1 Introduction

The Shapley value, regarded as the first axiomatization application to cooperative transferable utility (TU) games, is a well-known solution for cooperative TU games. As is known to all, Shapley (1953) characterized a unique expression, which verified the following four axioms: symmetry, efficiency, additivity, and anonymity. The Shapley value is usually used to address the problems about the profit sharing of cooperative coalitions (Liu et al. 2019; Sharma and Abhyankar 2017; Singh et al. 2012). What is more, some researchers study other applications of the Shapley value, such as the formation and stability of coalitions (Magaña and Carreras 2018) and the project scheduling (Arık et al. 2019). As a classical point-valued solution, the Shapley value, unquestionably, has played a huge role in the development history of cooperative TU games. Many researchers have made some improvements on the Shapley value (Abouou et al. 2019; Liu et al. 2018) under various situations or studied the relationship between it and other eminent solutions for cooperative TU games.

In this paper, first of all, motivated by the player’s productivity-based excess (called the player’s excess for short from now on) proposed by Sakawa and Nishizaki (1994), and the player's common excess and contribution excess proposed by Liu et al. (2020, 2021), we introduce player’s contribution-based excess, which will be an extremely important basis of the Shapley-like value and the efficient weighted Shapley-like value proposed in this paper. As we all know that, the notions of the excesses for cooperative TU games could be roughly divided into two categories: the excess of the player and that of the coalition. The excess of the player proposed by Sakawa and Nishizaki (1994) is a case in point for the former, while the excess of the coalition on the consequent payoff vector introduced by Ruiz et al. (1996) is a very good example for the latter. The existing researches show that several distinguished solutions for cooperative TU games are based on the excess, to be exact, the excess of the coalition, such as the core introduced by Gillies (1953), the (pre)nucleolus proposed by Schmeidler (1969), and the least square (pre)nucleolus given by Ruiz et al. (1996). Enlightened simultaneously by the excess of the player (Sakawa et al. 1994) and players’ marginal contributions to all sub-coalitions they belong to (Shapley 1953; Banzhaf 1965), we construct two quadratic programming models for solving players’ (weighted) least square contributions, which are used to substitute for players’ marginal contributions and seem to be the prerequisite of the improved Shapley values proposed in this paper.

Furthermore, the most basis assumption underlying many existing solutions for cooperative TU games is that all sub-coalitions in the grand coalition appear of equal importance, such as the Shapley value (Shapley 1953), and the Banzhaf value (Banzhaf 1965; Owen 1975). As far as cooperative TU games are concerned, in some cases, different coalitions should have different weights. Referring to Dubey et al. (1981) and Ruiz et al. (1998a, b), depending on the context, the weights of coalitions may have at least four interpretations as follows: the stability degrees of coalitions, the complexity of forming different coalitions, the importance degrees of sub-coalitions contained in the grand coalition, and the ability of coalitions in the procedure of bargaining. Accordingly, the weights of coalitions should not be ignored under particular circumstances. In other words, it is well worth exploring the efficient weighted Shapley-like value and proving its nice properties. In order to make the proposed efficient weighted Shapley-like value satisfy the symmetry, it is necessary to determine that coalitions with the same size will be assigned the same weight. The final results show that the efficient weighted Shapley-like value proposed in this paper has strong practicability and satisfies the four axioms just like the Shapley value.

The remainder of this paper is arranged as follows. In Sect. 2, we introduce some important and indispensable preliminaries related to the topic of this paper, such as the contribution-based excess and the player’s least square contribution depending on it. In Sect. 3, the efficient and efficient weighted Shapley-like values based on the player’s least square contribution are proposed, their nice properties are proved and the axioms which uniquely determine the efficient weighted Shapley-like value are considered. Section 4 demonstrates the superiority of the improved Shapley values herein through a real case about the collaborative profit sharing of the rural e-commerce. Section 5 gives some conclusions and discusses the possible further research.

2 Preliminaries

In this section, we introduce some basis notions related to the main concerns and contributions of this work.

2.1 Cooperative TU Games

A cooperative TU game, i.e., an n-person game shown in the form of characteristic function, is usually denoted by an ordered pair (\(N,\upsilon\)), where \(N = \{ 1,2, \ldots ,n\}\) is a finite set, consisting of all players joining the grand coalition, and \(\upsilon\) is a characteristic function defined on the reals from the power set of the finite set \(N\). To be clear, \(\upsilon :2^{N} \to {\text{R}}\) with \(\upsilon (\emptyset ) = 0\). For every coalition \(S\) (\(S \subseteq N\)), the real number \(\upsilon (S)\) denotes the profit that coalition \(S\) can win by working together. Generally, the number of players taking part in an arbitrary coalition S and the set of all cooperative TU games are denoted by \(s\) and \(G^{n}\), respectively.

A payoff vector \({\varvec{x}} = (x_{1} ,x_{2} , \ldots ,x_{n} ) \in {\text{R}}^{n}\), i.e., one kind of solution is a very important concept for a cooperative TU game. Let \(x(S) = \sum\nolimits_{i \in S} {x_{i} }\) denote the sum of the payoffs of players in \(S\). Generally speaking, as long as \(x(N) = \upsilon (N)\), the corresponding payoff vector \({\varvec{x}}\) is called as an efficient or a preimputation. What is more, it is said to be an imputation if, besides the aforementioned condition, \(x_{i} \ge \upsilon (i)\) (\(i \in N\)) is true all the time. For any cooperative TU game, we always devote ourselves to obtaining the imputation, which appears more satisfying for players and with more strong practicability.

