1 Introduction
A transferable utility cooperative game (TU game) formulates a situation in which a finite set of n players can cooperate with each other and generate joint revenues. The cooperative behavior among players is captured by the assumption that coalitions \(S\subseteq \{1,\dots ,n\}\) of players can form, and the worth of a coalition S represents the revenue that can be obtained by that coalition. Assuming that the grand coalition \(N=\{1,\dots ,n\}\) is being formed, one of the major questions in cooperative game theory is how the worth of the grand coalition should be distributed among the players. An answer to this question is a mapping of any given game to a payoff vector \((\psi _1,\dots ,\psi _n)\), which is called a value of a cooperative game. Arguably the most well-known value in cooperative game theory is the Shapley value (Shapley 1953b), which offers each player her expected marginal contribution when assuming all possible n! orders of the n players happen with the same probability.
It should be clear that the assumption of free cooperation between any subsets of players is sometimes not realistic. One concept to capture such situations is that players are partitioned into subgroups \(C_1,\dots ,C_m\), for example due to physical characteristics such as geographic location, or because players actively organize themselves into subgroups in order to improve their bargaining position, such as cartels and syndicates. Such pre-defined subgroups then mean that the cooperation among players can happen within a given subgroup \(C_k\), while cooperation outside the subgroups happens on the level of the subgroups themselves. The partition is usually called a coalition structure, and the subgroups \(C_k\) are referred to as unions.
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TU games with coalition structure were first considered by Aumann and Dreze (1974). They assume the grand coalition is divided into disjoint and independent unions, and there is no side payment between unions. In the Aumann-Drèze (AD) value (Aumann and Dreze 1974), every player receives the payoff allocated to her by the Shapley value in the subgame she is playing within her union. The possibility of cooperation between unions begins with Owen (1977), who interpreted the unions as “bargaining blocks”. Owen assumes that the coalition of all players is being formed, and hence, the worth of the grand coalition is distributed. The Owen value (Owen 1977) is defined by taking two levels of interaction among players into account, first among unions and then within each union: First the unions get the Shapley value of the game played in the so-called quotient game, which is the game where the unions are the players. Then, to distribute the each union’s Shapley payoff over its players, Owen defines an induced internal game in which he considers the worth of a coalition \(S\subseteq C_{k}\) to be the union’s Shapley value of the quotient game where the union \(C_{k}\) is replaced with S. The payoff is then again distributed using the Shapley value. Following the Owen procedure, several other values have been extended to TU games with coalition structure, including the Banzhaf value (Banzhaf 1965), the \(\tau\)-value (Tijs 1981), the equal division surplus value (Driessen and Funaki 1991), etc. We refer to Owen (1982), Casas-Méndez et al. (2003) and Alonso-Meijide et al. (2020) for these.
In this paper, along the lines of the previously mentioned papers, we suggest a new value for cooperative games with coalition structure. This value is closely related to another value for cooperative games with coalition structure that has been suggested by Kamijo (2009), the so-called two-step Shapley value. That value exhibits a certain conceptual simplicity, e.g., when compared to Owen’s value: In the first step, all players of a union equally share the Shapley net surplus of the union containing them, i.e., there is an equal distribution of the difference between the Shapley value obtained by this union in the quotient game, and the worth of it. That means the union is left with its worth, which is again distributed using the Shapley value. However, we believe that it lies in the nature of games with coalition structure that players within one union should exhibit a higher degree of solidarity, which is not captured by using the Shapley value for the game within unions. In the following, we elaborate a bit more on Kamijo’s value, as well as other closely related values from the literature, in order to motivate our proposal.
