76, u_{R} ({1, 2, 4, 5}) = 71.25, \\ u_{R} ({1, 3, 4, 5}) = 76.25, \\ u_{R} ({2, 3, 4, 5}) = 72, u_{R} ({1, 2, 3, 4, 5}) = 14 . \end{matrix}$

Moreover, based on the Equations (3.11) and (3.12), we have $\begin{matrix} x_{L 1} = 121574, x_{L 2} = 117572, x_{L 3} = 123898, \\ x_{L 4} = 119021, x_{L 5} = 117935, \end{matrix}$ $\begin{matrix} x_{R 1} = 142027, x_{R 2} = 138024, x_{R 3} = 143474, \\ x_{R 4} = 139059, x_{R 5} = 137416 . \end{matrix}$

Therefore, the interval imputations of five chemical enterprises are obtained as $\begin{matrix} {\tilde{x}}_{1} = [121574, 142027], {\tilde{x}}_{2} = [117572, 138024], \\ {\tilde{x}}_{3} = [123898, 143474], {\tilde{x}}_{4} = [119021, 139059], \\ {\tilde{x}}_{5} = [117935, 137416] . \end{matrix}$

The interval imputations of five chemical enterprises can be illustrated in Fig. 1 as follows.

Fig. 1

The interval imputations under the condition of all payoffs are known.

Case 2. The interval payoffs of coalitions are partly known

Influenced by the complexity of estimating payoffs, internal competitiveness and the information asymmetry, some coalitions often have unknown payoffs. The second chemical enterprise (player 2) and the fourth chemical enterprise (player 4) have not cooperated before, which makes it difficult to estimate the cost savings (payoffs) of coalition {2, 4}. Assume that the payoffs of each coalition are just as the same as the case 1, except for coalition {2, 4}. Based on Equations (4.1) and (4.2), we have $\begin{matrix} v_{L}^{-} ({2, 4}) = 14, v_{L}^{+} ({2, 4}) = 17, v_{R}^{-} ({2, 4}) = 22, \\ v_{R}^{+} ({2, 4}) = 24 . \end{matrix}$

Denoting $\tilde{v} ({2, 4}) = [η_{L}, η_{R}]$ , the programming model (M-6) is constructed as follows: $\begin{matrix} min [450.63 x_{L 1}^{2} + 563.87 x_{L 2}^{2} + 519.58 x_{L 3}^{2} \\ + 542.45 x_{L 4}^{2} + 489.53 x_{L 5}^{2} + 23.71 η_{L}^{2} + 473.98 x_{R 1}^{2} \\ + 553.69 x_{R 2}^{2} + 538.74 x_{R 3}^{2} + 517.38 x_{R 4}^{2} \\ + 496.58 x_{R 5}^{2} + 22.28 η_{R}^{2} - 4579.38 x_{L 1} - 5127.38 x_{L 2} \\ - 4957.94 x_{L 3} - 5726.38 x_{L 4} - 4897.64 x_{L 5} + 42.37 η_{L} \\ - 5025.67 x_{R 1} - 5436.17 x_{R 2} - 5138.64 x_{R 3} \\ - 5746.82 x_{R 4} - 5027.62 x_{R 5} + 48.69 η_{R} + 713.27 x_{L 1} x_{L 2} \\ + 452.34 x_{L 1} x_{L 3} + 486.25 x_{L 1} x_{L 4} + 476.39 x_{L 1} x_{L 5} \\ + 501.27 x_{L 2} x_{L 3} + 479.28 x_{L 2} x_{L 4} + 463.24 x_{L 2} x_{L 5} \\ + 516.98 x_{L 3} x_{L 4} + 507.34 x_{L 3} x_{L 5} + 476.35 x_{L 4} x_{L 5} \\ + η_{L} (51.23 x_{L 1} - 64.39 x_{L 2} + 54.38 x_{L 3} - 63.59 x_{L 4} \\ + 56.31 x_{L 5}) + 702.86 x_{R 1} x_{R 2} + 467.81 x_{R 1} x_{R 3} \\ + 497.36 x_{R 1} x_{R 4} + 465.37 x_{R 1} x_{R 5} + 512.85 x_{R 2} x_{R 3} \\ + 498.34 x_{R 2} x_{R 4} + 476.25 x_{R 2} x_{R 5} + 503.68 x_{R 3} x_{R 4} \\ + 514.39 x_{R 3} x_{R 5} + 472.36 x_{R 4} x_{R 5} + η_{R} (53.21 x_{R 1} \\ - 61.38 x_{R 2} + 56.98 x_{R 3} - 62.54 x_{R 4} + 54.39 x_{R 5})] \end{matrix}$ $s . t . {\begin{matrix} 14 \leq η_{L} \leq 17, \\ x_{L 1} + x_{L 2} + x_{L 3} + x_{L 4} + x_{L 5} = 60, \\ 22 \leq η_{R} \leq 24, \\ x_{R 1} + x_{R 2} + x_{R 3} + x_{R 4} + x_{R 5} = 70 . \end{matrix}$