2.2 Player’s Contribution-Based Excess

Inspired by the player’s excess, to be exact, the player’s productivity-based excess introduced by Sakawa et al. (1994) and the least square prenucleolus presented by Ruiz et al. (1996), we propose player’s contribution-based excess as follows:

$$ e^{C} (i,{\varvec{x}}) = \sum\limits_{S \subset N:i \in S} {e^{C} (S,{\varvec{x}})} = \sum\limits_{S \subset N:i \in S} {[(\upsilon (N) - \upsilon (N\backslash S)) - } \sum\limits_{j \in S} {x_{j}^{{}} } ] = \sum\limits_{S \subset N:i \in S} {\upsilon^{C} (S) - x(S) \, (i \in N)} $$
(1)

where \(e^{C} (S,{\varvec{x}})\) is called the contribution-based excess of coalition \(S\) (\(S \subseteq N\)) on the consequent payoff vector \({\varvec{x}}\). \(x_{j}^{{}}\) (\(j \in S\)) denotes the contribution-based payoff of player \(j\) contained in coalition \(S\) (\(S \subseteq N\)), and \(x(S)\) represents the sum of the contribution-based payoffs of all players in \(S\).

It is understandable that, \(e^{C} (i,{\varvec{x}})\) can be actually regarded as a kind of measure for the dissatisfaction of player \(i\) (\(i \in N\)) when \({\varvec{x}}\) is about to become the resulting payoff vector. What is more, on the basis of both the individual rationality principle and the group rationality principle for cooperative TU games, the following conclusion can be directly drawn: the smaller \(e^{C} (i,{\varvec{x}})\) is, the more satisfactory player \(i\) (\(i \in N\)) would feel. On the contrary, the greater \(e^{C} (i,{\varvec{x}})\) is, the more querulous player \(i\) would feel.

2.3 The Weighted Least Square Contribution Based on the Contribution-Based Excess

The least square contribution of player based on its contribution-based excess can be proved to be a kind of least square value and belongs to the family of the least square values proposed by Ruiz et al. (1998a, b). Therefore, the least square contribution of player, which is the basis of the improved Shapley values discussed in this paper, satisfies some nice properties and has a great reality-oriented meaning.

What follows is the quadratic programming model for solving the least square contribution of player \(i\) (\(i \in N\)), which selects the optimal payoff vector by minimizing the variance of the consequent contribution-based excesses of all \(n\) players in the grand coalition \(N\). In this sense, the key point of obtaining the least square contribution of player \(i\) (\(i \in N\)) is to make the maximal complaint minimized when facing \({\varvec{x}}\) as a possible resulting payoff vector.

Problem 1

Minimize \(\sum\limits_{i \in N} {(e^{C} (i,{\varvec{x}}) - \overline{e}^{C} (i,{\varvec{x}}))^{2} }\).

$${\text{s}}{\text{.t}}{. }\sum\limits_{i \in N} {x_{i}^{{}} } = \upsilon (N)$$

where \(\overline{e}^{C} (i,{\varvec{x}})\) represents the mean contribution-based excess of all \(n\) players at the final payoff vector \({\varvec{x}}\), given by

$$\overline{e}^{C} (i,{\varvec{x}}) = \frac{1}{n}\sum\limits_{i \in N} {e^{C} (i,{\varvec{x}})}$$

Lemma 2

For an arbitrary cooperative TU game, the sum of the contribution-based excesses of all \(n\) players is identical for any payoff vector satisfying the efficiency.

Proof

Suppose that \({\varvec{x}}\) is any efficient payoff vector, and then

$$ \begin{gathered} \sum\limits_{i \in N} {e^{C} (i,{\varvec{x}})} = \sum\limits_{i \in N} {\sum\limits_{S \subseteq N,i \in S} {\upsilon^{C} (S) - x(S)} } = \sum\limits_{S \subseteq N,S \ne \emptyset } {s(\upsilon^{C} (S) - x(S)} ) \, = \hfill \\ \sum\limits_{S \subseteq N,S \ne \emptyset } {s\upsilon^{C} (S)} - \sum\limits_{S \subseteq N,S \ne \emptyset } {sx(S)} = \sum\limits_{S \subseteq N,S \ne \emptyset } {s\upsilon^{C} (S)} - \sum\limits_{s = 1}^{n} {s\left( \begin{gathered} n - 1 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} \upsilon (N), \hfill \\ \end{gathered} $$

owing to the efficiency.

In other words, for a specific cooperative TU game, \(\overline{e}^{C} (i,{\varvec{x}})\) is a constant as long as the resulting payoff vector \({\varvec{x}}\) is efficient. In this sense, \(\overline{e}^{C} (i,{\varvec{x}})\) can be rewritten as \(\overline{e}^{C} (i)\) for short. Formally, we consider the following quadratic programming model.

Problem 3

Minimize \(\sum\limits_{i \in N} {e^{C} (i,{\varvec{x}})^{2} }\)

$${\text{s}}{\text{.t}}{. }\sum\limits_{i \in N} {x_{i}^{{}} } = \upsilon (N)$$

Clearly, the optimal solution of Problem 3 is equal to that of Problem 1 depending on Remark 4.

Remark 4

The optimal solution of Problem 1 always is identical when \(\overline{e}^{C} (i,{\varvec{x}})\) is substituted by any other constants including the value 0.

Proof

Let \(k\) (\(k \in {\text{R}}\)) be an arbitrary constant. Then, we have

$$ \sum\limits_{i \in N} {(e^{C} (i,{\varvec{x}}) - k)^{2} } = \sum\limits_{i \in N} {e^{C} (i,{\varvec{x}})^{2} } - 2k\sum\limits_{i \in N} {e^{C} (i,{\varvec{x}}) + nk^{2} .} $$

According to Lemma 2, \(\sum\limits_{i \in N} {e^{C} (i,{\varvec{x}})}\) will keep identical with each other if all considered payoff vectors are efficient. Hence, the optimal solution of Problem 1 always remains unchanged no matter \(\overline{e}^{C} (i)\) will be substituted by which constant.