Kamijo’s two-step Shapley value actually establishes an “interpolation” between the approaches suggested by Owen and Aumann and Drèze. On the one hand, it affirms Owen’s assumption that the grand coalition is being formed, and unions first play the quotient game to distribute the worth of the grand coalition among them. This is the same as in the first step of Owen’s approach. On the other hand, it also retains the idea of separation between unions, because cooperation of the players within a union is simply modelled by the corresponding subgame restricted to the players of a union, as by Aumann and Drèze. In the second step, every player is just assigned the Shapley value of the respective subgame, and the sum of the two parts gives the two-step Shapley value (Kamijo 2009). With a change to the weighted Shapley value (Shapley 1953a) in the first step, this was generalized even further to the so-called collective value (Kamijo 2013).
Observe that solidarity among players within one union is embedded in Kamijo’s approach only from the perspective of interaction among unions, as the union’s Shapley surplus is equally divided among its members in the first step. However, solidarity is not really reflected by using the Shapley value for the games within the unions, because the Shapley value is known to be a purely performance-based value, which has been formalized elegantly by an axiomatization based on marginality as given by Young (1985). Here, marginality refers to the fact that a player’s payoff only depends on her own marginal contribution.
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Almost parallel to our work, Hu (2020) also suggested to address this issue, i.e., incorporating a higher degree of solidarity among the players within a union, by using the equal division value for the subgames per union, which then gives rise to the so-called weighted Shapley-egalitarian value (Hu 2020). The equal division value, however, is totally independent of players’ performance. Another value that suggests itself in this context is the solidarity value by Sprumont (1990) and Nowak and Radzik (1994). Unlike equal division, the solidarity value takes individuals’ differences into consideration, yet implements the solidarity principle as well: It employs the average marginal contribution instead of marginal contribution as in the Shapley value, and in this way, the value is equipped with the feature of solidarity by providing support to “weaker” players: a player who contributes less than the average marginal contribution is supported by stronger partners.
That said, it should be mentioned that the solidarity value has previously been adopted into games with coalition structure by Calvo and Gutiérrez (2013). They also use the Shapley value for the quotient game of all unions, but following Owen’s procedure, consider the induced internal games to distribute the unions’ Shapley payoffs among the players, and this is based on the solidarity value. The resulting value is the Shapley-solidarity value.
In this paper, inspired by the conceptual simplicity of Kamijo’s two-step approach, and the idea to incorporate a realistic level of solidarity among the players of a given union, we suggest to marry Kamijo’s two-step approach with the idea to use the solidarity value for the game played by players within any given union. Note that this is different from Owen’s value as well as Calvo and Gutiérrez’s Shapley-solidarity value, as we follow Kamijo’s approach and only distribute the net surplus of the unions’ Shapley payoffs in the first step. Hence, for the second step, there is no need to revert to Owen’s induced internal game, and we simply use the solidarity value to distribute the union’s worth among its players. Arguably, this is conceptually simpler.
In lack of a better name and to avoid confusion with the values proposed earlier, we call this new value the two-step Shapley-solidarity value. Our main contributions are an intuitive procedural interpretation for this new value, and to give three axiomatizations that highlight the differences between this value and specifically the two-step Shapley value. As to the technical contribution of the paper, in order to get our axiomatizations done, we use a property that we call coalitional A-null player property, and moreover, we have to revert to a new basis of the space of all games.
The rest of the paper is organized as follows. Section 2 contains some preliminaries, fixes the notation used, and gives a quick recap of the most relevant values that have been proposed earlier. After introducing the two-step Shapley-solidarity value in Sect. 3, we provide a procedural interpretation for it in Sect. 4. Finally, three axiomatic characterizations are presented in Sect. 5. Section 6 gives some final conclusions.
2 Definitions and notations
2.1 TU games and values
A cooperative game with transferable utility (or TU game) is a pair (N, v) consisting of a nonempty and finite set of players N and the characteristic function \(v:2^{N}\rightarrow {\mathbb {R}}\) such that \(v(\emptyset )=0\). An element \(i\in N\) and a subset S of N are called a player and a coalition, respectively. Especially, we call N the grand coalition. With some abuse of notation, we omit the braces for singletons. Thus, we write \(S\cup i\) for \(S\cup \{i\}\), \(S{\setminus } i\) for \(S{\setminus } \{i\}\), etc. For each \(S\subseteq N\), v(S) is the worth of coalition S. The cardinality of S is denoted by the corresponding lower case letter s or \(|S |\).