By solving the above programming model, the optimal solution of model (M-6) is derived as follows: $\begin{matrix} x_{L 1} = 121148, x_{L 2} = 117954, x_{L 3} = 124028, \\ x_{L 4} = 118723, \\ x_{L 5} = 118147, η_{L} = 15.63, x_{R 1} = 141837, \\ x_{R 2} = 138764, \\ x_{R 3} = 143860, x_{R 4} = 140269, x_{R 5} = 135270, \\ η_{R} = 22.39 . \end{matrix}$

Accordingly, the interval imputations of five chemical enterprises are obtained as $\begin{matrix} {\tilde{x}}_{1} = [121148, 141837], {\tilde{x}}_{2} = [117954, 138764], \\ {\tilde{x}}_{3} = [124028, 143860], {\tilde{x}}_{4} = [118723, 140269], \\ {\tilde{x}}_{5} = [118147, 135270] . \end{matrix}$

The interval imputations of five chemical enterprises are illustrated in Fig. 2 as follows.

Fig. 2

The interval imputations under the condition of payoffs are partly known.

The above application of least square nucleolus in two cases helps to explain and clarify the process of cooperation and final distribution of cost savings in land pollution control. It can provide necessary quantitative basis and data support for the income (cost savings) distribution of players (chemical enterprises).

6 Comparison and analysis

Shapley value is one of the widely used solution concepts in cooperative game. In order to characterize the interval uncertainty of the payoffs, Alparslan Gök et al. [37] extended the Shapley value to cooperative games with interval payoffs as follows:

$\begin{matrix} φ_{i} (\tilde{v}) = \sum_{S \overset{`}{I} N ∖ {i}} \frac{| S | (| N | - | S | - 1)!}{| N |!} \\ (\tilde{v} (S \cup {i}) -_{Alparslan} \tilde{v} (S)) . \end{matrix}$ (6.1) where “-_Alparslan” is the subtraction operation between interval numbers. Because there will be logical errors with subtraction “-_Alparslan” in some cases, Han [38] improved the Equation (6.1) by introducing a new subtraction operation “-_Han”. For example, $\forall \tilde{a} = [a_{L}, a_{R}]$ and $\tilde{b} = [b_{L}, b_{R}]$ , we have $\tilde{a} -_{Alparslan} \tilde{b} = [a_{L} - b_{L}, a_{R} - b_{R}]$ , $\tilde{a} -_{Han} \tilde{b} = [a_{L} - b_{R}, a_{R} - b_{L}]$ .

6.1 The interval imputation derived from interval Shapley value

Let N = {1, 2, 3} be the player set. Suppose that the payoffs of coalitions are given as follows: $\begin{matrix} \tilde{v} ({1}) = [1, 5], \tilde{v} ({2}) = [1, 5], \tilde{v} ({3}) = [1, 5], \\ \tilde{v} ({1, 2}) = [3, 8], \end{matrix}$ $\begin{matrix} \tilde{v} ({1, 3}) = [3, 8], \tilde{v} ({2, 3}) = [3, 8], \\ \tilde{v} ({1, 2, 3}) = [7, 11] . \end{matrix}$

The marginal contributions of players are calculated as: $\begin{matrix} \tilde{v} ({1, 2}) -_{Han} \tilde{v} ({2}) = [- 2, 7], \\ \tilde{v} ({1, 3}) -_{Han} \tilde{v} ({3}) = [- 2, 7], \\ \tilde{v} ({1}) -_{Han} \tilde{v} (ø) = [1, 5], \\ \tilde{v} ({1, 2, 3}) -_{Han} \tilde{v} ({2, 3}) = [- 1, 8], \\ \tilde{v} ({1, 2}) -_{Han} \tilde{v} ({1}) = [- 2, 7], \\ \tilde{v} ({2, 3}) -_{Han} \tilde{v} ({3}) = [- 2, 7], \\ \tilde{v} ({2}) -_{Han} \tilde{v} (ø) = [1, 5], \\ \tilde{v} ({1, 2, 3}) -_{Han} \tilde{v} ({1, 3}) = [- 1, 8], \\ \tilde{v} ({1, 3}) -_{Han} \tilde{v} ({1}) = [- 2, 7], \\ \tilde{v} ({2, 3}) -_{Han} \tilde{v} ({2}) = [- 2, 7], \\ \tilde{v} ({3}) -_{Han} \tilde{v} (ø) = [1, 5], \\ \tilde{v} ({1, 2, 3}) -_{Han} \tilde{v} ({1, 2}) = [- 1, 8] . \end{matrix}$

Using the Shapley value proposed by Han [33], from Equation (6.1), we have $φ_{1} (\tilde{v}) = [- \frac{2}{3}, \frac{20}{3}]$ , $φ_{2} (\tilde{v}) = [- \frac{2}{3}, \frac{20}{3}]$ and $φ_{3} (\tilde{v}) = [- \frac{2}{3}, \frac{20}{3}]$ .