What follows is the solving process of Problem 3. It is obvious that the Lagrangian function of Problem 3 can be shown as

$$ L(x,\lambda ) = \sum\limits_{{i \in N}} {\sum\limits_{{S \subset N:i \in S}} {(\upsilon ^{C} (S) - x(S)} )^{2} } + \lambda ({\mkern 1mu} \sum\limits_{{i \in N}} {x_{i}^{{}} } - \upsilon (N). $$

Let the partial derivatives of \(L(x,\lambda )\) with regard to the variables \(x_{i}\) (\(i \in S \subset N\)) be equal to 0, i.e.,

$$ \frac{dL(x,\lambda )}{{dx_{i} }} = 2\sum\limits_{s = 1}^{n - 1} {s\left( \begin{gathered} n - 2 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} x_{i}^{ * } + 2\sum\limits_{s = 2}^{n} {s\left( \begin{gathered} n - 2 \hfill \\ s - 2 \hfill \\ \end{gathered} \right)} \upsilon (N) - 2\sum\limits_{S \subset N:i \in S} {s\upsilon^{C} (S)} + \lambda^{ * } = 0 $$

and then, we have

$$x_{i}^{ * } = \frac{{2\sum\limits_{S \subset N:i \in S} {s\upsilon^{C} (S)} - 2\sum\limits_{s = 2}^{n} {s\left( \begin{gathered} n - 2 \hfill \\ s - 2 \hfill \\ \end{gathered} \right)} \upsilon (N) - \lambda^{ * } }}{{2\sum\limits_{s = 1}^{n - 1} {s\left( \begin{gathered} n - 2 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} }}.$$

Obviously, the key of solving Problem 3 turns to obtain the expression of \(\lambda^{ * }\). According to the constraint equation, i.e., \(\sum\limits_{i \in N} {x_{i}^{{}} } = \upsilon (N)\), it is clear that

$$\sum\limits_{i \in N}^{{}} {\frac{{2\sum\limits_{S \subset N:i \in S} {s\upsilon^{C} (S)} - 2\sum\limits_{s = 2}^{n} {s\left( \begin{gathered} n - 2 \hfill \\ s - 2 \hfill \\ \end{gathered} \right)} \upsilon (N) - \lambda^{ * } }}{{2\sum\limits_{s = 1}^{n - 1} {s\left( \begin{gathered} n - 2 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} }}} = \upsilon (N), $$

which is equivalent to

$$2\sum\limits_{j \in N} {\sum\limits_{S \subset N:j \in S} {s\upsilon^{C} (S)} } - 2n\sum\limits_{s = 2}^{n} {s\left( \begin{gathered} n - 2 \hfill \\ s - 2 \hfill \\ \end{gathered} \right)} \upsilon (N) - n\lambda^{ * } = 2\sum\limits_{s = 1}^{n - 1} {s\left( \begin{gathered} n - 2 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} \upsilon (N)$$

Hence,

$$\lambda^{ * } = \frac{2}{n}\sum\limits_{j \in N} {\sum\limits_{S \subset N:j \in S} {s\upsilon^{C} (S)} } - 2\sum\limits_{s = 2}^{n} {s\left( \begin{gathered} n - 2 \hfill \\ s - 2 \hfill \\ \end{gathered} \right)} \upsilon (N) - \frac{2}{n}\sum\limits_{s = 1}^{n - 1} {s\left( \begin{gathered} n - 2 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} \upsilon (N)$$

Ultimately, the unique optimal solution of Problem 3 can be given by

$$ \begin{aligned} x_{i}^{ * } = \frac{{2\sum\limits_{S \subset N:i \in S} {s\upsilon^{C} (S)} - 2\sum\limits_{s = 2}^{n} {s\left( \begin{aligned} n - 2 \hfill \\ s - 2 \hfill \\ \end{aligned} \right)} \upsilon (N) - \lambda^{ * } }}{{2\sum\limits_{s = 1}^{n - 1} {s\left( \begin{aligned} n - 2 \hfill \\ s - 1 \hfill \\ \end{aligned} \right)} }} = \frac{\upsilon (N)}{n} + \frac{{\sum\limits_{S \subset N:i \in S} {s\upsilon^{C} (S)} - \frac{1}{n}\sum\limits_{j \in N} {\sum\limits_{S \subset N:j \in S} {s\upsilon^{C} (S)} } }}{{\sum\limits_{s = 1}^{n - 1} {s\left( \begin{aligned} n - 2 \hfill \\ s - 1 \hfill \\ \end{aligned} \right)} }} \hfill \\ \, = \frac{1}{n}\upsilon (N) + k^{ - 1} [a_{i} (\upsilon ) - \frac{1}{n}\sum\limits_{j \in N} {a_{j} (\upsilon )} ], \hfill \\ \end{aligned} $$
(2)

where \(k = \sum\limits_{s = 1}^{n - 1} {s\left( \begin{gathered} n - 2 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} ,\) and \(a_{i} (\upsilon ) = \sum\limits_{S \subset N:i \in S} {s\upsilon^{C} (S)}\).

Therefore, \({\varvec{x}}^{ * } = (x_{1}^{ * } ,x_{2}^{ * } , \ldots ,x_{n}^{ * } )\) with its component \(x_{i}^{ * }\) (\(i \in N\)) shown by Eq. (2) is the optimal payoff vector (unique optimal solution) of Problem 3. The least square contribution of player \(i\) (\(i \in N\)) is very important to the improved Shapley values proposed in this paper.

However, in most cases, different coalitions should be assigned different weights, because of the various specific meanings of coalitional weight. By referring to (Ruiz et al. 1998a, b) and (Grotte 1976), the coalitional weight has at least three interpretations as follows: a measure with respect to the capacity of the coalition during the procedure of bargaining, the possibility of the members in coalition \(S\) to form a coalition and work together, or the stability of coalition \(S\) (i.e., the overall satisfaction of the members in\(S\)). In fact, some existing well-known solutions for cooperative TU games have actually taken into account the impact of the coalitional weight on the optimal distribution strategy. A case in point is the family of the semivalues introduced by Dubey et al. (1981). In order to embody the influence of the coalitional weight on the distribution outcome, for any cooperative TU game\(\upsilon \in G^{n}\), we formally consider the following quadratic programming model. Its unique optimal solution can directly reflect the effect of coalitional weight on the optimal distribution strategy.