Let \({\mathcal {G}}^{N}\) be the family of all TU games over N. For any two TU games \((N,v),(N,w)\in {\mathcal {G}}^{N}\), \(\alpha \in {\mathbb {R}}\), the characteristic functions of TU games \((N,v+w)\) and \((N,\alpha v) \in {\mathcal {G}}^{N}\) are respectively given by \((v+w)(S)=v(S)+w(S)\) and \((\alpha v)(S)=\alpha v(S)\) for all \(S\subseteq N\). For \(T\in 2^{N}\backslash \emptyset\), the subgame of (N, v) when the player set is restricted to T is denoted by TU game \((T,v_{|T})\in {\mathcal {G}}^{T}\), where \(v_{|T}(S)=v(S)\) for all \(S\subseteq T\). The TU game \((N,u_{T})\in {\mathcal {G}}^{N}\), where \(u_{T}(S)=1\) if \(T\subseteq S\) and \(u_{T}(S)=0\) otherwise, is called a unanimity game. Any TU game can be uniquely represented by unanimity games,where \(c_{T}=\sum _{S\subseteq T}(-1)^{t-s}v(S)\) is called the Harsanyi dividend (Harsanyi 1963).
$$\begin{aligned} v=\sum _{T\subseteq N, T\ne \emptyset }c_{T}u_{T}, \end{aligned}$$
(1)
A player \(i\in N\) is a null player in \((N,v)\in {\mathcal {G}}^{N}\) if \(v(S)=v(S\cup i)\) for all \(S\subseteq N{\setminus } i\) and a dummy player if \(v(S\cup i)=v(S)+v(i)\) for all \(S\subseteq N\setminus i\). A player \(i\in N\) is an A-null player in \((N,v)\in {\mathcal {G}}^{N}\) if the average marginal contribution of the singleton players in any coalition \(S\ni i\) is zero, that is, \(\frac{1}{s}\sum _{j\in S}(v(S)-v(S{\setminus } j))=0\) for all \(S\subseteq N\) with \(i\in S\). Replacing individual marginal contributions by average marginal contributions can be seen as a means of solidarity among the players in a coalition. An A-null player does not contribute anything to any coalition in this average sense. Two players \(i,j \in N\) are symmetric in \((N,v)\in {\mathcal {G}}^{N}\) if \(v(S\cup i)=v(S\cup j)\) for all \(S\subseteq N{\setminus } \{i,j\}\). A permutation on N is a mapping \(\pi : N\rightarrow N\) that associates every player i with a position \(\pi (i)\). Let \(\Pi _{N}\) denote the collection of all n! permutations on N. Given \(\pi \in \Pi _{N}\), the set of players that are in positions before player i, called the predecessors of i, is denoted by \(P^{\pi }(N, i)=\{j\in N \, |\, \pi (j)\le \pi (i)\}\).
A value on \({\mathcal {G}}^{N}\) is an operator that assigns a payoff vector \(\varphi (N,v)=(\varphi _{i}(N,v))_{i\in N}\in {\mathbb {R}}^{n}\) to every TU game \((N,v)\in {\mathcal {G}}^{N}\). The Shapley value (Shapley 1953b) is probably the best known value. For any \((N,v)\in {\mathcal {G}}^{N}\), it is given bywhile the solidarity value (Sprumont 1990) is given by
$$\begin{aligned} Sh_{i}(N,v)= \sum _{S\subseteq N\backslash i}\frac{s!(n-s-1)!}{n!}\left[ v(S\cup \{i\})-v(S)\right] , \; i\in N, \end{aligned}$$
$$\begin{aligned} Sol_{i}(N,v)= \sum _{S\ni i}\frac{(s-1)!(n-s)!}{n!}\left[ \frac{1}{s}\sum _{j\in S}(v(S)-v(S\setminus j))\right] , \; i\in N. \end{aligned}$$
In order to describe values of games by their characteristic properties, consider the following axioms of a value \(\varphi\) on \({\mathcal {G}}^{N}\):While the Shapley value is characterized by E, A, S and NP, the characterization of the solidarity value can be obtained by replacing NP with ANP.