From the above calculation results, it can be seen that there are negative numbers in the Shapley values of the players, which are not consistent with the reality. Moreover, we have $\sum_{i = 1}^{3} φ_{i} (\tilde{v}) = [- 2, 20] \neq \tilde{v} ({1, 2, 3})$ . It means that the calculated result of interval Shapley value does not satisfy the efficiency.

If subtraction operation “-_Alparslan” is adopted to calculate the marginal contributions of players, then we have $\begin{matrix} \tilde{v} ({1, 2, 3}) -_{Alparslan} \tilde{v} ({2, 3}) = [4, 3], \\ \tilde{v} ({1, 2, 3}) -_{Alparslan} \tilde{v} ({1, 3}) = [4, 3], \\ \tilde{v} ({1, 2, 3}) -_{Alparslan} \tilde{v} ({1, 2}) = [4, 3] . \end{matrix}$

It is clear that the left endpoint of the above marginal contribution values is greater than its right endpoint. Thus, these marginal contribution values are no longer interval numbers. In this case, the Shapley value calculated by Alparslan Gök’s formula is illogical and unreasonable.

6.2 The interval imputation derived from least square pre-nucleolus

For comparing the least square pre-nucleolus with Shapley value, parameter θ = 0 is selected. By Equation (3.9), we have $\begin{matrix} u_{L} ({1}) = 5, u_{L} ({2}) = 5, u_{L} ({3}) = 5, \\ u_{L} ({1, 2}) = 9, \end{matrix}$ $\begin{matrix} u_{L} ({1, 3}) = 9, u_{L} ({2, 3}) = 9, u_{L} ({1, 2, 3}) = 14 . \end{matrix}$

Moreover, from Equation (3.10), the following results are calculated: $\begin{matrix} u_{R} ({1}) = 8, u_{R} ({2}) = 8, u_{R} ({3}) = 8, \\ u_{R} ({1, 2}) = 14, \\ u_{R} ({1, 3}) = 14, u_{R} ({2, 3}) = 14, \\ u_{R} ({1, 2, 3}) = 22 . \end{matrix}$

By using Equations (3.11) and (3.12), we have $\begin{matrix} x_{L 1} = \frac{7}{3}, x_{L 2} = \frac{7}{3}, x_{L 3} = \frac{7}{3}, x_{R 1} = \frac{11}{3}, \\ x_{R 2} = \frac{11}{3}, x_{R 3} = \frac{11}{3} . \end{matrix}$

Therefore, the interval imputations of three players can be expressed as ${\tilde{x}}_{1} = [\frac{7}{3}, \frac{11}{3}], {\tilde{x}}_{2} = [\frac{7}{3}, \frac{11}{3}], {\tilde{x}}_{3} = [\frac{7}{3}, \frac{11}{3}] .$

It follows that $\sum_{i = 1}^{3} {\tilde{x}}_{i} = [7, 11] = \tilde{v} ({1, 2, 3})$ . Thus, the least square pre-nucleolus of interval cooperative games satisfies the efficiency.

Alparslan Gök’s formula [37] only focused on keeping the reversibility of subtraction operation, and the order relation of the interval endpoints was ignored. Thus, it leads to the case that the interval marginal contribution values are no longer interval numbers. The reason why the interval Shapley value proposed in [38] is illogical is that the subtraction operation between interval numbers is irreversible. The value range of Shapley value is artificially expanded, which causes the potential for negative imputations. As a result, the interval Shapley value loses its efficiency.

7 Conclusions

Cooperative game plays an important role in dealing with income distribution and cost allocation. This paper has proposed a novel solution concept which is called the least square pre-nucleolus for cooperative games with interval payoffs. Aiming at two cases of payoffs information, the effective methods for solving least square nucleolus have been given respectively. The main advantages of this paper are shown as follows:

The least square pre-nucleolus of interval cooperative games considers both the cardinal characteristics and double powers of coalitions, and helps us to overcome some drawbacks of the existing solution concepts.

This study takes two cases of payoff information (complete known and partly known) into account, which makes the least square pre-nucleolus more flexible than other solution concepts.

The analytic expression of least square pre-nucleolus is obtained by using Lagrange multiplier method. The analytic expression is helpful to the application and popularization of least square pre-nucleolus.

Moreover, the above solution concepts of interval cooperative games have been applied to land pollution control, which reflect the effectiveness and practicability of the least square nucleolus. In the following research, we intend to make further extensions of the least square nucleolus for cooperative games with fuzzy payoffs. We will consider several applications of the least square nucleolus in actual fields such as energy management, disaster control and medical service.

8 Conflicts of Interest

The authors declare that they have no conflicts of interest.

9 Data availability statement

The data used to support the findings of this study are included within the article.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 71601049), the Humanities and Social Sciences Fund of the Ministry of Education (No. 16YJC630064), the Foundation of Beijing Intelligent Logistics System Collaborative Innovation Center (No. BILSCIC-2019KF-16) and the University Training Program in Scientific Research for Outstanding Young Talents of Fujian Province: cost sharing strategy of incomplete cooperative game and its application in water pollution control.

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