Problem 5

Minimize \(\sum\limits_{i \in N} {(e^{C\omega } (i,{\varvec{x}}) - \overline{e}^{C\omega } (i,{\varvec{x}}))^{2} }\)

$${\text{s}}{\text{.t}}{. }\sum\limits_{i \in N} {x_{i}^{\omega } } = \upsilon (N), $$

where \(\overline{e}^{C\omega } (i,{\varvec{x}}) = \frac{1}{n}\sum\limits_{i \in N} {e^{C\omega } (i,{\varvec{x}})}\) and \(e^{C\omega } (i,{\varvec{x}}) = \sum\limits_{S \subseteq N,i \in S} {\omega (s)[\upsilon^{C} (S) - x^{\omega } (S)]}\).

Therein, \(s\) represents the size of coalition \(S\) and \(\omega (s)\) is called a class of weight function with \(\omega (s) > 0\) for all \(S \subseteq N\) (\(S \ne \emptyset\)). In this sense, the coalitions with the same size should be assigned the same weight. To be exact, unless otherwise stated, the weight function \(\omega (s)\) is assumed to be symmetric and positive in this paper.

According to the context, the weight function \(\omega (s)\) can be regarded as the chance of forming a coalition\(S\), the stability degree, the overall satisfaction or the bargaining ability of a coalition\(S\). Just like \(\overline{e}^{C} (i,{\varvec{x}})\) mentioned in Problem 1, \(\overline{e}^{C\omega } (i,{\varvec{x}})\) remains the same for all efficient payoff vectors. That is to say, \(\overline{e}^{C\omega } (i,{\varvec{x}})\) is a constant when the considered payoff vector \({\varvec{x}}\) is efficient.

Remark 6

The unique optimal solution of Problem 5 will remain unchanged when \(\overline{e}^{C\omega } (i,{\varvec{x}})\) is replaced by any constant even 0.

Proof

For any constant \(k\) (\(k \in {\text{R}}\)), it is always true that

\(\sum\limits_{i \in N} {(e^{C\omega } (i,{\varvec{x}}) - k)^{2} } = \sum\limits_{i \in N} {e^{C\omega } (i,{\varvec{x}})^{2} } - 2k\sum\limits_{i \in N} {e^{C\omega } (i,{\varvec{x}}) + nk^{2} }\).

It follows that, if the consequent payoff vector \({\varvec{x}}\) is efficient,

$$ \begin{aligned} \sum\limits_{i \in N} {e^{C\omega } (i,{\varvec{x}})} = \sum\limits_{i \in N} {\sum\limits_{S \subseteq N,i \in S} {\omega (s)[\upsilon^{C} (S) - x^{\omega } (S)]} } = \sum\limits_{S \subseteq N,S \ne \emptyset } {s\omega (s)(\upsilon^{C} (S) - x^{\omega } (S)} ) \hfill \\ \, = \sum\limits_{S \subseteq N,S \ne \emptyset } {s\omega (s)\upsilon^{C} (S)} - \sum\limits_{S \subseteq N,S \ne \emptyset } {s\omega (s)x^{\omega } (S)} = \sum\limits_{S \subseteq N,S \ne \emptyset } {s\omega (s)\upsilon^{C} (S)} - \sum\limits_{s = 1}^{n} {s\omega (s)\left( \begin{aligned} n - 1 \hfill \\ s - 1 \hfill \\ \end{aligned} \right)} \upsilon (N). \hfill \\ \end{aligned} $$

Accordingly, if we replace \(\overline{e}^{C\omega } (i,{\varvec{x}})\) with any constant \(k\) (\(k \in {\text{R}}\)) in Problem 5, the consequent target function will be different on a constant on the feasible set.

Particularly, considering the situation where\(k = 0\), Problem 5 can be greatly simplified to Problem 7. Clearly, the optimal solution of Problem 5 is the same as that of Problem 7.

Problem 7

Minimize \(\sum\limits_{i \in N} {e^{C\omega } (i,{\varvec{x}})^{2} }\)

$${\text{s}}{\text{.t}}{. }\sum\limits_{i \in N} {x_{i}^{\omega } } = \upsilon (N).$$

Obviously, the Lagrangian function of Problem 7 is given by

$$ L(x^{\omega } ,\lambda ^{\omega } ) = \sum\limits_{{i \in N}} {\left[ {\sum\limits_{{S \subset N:i \in S}} {\omega (s)} \left( {\upsilon ^{C} (S) - x^{\omega } (S)} \right)} \right]} ^{2} + \lambda ^{\omega } \left( {\sum\limits_{{i \in N}} {x_{i}^{\omega } } - \upsilon (N)} \right) $$

To be exact, the Lagrange conditions of Problem 7 are shown as follows:

$$ \left\{ \begin{gathered} \sum\limits_{i \in N} {x_{i}^{\omega * } } = \upsilon (N) \hfill \\ 2\sum\limits_{s = 1}^{n - 1} {\omega (s)s\left( \begin{gathered} n - 2 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} x_{i}^{\omega * } + 2\sum\limits_{s = 2}^{n} {\omega (s)s\left( \begin{gathered} n - 2 \hfill \\ s - 2 \hfill \\ \end{gathered} \right)} \upsilon (N) - 2\sum\limits_{S \subset N:i \in S} {\omega (s)s\upsilon^{C} (S)} + \lambda^{\omega * } = 0 \hfill \\ \end{gathered} \right. $$

We can solve Problem 7 and obtain its unique optimal solution just through simple calculations, which is given by

$$ x_{i}^{\omega * } = \frac{\upsilon (N)}{n} + \frac{{\sum\limits_{S \subset N:i \in S} {\omega (s)s\upsilon^{C} (S)} - \frac{1}{n}\sum\limits_{j \in N} {\sum\limits_{S \subset N:j \in S} {\omega (s)s\upsilon^{C} (S)} } }}{{\sum\limits_{s = 1}^{n - 1} {\omega (s)s\left( \begin{gathered} n - 2 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} }} = \frac{1}{n}\upsilon (N) + k^{ - 1} [a_{i} (\upsilon ) - \frac{1}{n}\sum\limits_{j \in N} {a_{j} (\upsilon )} ], $$
(3)

where \(k = \sum\limits_{s = 1}^{n - 1} {\omega (s)s\left( \begin{gathered} n - 2 \hfill \\ s - 1 \hfill \\ \end{gathered} \right)} ,\) and\(a_{i} (\upsilon ) = \sum\limits_{S \subset N:i \in S} {\omega (s)s\upsilon^{C} (S)}\).