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Efficiency (E): For all \((N,v)\in {\mathcal {G}}^{N}\), \(\sum _{i\in N}\varphi _{i}(N,v)=v(N)\).
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Symmetry (S): For all \((N,v)\in {\mathcal {G}}^{N}\) and \(\{i,j\}\subseteq N\), if i, j are symmetric, then \(\varphi _{i}(N,v) = \varphi _{j}(N,v)\).
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Additivity (A): For all \((N,v), (N,w)\in {\mathcal {G}}^{N}\), \(\varphi _{i}(N,v+w)=\varphi _{i}(N,v)+\varphi _{i}(N,w)\).
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Null player axiom (NP): For all \((N,v)\in {\mathcal {G}}^{N}\) and \(i\in N\), if i is a null player in (N, v), then \(\varphi _{i}(N,v) = 0\).
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A-Null player axiom (ANP): For all \((N,v)\in {\mathcal {G}}^{N}\) and \(i\in N\), if i is an A-null player in (N, v), then \(\varphi _{i}(N,v) = 0\).
Theorem 1
(Shapley 1953b) A value \(\varphi\) on \({\mathcal {G}}^{N}\) satisfies efficiency, additivity, symmetry and null player axiom if and only if \(\varphi (N,v)=Sh(N,v)\) for each \((N,v)\in {\mathcal {G}}^{N}\).
Theorem 2
(Nowak and Radzik 1994) A value \(\varphi\) on \({\mathcal {G}}^{N}\) satisfies efficiency, additivity, symmetry and A-null player axiom if and only if \(\varphi (N,v)=Sol(N,v)\) for each \((N,v)\in {\mathcal {G}}^{N}\).
2.2 TU games with coalition structure and some coalitional values
Given a finite set of players N, a coalition structure over N is a partition of the player set N, i.e., \({\mathcal {C}}=\{C_{1},C_{2},\ldots ,C_{m}\}\) is a coalition structure if \(\bigcup _{h\in M}C_{h}=N\), where \(M=\{1,2,\ldots ,m\}\), and \(C_{h}\cap C_{r}=\emptyset\) when \(h\ne r\). We call an element \(C_{h}\in {\mathcal {C}}\) a union. There are two trivial coalition structures, namely \({\mathcal {C}}^{N}=\{N\}\) and \({\mathcal {C}}^{n}=\{\{i\}\,|\,i\in N\}\), where only the grand coalition forms in \({\mathcal {C}}^{N}\) and each union is a singleton in \({\mathcal {C}}^{n}\). Denote by \({\mathcal {C}}_{N}\) the set of all possible coalition structures over N. For any \(S\subseteq N\), we denote the restriction of \({\mathcal {C}}\) on the player set S as \({\mathcal {C}}_{|S}\), i.e., \({\mathcal {C}}_{|S}=\{C_{h}\cap S \, |\,C_{h}\in {\mathcal {C}} \; \text {and} \; C_{h}\cap S\ne \emptyset \}\).
For a given coalition structure \({\mathcal {C}}\), a permutation \(\pi \in \Pi _{N}\) is consistent with \({\mathcal {C}}\) if \(i\in C_{h} \in {\mathcal {C}}\) and \(j\in C_{h} \in {\mathcal {C}}\) and \(k\in N\), \(\pi (i)<\pi (k)<\pi (j)\) implies that the player k also belongs to union \(C_{h}\). The set of all the permutations on N that are consistent with \({\mathcal {C}}\) is denoted by \(\Pi _{N,{\mathcal {C}}}\).