3 Improved Shapley Values Based on Players’ Least Square Contributions

As is known to all, the Shapley value is famous as a point-value solution for cooperative TU games. What is more, it is universally acknowledged that the axiomatic method applied to cooperative TU games for the first time in (Shapley 1953). However, the marginal contribution, regarded as the foremost factor for the Shapley value, can still be improved. In the following, the marginal contribution is replaced with the weighted least square contribution and some improved Shapley values are generated.

3.1 The Weighted Shapley-Like Value

As mentioned in Sect. 2.3, the coalitional weight has several important meanings and usually appears non-negligible. In this section, we try to substitute the player’s marginal contribution by its weighted least square contribution, propose a novel weighted Shapley-like value and verify its properties. We formally consider the following function based on the weighted least square contribution, which is called the weighted Shapley-like value.

$$ {\text{Sh}-\text{L}}_{i}^{\omega * } (\upsilon ) = \sum\limits_{S \subseteq N:i \in S} {\frac{(s - 1)!(n - s)!}{{n!}}x_{i}^{\omega * } (S)} , $$
(4)

where

$$ x_{i}^{\omega * } (S) = \frac{1}{s}\upsilon (S) + \frac{{\sum\limits_{{S{\prime} \subset S:i \in S{\prime}}} {\omega (s{\prime})s{\prime}\upsilon^{C} (S{\prime})} - \frac{1}{s}\sum\limits_{j \in S} {\sum\limits_{{S{\prime} \subset S:j \in S{\prime}}} {\omega (s{\prime})s{\prime}\upsilon^{C} (S{\prime})} } }}{{\sum\limits_{{s{\prime} = 1}}^{s - 1} {\omega (s{\prime})s{\prime}\left( \begin{gathered} s - 2 \hfill \\ s{\prime} - 1 \hfill \\ \end{gathered} \right)} }} = \frac{1}{s}\upsilon (S) + k{\prime}^{ - 1} [a{\prime}_{i} (\upsilon ) - \frac{1}{n}\sum\limits_{j \in S} {a{\prime}_{j} (\upsilon )} ] $$

with \(s^{\prime} = \left| {S^{\prime}} \right|\), \(k^{\prime} = \sum\limits_{{s^{\prime} = 1}}^{s - 1} {\omega (s^{\prime})s^{\prime}\left( \begin{gathered} s - 2 \hfill \\ s^{\prime} - 1 \hfill \\ \end{gathered} \right)}\), and \(a^{\prime}_{i} (\upsilon ) = \sum\limits_{{S^{\prime} \subset S:i \in S^{\prime}}} {\omega (s^{\prime})s^{\prime}\upsilon^{C} (S^{\prime})}\).

That is to say, \(x_{i}^{\omega * } (S)\) can be directly derived from Eq. (3) under special circumstances, where the coalition (subset) \(S\) (\(S \subseteq N\)) is temporarily regarded as the grand cooperative coalition and \(S^{\prime}\) represents the sub-coalition (subset) contained in \(S\).

Lemma 8

For any cooperative TU game\(\upsilon \in G^{n}\), the sum of all players’ payoffs, allocated by the weighted Shapley-like value (Eq. (4)), is constant and independent of the weighted least square contribution itself.

Proof

Owing to the efficiency, i.e., \(\sum\limits_{i \in S} {x_{i}^{\omega * } (S)} = \upsilon (S)\) (\(S \subseteq N\)), it directly follows that

$$ \sum\limits_{i \in N} {{\text{Sh - L}}_{i}^{\omega * } (\upsilon )} = \sum\limits_{S \subset N} {\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)} + \frac{1}{n}\upsilon (N). $$
(5)

Hence, Lemma 8 has been proved.

\(\Delta (\upsilon )\) denotes the difference between \(\sum\limits_{i \in N} {{\text{Sh} - \text{L}}_{i}^{\omega * } (\upsilon )}\) and \(\upsilon (N)\), and so

$$\Delta (\upsilon ) = \sum\limits_{i \in N} {{\text{Sh - L}}_{i}^{\omega * } (\upsilon )} - \upsilon (N) = \sum\limits_{S \subset N} {\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)} - \frac{n - 1}{n}\upsilon (N)$$

Clearly, \(\Delta (\upsilon )\) is not more than 0 owing to the obvious conclusion that

$$\sum\limits_{S \subset N} {\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)} \le \frac{n - 1}{n}\upsilon (N), $$

which can be directly derived from the superadditivity for cooperative TU games.

Remark 9

The weighted Shapley-like value satisfies the efficiency if and only if \(\sum\limits_{S \subset N} {\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)} = \frac{n - 1}{n}\upsilon (N)\). Therefore, for the weighted Shapley-like value, there usually exists a cooperative surplus (\(\frac{n - 1}{n}\upsilon (N) - \sum\limits_{S \subset N} {\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)}\)).

As is known to all, the efficiency is an essential prerequisite for a payoff vector to become a preimputation. Without loss of generality, we make some improvement on the weighted Shapley-like value and assure that it satisfies the efficiency. Formally, we consider the following function.

$$ \begin{aligned} {\text{Sh - L}}_{i}^{\omega * }{\prime } (\upsilon ) &= \sum\limits_{S \subseteq N:i \in S} {\frac{(s - 1)!(n - s)!}{{n!}}x_{i}^{\omega * } (S)} + \frac{n - 1}{{n^{2} }}\upsilon (N) - \sum\limits_{S \subset N:i \in S} {\frac{1}{s}\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)} \hfill \\ \, &= \sum\limits_{S \subseteq N:i \in S} {\frac{(s - 1)!(n - s)!}{{n!}}[x_{i}^{\omega * } (S) - \frac{1}{s}\upsilon (S)]} + \frac{2n - 1}{{n^{2} }}\upsilon (N), \hfill \\ \end{aligned} $$
(6)

which is called the efficient weighted Shapley-like value.