A cooperative game with a coalition structure is a triple \((N,v,{\mathcal {C}})\) where (N, v) is a TU game and \({\mathcal {C}}\) is a coalition structure over N. We denote by \(\mathcal{CG}\mathcal{}^{N}\) the collection of all TU games with coalition structure over player set N, and by \(\mathcal{CG}\mathcal{}\) the collection of all TU games with coalition structure. Given a non-empty coalition S, denote the restriction of \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\) to S as the TU game with coalition structure \((S,v_{|S},{\mathcal {C}}_{|S})\). Given a TU game with a coalition structure \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), the quotient game \((M,v^{{\mathcal {C}}})\) is defined asWe say that \(C_{h}\in {\mathcal {C}}\) is a null coalition in \((N,v,{\mathcal {C}})\) if h is a null player in \((M,v^{{\mathcal {C}}})\), and \(C_{h},C_{r}\in {\mathcal {C}}\) are symmetric coalitions in \((N,v,{\mathcal {C}})\) if h and r are symmetric players in \((M,v^{{\mathcal {C}}})\).
$$\begin{aligned} v^{{\mathcal {C}}}(Q)=v(\cup _{h\in Q}C_{h}), \; Q\subseteq M. \end{aligned}$$
A coalitional value is a function \(\psi\): \(\mathcal{CG}\mathcal{}^{N} \rightarrow {\mathbb {R}}^{n}\) that assigns to each cooperative game with coalition structure \((N,v,{\mathcal {C}})\) a payoff vector.
The Shapley-solidarity value (SS value) employs different rules between guiding cooperation among the players within a union and interaction among unions. Firstly, at the union level, all unions play the corresponding quotient game and get their payoffs prescribed by the Shapley value, i.e., for each \((N,v,{\mathcal {C}})\) and \(C_{k}\in {\mathcal {C}}\),For players within a union \(C_{k}\in {\mathcal {C}}\), they play an induced internal game \((C_{k},v_{C_{k}})\) in which the worth of a coalition is reassessed. For coalition \(S\subseteq C_{k}\), assume that its complement \({\bar{S}}\), where \({\bar{S}}=C_{k}\setminus S\), leaves the game. The worth of S in the induced internal game is specified by the Shapley value of the quotient game when replacing the union \(C_{k}\) with coalition S. That isThen, a player \(i\in C_{k}\) gets payoff according to the solidarity value (Calvo and Gutiérrez 2013),Unlike the two-step procedure above, Kamijo (2009) introduced the two-step Shapley value with a different, arguably simpler, two-step approach. To be more specific, one considers the surplus \(Sh_{k}(M,v^{{\mathcal {C}}})-v(C_{k})\) in the first step, which is equally divided among the players in \(C_k\). Then, there is no need to resort to a new worth to assess a coalition’s power when considering the intra-union bargaining. For \(C_{k}\in {\mathcal {C}}\), one can just consider the corresponding subgame \((C_{k},v_{|C_{k}})\) and Kamijo proposes to use the Shapley value to distribute \(v(C_k)\). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\) and \(i\in C_{k}\), the two-step Shapley value (Kamijo 2009) is therefore given by
$$\begin{aligned} \sum _{i\in C_{k}}SS_{i}(N,v,{\mathcal {C}})=Sh_{k}\left(M,v^{{\mathcal {C}}}\right). \end{aligned}$$
$$\begin{aligned} v_{{C_{k} }} (S) = Sh_{k} \left( {M,(v_{{|N{ \setminus }\bar{S}}} )^{{{\mathcal{C}}_{{|N{ \setminus }\bar{S}}} }} } \right). \end{aligned}$$
$$\begin{aligned} SS_{i}(N,v,{\mathcal {C}})=Sol_{i}(C_{k},v_{C_{k}}). \end{aligned}$$
$$\begin{aligned} TSh_{i} (N,v,{\mathcal{C}}) = Sh_{i} (C_{k} ,v_{{|C_{k} }} ) + \frac{{Sh_{k} \left( {M,v^{{\mathcal{C}}} } \right) - v\left( {C_{k} } \right)}}{{|C_{k} |}}. \end{aligned}$$
3 The two-step Shapley-solidarity value
Similar to the two-step Shapley value and the collective value, the two-step Shapley-solidarity value proposed here also distributes the worth of the grand coalition in two steps. Firstly, players within one union act collectively to bargain with other unions, all unions play the quotient game and obtain a payoff prescribed by the Shapley value. The surplus of the difference between the obtained payoff and the worth of the union is distributed equally among union members. Then, players within one union negotiate the worth that they can guarantee on their own, namely the worth of the union they belong to, and they obtain the solidarity value for the subgame restricted on the corresponding union.
Definition 1
For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\) and \(i\in C_{k}\in {\mathcal {C}}\), the two-step Shapley-solidarity value is given by
$$\begin{aligned} TSS_{i}(N,v,{\mathcal {C}})=Sol_{i}(C_{k},v_{|C_{k}})+\frac{Sh_{k}(M,v^{{\mathcal {C}}})-v(C_{k})}{|C_{k}|}. \end{aligned}$$
(2)
Clearly, the so defined two-step Shapley-solidarity value will degenerate to the Shapley value, respectively the solidarity value in the two extreme cases when the coalition structure is either all singleton players, or the grand coalition.
Remark 1
For \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\) and if \({\mathcal {C}}={\mathcal {C}}^{N}\), \(TSS(N,v,{\mathcal {C}})=Sol(N,v)\), and for \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\) and \({\mathcal {C}}={\mathcal {C}}^{n}\), \(TSS(N,v,{\mathcal {C}})=Sh(N,v)\).
In that sense, the level of solidarity increases with more players joining unions. The same property is shared by the Shapley-solidarity value of Calvo and Gutiérrez (2013). Except for the equal distribution of the surplus of a union which is not present there, the main difference lies in another intra-union game, i.e., Owen’s induced internal game. In this game the unions’ internal behavior is actually re-assessed from a “non-solidarity” perspective. Intuitively speaking, a coalition S contained in one union \(C_k\) takes into consideration that the remaining members \(C_k\setminus S\) might defect. Hence, they re-evaluate their worth to be what they can earn in the quotient game while assuming the remaining members are breaking away from their union.
Compared with the two close relatives, the two-step Shapley value and the Shapley-solidarity value, the two-step Shapley-solidarity value embeds more of a solidarity principle in the intra-union game, as it avoids the possible divergence among union members for the evaluation of their internal cooperation, and as it uses the solidarity value instead of the Shapley value. In this sense, the outcome should reflect a larger level of solidarity within unions. This can also be illustrated with the following, simple example of a four player game.
Example 1
Consider player set \(N=\{1,2,3,4\}\) and TU game (N, v) where the characteristic function v is given by \(v(\{3\})=v(\{1,2,3\})=1\), \(v(\{4\})=v(\{3,4\})=v(\{1,2,4\})=v(\{1,3,4\})=v(\{2,3,4\})=v(N)=L\) (\(L\in {\mathbb {R}} _{+}\)), and \(v(S)=0\) otherwise. Now, players 1, 2 and 3 form the union \(C_{I}\) and player 4 remains alone, which gives rise to the coalition structure \({\mathcal {C}}=\{C_{I}=\{1,2,3\}, C_{II}=\{4\}\}\).