Let us consider the following four axioms (i)–(iv), which can uniquely determine the efficient weighted Shapley-like value. For any two cooperative TU games \(\upsilon \in G^{n}\) and \(v \in G^{n}\),

Axiom (i) (Symmetry): \( {\text{Sh}} - L_{i}^{\omega * ^\prime } (\upsilon ) = {\text{Sh}} - L_{i}^{\omega * ^\prime } (\upsilon )\;\;\left( {i,j \in N} \right) \).

Owing to the assumption that player \(i\) is symmetric with player\(j\), the conclusion can be easily drawn from Eq. (6) that\( {\text{Sh}} - {\text{L}}_{i}^{\omega *^\prime } (\upsilon ) = {\text{Sh}} - {\text{L}}_{i}^{\omega*^\prime } (\upsilon )\;\;\left( {i,j \in N} \right) \).

Axiom (ii) (Efficiency):\( \sum\limits_{{i \in N}} {{\text{Sh}} - {\text{L}}_{i}^{{\omega *}^{\prime}} (\upsilon )} = \upsilon (N) \).

The sum of all players’ payoffs allocated by Eq. (6) is given by

$$ \begin{aligned} \sum\limits_{i \in N} {{\text{Sh - L}}_{i}^{\omega * }{\prime } (\upsilon )} = \sum\limits_{i \in N} {[\sum\limits_{S \subseteq N:i \in S} {\frac{(s - 1)!(n - s)!}{{n!}}[x_{i}^{\omega * } (S) - \frac{1}{s}\upsilon (S)]} + \frac{2n - 1}{{n^{2} }}\upsilon (N)} ] \hfill \\ \, = \sum\limits_{S \subset N} {\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)} + \frac{1}{n}\upsilon (N) + \frac{n - 1}{n}\upsilon (N) - \sum\limits_{S \subset N} {\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)} \hfill \\ \, = \upsilon (N), \hfill \\ \end{aligned} $$

which means Eq. (6) satisfies the efficiency.

Axiom (iii) (Additivity):\({\text{Sh} - \text{L}}_{i}^{\omega * \prime } (\upsilon + v) = {\text{Sh} - \text{L}}_{i}^{\omega * \prime } (\upsilon ) + {\text{Sh}- \text{L}}_{i}^{\omega *\prime } (v)\).

According to Eq. (6), it is obvious that

$ \begin{gathered} {\text{Sh}} - {\text{L}}_{i}^{{\omega *}} \prime (\upsilon + v) = \sum\limits_{{S \subseteq N:i \in S}} {\frac{{(s - 1)!(n - s)!}}{{n!}}[x_{i}^{{\omega *}} (S) - \frac{1}{s}\upsilon (S) + x_{i}^{{\omega *}} (S) - \frac{1}{s}v(S)]} + \frac{{2n - 1}}{{n^{2} }}[\upsilon (N) + v(N)] \hfill \\ {\mkern 1mu} = \sum\limits_{{S \subseteq N:i \in S}} {\frac{{(s - 1)!(n - s)!}}{{n!}}[x_{i}^{{\omega *}} (S) - \frac{1}{s}\upsilon (S)]} + \frac{{2n - 1}}{{n^{2} }}\upsilon (N) \hfill \\ {\mkern 1mu} + \sum\limits_{{S \subseteq N:i \in S}} {\frac{{(s - 1)!(n - s)!}}{{n!}}[x_{i}^{{\omega *}} (S) - \frac{1}{s}v(S)]} + \frac{{2n - 1}}{{n^{2} }}v(N) \hfill \\ {\mkern 1mu} = {\text{Sh}} - {\text{L}}_{i}^{{\omega *}} \prime (\upsilon ) + Sh{\text{ }} - {\text{ }}L_{i}^{{\omega *}} \prime (v) \hfill \\ \end{gathered} $

Axiom (iv) (Quasi-null player): \( {\text{Sh}} - {\text{L}}_{i}^{{\omega *}} \prime (\upsilon ) = \frac{{n - 1}}{{n^{2} }}\upsilon (N) - \sum\limits_{{S \subset N:i \in S}} {\frac{1}{s}\frac{{(s - 1)!(n - s)!}}{{n!}}\upsilon (S)} \) if \(i \in N\) is a null player in a game \(\upsilon\).

The efficient weighted Shapley-like value is an extension of the Shapley value, which is improved through considering both the efficiency principle (the least square contributions) and the fairness principle. Namely, if \(i \in N\) is a null player, then the efficient weighted Shapley-like value will allocate the player \(i \in N\) the value \(\frac{n - 1}{{n^{2} }}\upsilon (N) - \sum\limits_{S \subset N:i \in S} {\frac{1}{s}\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)}\) rather than 0 allocated by the Shapley value.

Theorem 10

The efficient weighted Shapley-like value is the unique value over \(G^{n}\) satisfying the above axioms (i)–(iv).

Proof

Let \(f\) be a value over \(G^{n}\) satisfying the axioms (i)–(iv). Clearly, every cooperative TU game \((N,\upsilon ) \in G^{n}\) admits a unique decomposition into unanimity games \((N,\mu_{T} ) \in G^{n}\) (\(T \in 2^{N} \backslash \emptyset\)):

\(\upsilon = \sum\limits_{{T \in 2^{N} \backslash \emptyset }} {\Delta_{\upsilon } } (T)\mu_{T}\),where \(\mu_{T}\) is defined by \(\mu_{T} (S) = 1\) if \(T \subseteq S\) and \(\mu_{T} (S) = 0\) otherwise.

(1) Based on the axiom (iv), we have \( {\text{Sh}} - {\text{L}}_{i}^{\omega * \prime} (\mu _{T} ) = \frac{{n - 1}}{{n^{2} }}\mu _{T} (N) - \sum\limits_{{S \subset N:i \in S}} {\frac{1}{s}\frac{{(s - 1)!(n - s)!}}{{n!}}\mu _{T} (S)} \) (\(i \in N\backslash T\)) if \(i \in N\backslash T\) is a null player.