The above description gives rise to the TU game with coalition structure \((N,v,{\mathcal {C}})\). Note that players 1 and 2 can be considered the “weak” players for union \(C_{I}\), because they cannot generate any worth on their own, no matter if they choose to act alone or cooperate. The corresponding quotient game \((M,v^{{\mathcal {C}}})\) is a two-person TU game where \(M=\{I, II\}\). Hence, it is easy to get thatFollowing Owen’s procedure, we get the induced internal game for union \(C_{I}\), namely \((C_{I}, v_{C_{I}})\), where \(v_{C_{I}}(\{1\})= v_{C_{I}}(\{2\})=-{L}/{2}\), \(v_{C_{I}}(\{3\})={1}/{2}\), \(v_{C_{I}}(\{1,2\})= v_{C_{I}}(\{1,3\})\) \(= v_{C_{I}}(\{2,3\})=0\) and finally, \(v_{C_{I}}(\{1,2,3\})={1}/{2}\). Obviously, players 1 and 2 are symmetric in \((C_{I}, v_{C_{I}})\). For Kamijo’s two-step approach, players 1, 2 and 3 bargain with their union worth based on the restricted subgame \((C_{I}, v_{|C_{I}})\), and the symmetric relationship between player 1 and 2 holds in this subgame as well. Meanwhile, there is no need to consider the intra-bargaining for union \(C_{II}\) since it only contains player 4. Then, we can compute the three coalitional values for the TU game with the coalition structure in Example 1 as shown in Table 1.
$$\begin{aligned} Sh_{I}(M,v^{{\mathcal {C}}})=\frac{1}{2}, \; Sh_{II}\left(M,v^{{\mathcal {C}}}\right)=L-\frac{1}{2}. \end{aligned}$$
Table 1
Three payoff vectors for Example 1
Values | \(TSh(N,v,{\mathcal {C}})\) Kamijo (2009) | \(SS(N,v,{\mathcal {C}})\) Calvo and Gutiérrez (2013) | \(TSS(N,v,{\mathcal {C}})\) |
|---|---|---|---|
Payoffs | \((0,0,\frac{1}{2},L-\frac{1}{2})\) | \((\frac{1}{8}-\frac{L}{24},\frac{1}{8}-\frac{L}{24},\frac{1}{4}+\frac{L}{12},L-\frac{1}{2})\) | \((\frac{1}{12},\frac{1}{12},\frac{1}{3},L-\frac{1}{2})\) |
Player 4 obtains the Shapley value in the quotient game as the final payoff since she forms a union alone, and there exists no difference in her payoff assigned by the three coalitional values. We focus on the payoffs of players in union \(C_{I}\): The symmetry of players 1 and 2 accounts for their same payoff in all three coalitional values. Hence, the payoff difference between them and player 3 directly reflects the level of solidarity of union \(C_{I}\). For this example, the difference is 1/4 within players of union \(C_{I}\) for the two-step Shapley-solidarity value, compared to 1/2 for the two-step Shapley value. We see the same effect also when compared to the Shapley-solidarity value, as long as \(L>1\). Moreover, it turns out that the Shapley-solidarity value has a payoff difference of \({(L+1)}/{8}\) within the players of union \(C_{I}\) which grows linearly in L, even though the subgame within union \(C_{I}\) has worths 0 and 1 only.
4 Procedural characterization of the two-step Shapley-solidarity value
Along the lines of Shapley’s procedural characterization of the Shapley value via average marginal contributions for all n! permutations of players, we here provide a corresponding characterization of the two-step Shapley-solidarity value. First, with the restriction of coalition structure, it is assumed that the grand coalition forms in a consistent permutation, which indicates that the players within the same union enter the grand coalition consecutively. For each \(\pi _{c}\in \Pi _{N,{\mathcal {C}}}\) and \(i\in C_{k}\), we denote by \(p^{\pi _{c}}(N,i)\) and \(p^{\pi _{c}}(C_{k},i)\) the predecessors of player i with respect to N and \(C_{k}\) respectively, i.e., \(p^{\pi _{c}}(N,i)=\{j\in N \, |\, \pi _{c}(j)\le \pi _{c}(i)\}\), \(p^{\pi _{c}}(C_{k},i)=\{j\in C_{k} \,|\, \pi _{c}(j)\le \pi _{c}(i)\}\). The predecessors of a union \(C_{k}\in {\mathcal {C}}\) is denoted by \(p^{\pi _{c}}(N,C_{k})=\{j\in N \, |\, \pi _{c}(j)< \min _{i\in C_{k}}\pi _{c}(i)\}\). In the following, we present a procedure in which the allocation scenario is envisaged to generate the two-step Shapley-solidarity value.