(2) Based on the axioms (i) and (ii), if \(i \in T\) satisfies the symmetry, we have \({\text{Sh}} - {\text{L}}_{i}^{\omega *\prime } (\mu_{T} ) = \frac{{\mu_{T} (N) - \sum\limits_{j \in N\backslash T} {{\text{Sh - L}}_{j}^{\omega *\prime } (\mu_{T} )} }}{\left| T \right|}\).

Remark 11

The axioms (i)–(iv) for the efficient weighted Shapley-like value are independent from each other. As stated earlier, the efficient weighted Shapley-like value is an extension of the Shapley value. Therefore, the axioms (i)–(iii) are independent from each other as the Shapley value’s three axioms. It is easy to see that the Shapley value satisfies the axioms (i)–(iii) except the axiom (iv): quasi-null player. Therefore, the efficient weighted Shapley-like value satisfies the axioms (i)–(iii) except the axiom (iv) due to the fact that the Shapley value is a special case of the efficient weighted Shapley-like value.

3.2 The Efficient Shapley-Like Value

Although the coalitional weight appears important in many cases, it is sometimes ignored. That is to say, the coalitional value function can be taken as an identical function in real-world applications, i.e., it assigns the value 1 to coalitions of all sizes from 1 to \(n\) and 0 for size 0.

Remark 12

The weighted Shapley-like value (Eq. (4)) can be reduced to (simplified as) the Shapley-like value given in Eq. (7) when the coalitional weights are not taken into account or taken as identical values.

$${\text{Sh}} - {\text{L}}_{i}^{ * } (\upsilon ) = \sum\limits_{S \subseteq N:i \in S} {\frac{(s - 1)!(n - s)!}{{n!}}x_{i}^{ * } (S)}. $$
(7)

where

$$ x_{i}^{ * } (S) = \frac{1}{s}\upsilon (S) + \frac{{\sum\limits_{{S^{\prime} \subset S:i \in S^{\prime}}} {s^{\prime}\upsilon^{C} (S^{\prime})} - \frac{1}{s}\sum\limits_{j \in S} {\sum\limits_{{S{\prime} \subset S:j \in S^{\prime}}} {s{\prime}\upsilon^{C} (S{\prime})} } }}{{\sum\limits_{{s{\prime} = 1}}^{s - 1} {s{\prime}\left( \begin{gathered} s - 2 \hfill \\ s^{\prime} - 1 \hfill \\ \end{gathered} \right)} }} = \frac{1}{s}\upsilon (S) + k{\prime}^{ - 1} [a{\prime}_{i} (\upsilon ) - \frac{1}{n}\sum\limits_{j \in S} {a{\prime}_{j} (\upsilon )} ] $$

with \(s^{\prime} = \left| {S^{\prime}} \right|\), \(k^{\prime} = \sum\limits_{{s^{\prime} = 1}}^{s - 1} {s^{\prime}\left( \begin{gathered} s - 2 \hfill \\ s^{\prime} - 1 \hfill \\ \end{gathered} \right)} ,\) and \(a^{\prime}_{i} (\upsilon ) = \sum\limits_{{S^{\prime} \subset S:i \in S^{\prime}}} {s^{\prime}\upsilon^{C} (S^{\prime})}\).

Remark 13

Generally speaking, the Shapley-like value is usually not efficient except \(\sum\nolimits_{S \subset N} {\frac{(s - 1)!(n - s)!}{{n!}}\upsilon (S)} = \frac{n - 1}{n}\upsilon (N)\), i.e., \(\sum\nolimits_{j \in S} {\upsilon_{j}^{C} (S) = } \upsilon (S)\) for all coalition \(S\) (\(S \subseteq N\)). To be exact, the Shapley-like value (Eq. (7)) is equal to the Shapley value when \(\sum\nolimits_{j \in S} {\upsilon_{j}^{C} (S) = } \upsilon (S)\) (\(S \subseteq N\)).

The conclusion can be drawn from Remark 14 that, the payoff vector assigned by Eq. (7) is efficient in certain situations. In most cases, however, Eq. (7) fails to satisfy the efficiency. Hence, it is necessary to make some improvements on it. What follows is the efficient Shapley-like value.

$$ {\text{Sh} - \text{L}}_{i}^{ * \prime } (\upsilon ) = \sum\limits_{S \subseteq N:i \in S} {\frac{1}{{nC_{n - 1}^{s - 1} }}x_{i}^{ * } (S)} + \frac{n - 1}{{n^{2} }}\upsilon (N) - \sum\limits_{S \subset N:i \in S} {\frac{1}{s}\frac{1}{{nC_{n - 1}^{s - 1} }}\upsilon (S)} = \sum\limits_{S \subseteq N:i \in S} {\frac{1}{{nC_{n - 1}^{s - 1} }}[x_{i}^{ * } (S) - \frac{1}{s}\upsilon (S)]} + \frac{2n - 1}{{n^{2} }}\upsilon (N) $$
(8)

Theorem 14

The efficient Shapley-like value (Eq. (8)) is the unique value over \(G^{n}\) satisfying the four axioms (i)–(iv): the symmetry, the efficiency, the additivity, and the quasi-null player.

Remark 15

The axioms (i)-(iv) mentioned in Theorem 14 for the efficient Shapley-like value (Eq. (8)) are independent from each other due to the fact that the efficient Shapley-like value given in Eq. (8) is a special case of the efficient weighted Shapley-like value given in Eq. (6) if the coalitional weights are not considered. To be exact, the efficient Shapley-like value is a special case of the efficient weighted Shapley-like value when the weight functions \(\omega (s^{\prime})\) remain identical for all possible coalitions in \(S\) (\(S \subseteq N\)). The proofs of the uniqueness and the independence of the efficient Shapley-like value given are very similar to those of the efficient weighted Shapley-like value. For the sake of conciseness, they are omitted here.

4 Case Study and Comparative Analysis

Case 1 Considering a rural e-commerce coalition including three diverse electric commercial enterprises, named Enterprise 1, Enterprise 2, and Enterprise 3, respectively. All of them have own advantages. Therein, Enterprise 1 has abundant funds and scientific management, and Enterprise 2 is equipped with strong technical force. Nevertheless, Enterprise 3 owns rich supply of agricultural products. If they operate the business by their own, they will receive the following gains per year (unit: ten thousand yuan):\(\upsilon (1) = 6\), \(\upsilon (2) = 15\), \(\upsilon (3) = 30\). In order to better resist the market risk and make more profits, they determine to form a cooperative coalition and work together. The characteristic functions (collaborative profits after cooperation) are shown as follows:\(\upsilon (12) = 60\), \(\upsilon (13) = 62\), \(\upsilon (23) = 64\), \(\upsilon (123) = 120\). The marginal contributions of all subsets except \(\emptyset\) to the grand cooperative coalition\(N\), consequently, can be given by\(\upsilon^{C} (1) = 56\), \(\upsilon^{C} (2) = 58\), \(\upsilon^{C} (3) = 60\), \(\upsilon^{C} (12) = 90\), \(\upsilon^{C} (13) = 105\), \(\upsilon^{C} (23) = 114\), \(\upsilon^{C} (123) = 120\), respectively.

As is known to all, the stability of a cooperative coalition mainly depends on the consequent collaborative profit allocation scheme. Several eminent point-valued solutions for cooperative TU games, beyond all doubt, can apply to address this problem, such as the Shapley value (Shapley 1953), the Banzhaf value (Banzhaf 1965; Owen 1975), and the equal surplus division value (ES-value) (Driessen and Funaki 1991). The improved Shapley values proposed in this paper, however, have many advantages in this class of cooperative TU games. The advantages can be naturally demonstrated by the collaborative profit allocation results from different solutions of Case 1, which is shown in Table 1.

Table 1 Collaborative profit allocation results from different solutions
Full size table

The Banzhaf value, as we all know, usually fails to satisfy the efficiency. Case 1 is an example in point. However, it is still one eminent point-valued solution for cooperative TU games and the basis of the improved Banzhaf value which is called the multiplicative normalization of the Banzhaf value (\({\varvec{B}}^{m}\) value) (Hammer and Holzman 1992). The sum of three enterprises’ allocated profits is 123.9 (34.8 + 40.3 + 48.8) according to the Banzhaf value, which is obviously more than the total collaborative profit (i.e., 120). The profit allocation scheme based on the Banzhaf value, accordingly, is difficult to realize. Therefore, the \({\varvec{B}}^{m}\) value is used to address this problem and then an efficient payoff vector is obtained, which is given by 33.7, 39.1 and 47.2.

The ES-value satisfies the efficiency. However, it just takes account of players’ productivities when working alone, regardless of their contributions to the grand coalition\(N\). In fact, not only players’ productivities, but also their contributions are equally important for the collaborative profit allocation scheme. Taking Enterprise 1 as an example, its marginal contribution to the grand coalition \(N\) is 56, which is extremely close to those of Enterprise 2 and Enterprise 3. Nevertheless, the gap among the allocated collaborative profits of the three enterprises is bigger than that based on the improved Shapley values proposed in this paper. The main reason for this unreasonable allocation result is that the ES-value does not consider the players’ contributions to the grand coalition\(N\).

The Shapley value, known as a well-known solution for cooperative TU games, mainly considers the influence of players’ marginal contributions. What is more, it ignores the significance of coalitional weights. Generally speaking, the coalitional weights have several reasonable interpretations, such as the stability of a cooperative coalition, the complexity to form a cooperative coalition, and the probability of a cooperative coalition. The coalitional weights, beyond all doubt, should not be neglected. To better show the superiorities of the efficient weighted Shapley-like value proposed herein, the weighted Shapley value (Kalai and Samet 1987) is mentioned in Table 1. Clearly, the weight function of the latter is related to the players, while that of the former is in relation to the coalitions.

All the improved Shapley values proposed in paper base the collaborative profit allocation scheme not only on players’ productivities, but also on their least square contributions. Furthermore, we can choose the appropriate coalitional weights according to the actual demand and the specific context. Compared with some extant eminent solutions for cooperative TU games, such as the Shapley value, the Banzhaf value, and the ES-value, the improved Shapley values herein have some advantages, as shown in Table 1. Figure 1 gives the variation trend of the collaborative profit allocation schemes with different coalitional weights when Eq. (8) is applied.

Fig. 1
figure 1

Variation trend of the collaborative profit allocation schemes with different coalitional weights

Full size image

5 Conclusions

Inspired by the idea of player’s productivity-based excess, we firstly propose the concept of player’s contribution-based excess. A baseline quadratic programming model is constructed for solving the least square value based on player’s contributions to all coalitions containing itself. The optimal solution of the baseline model is called the Shapley-like value. In order to further explore the relationship between the Shapley value and the least square family, we substitute player’ least square contribution for its marginal contribution, and prove that the Shapley-like value herein is equal to the Shapley value under specified circumstances. Without loss of generality, we make some necessary improvements on the Shapley-like value and propose the efficient Shapley-like value.

What is more, by taking into account coalitional weights, the weighted Shapley-like value is generated. Similar to the Shapley-like value, the weighted Shapley-like value is made some improvements to satisfy more nice axioms or properties. As a result, we propose the efficient weighted Shapley-like value, which can be characterized by the four independent axioms such as the symmetry, the efficiency, the additivity, and the quasi-null player, which are similarly used to characterize the Shapley value. The only difference is that the anonymity in latter is replaced with the quasi-null player in the former.

We provide a new and an original view for studying the relationship between the Shapley value and the family of least square values for cooperative TU games in this paper. Enlightened by the idea that the Banzhaf value is also based on player’s marginal contributions to the related coalitions containing itself, in the near future, maybe the relationship between the Banzhaf value and the family of least square values for cooperative TU games will be another research hotspot and attract our attention.