Given a TU game with coalition structure \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), the procedure consists of the following steps:
Step 1
The players enter the grand coalition in a consistent permutation, and all consistent permutations have the same probability.
Step 2
Every entering player \(i\in C_{k}\in {\mathcal {C}}\) joins in and forms the new coalition \(p^{\pi _{c}}(N,i)\). The player brings the marginal contribution \(M^{\pi _{c}}_{i}(N):=v(p^{\pi _{c}}(N,i))-v(p^{\pi _{c}}(N,i)\setminus i)\). With a near-sighted union solidarity principle in mind, the player takes her marginal contribution with respect to the union she belongs to, namely \(M^{\pi _{c}}_{i}(C_{k}):=v(p^{\pi _{c}}(C_{k},i))-v(p^{\pi _{c}}(C_{k},i)\setminus i)\), and splits it equally among her union predecessors \(p^{\pi _{c}}(C_{k},i)\).
Step 3
The residual (negative or positive) brought by player i’s joining, \(M^{\pi _{c}}_{i}(N)-M^{\pi _{c}}_{i}(C_{k})\), is equally shared by the union successors of player i, i.e., \(s^{\pi _{c}}(C_{k},i)=\{j\in C_{k} \, |\, \pi _{c}(j)> \pi _{c}(i)\}\).
Step 4
The last player of a union \(i\in C_{k}\in {\mathcal {C}}\), so when \(|p^{\pi _{c}}(C_{k},i)|=|C_{k} |\), is then to be treated in a special way, and obtains a residual of \(M^{\pi _{c}}_{i}(N)-M^{\pi _{c}}_{i}(C_{k})-v(p^{\pi _{c}}(N,C_{k}))\), which is denoted by \(\gamma ^{\pi _{c}}_{i}\).
As shown in the procedure, each player focuses only on the corresponding union members under the restriction of the coalition structure. Either one player’s marginal contribution or the residual is shared only by the players who are in the same union. This is exactly an embodiment of solidarity within a union. Besides, with the last union member joining in, this union is complete and the last player of the union thereby affords a payment to the union’s predecessors to prevent their coalition’s worth from being infringed. In view of the fact that the last union player has no union successors, a residual of \(M^{\pi _{c}}_{i}(N)-M^{\pi _{c}}_{i}(C_{k})-v(p^{\pi _{c}}(N,C_{k}))\) is shared “by herself”.
In order to show that this procedural description coincides with the two-step Shapley-solidarity value, note that for each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\) and each \(\pi _{c} \in \Pi _{N,{\mathcal {C}}}\), Steps 1–4 determine a payoff \(\psi ^{\pi _{c}}_{i}(N,v,{\mathcal {C}})\) for each \(i\in C_{k} \in {\mathcal {C}}\) as follows:whereandThen, for \(i\in C_{k}\in {\mathcal {C}}\) the procedural outcome is given byNext, we will show the coincidence between the procedural outcome and the two-step Shapley-solidarity value.
$$\begin{aligned} \psi ^{\pi _{c}}_{i}(N,v,{\mathcal {C}})=\left\{ \begin{array}{ll} M^{\pi _{c}}_{i}(C_{k})+\beta ^{\pi _{c}}_{i},\ \, &{} \, \pi _{c}(i)=|p^{\pi _{c}}(N,C_{k})|+1; \\ \frac{M^{\pi _{c}}_{i}(C_{k})}{|p^{\pi _{c}}(C_{k},i)|}+\beta ^{\pi _{c}}_{i}+\alpha ^{\pi _{c}}_{i}, \ \ &{} 1<\pi _{c}(i)-|p^{\pi _{c}}(N,C_{k})|<|C_{k}|; \\ \frac{M^{\pi _{c}}_{i}(C_{k})}{|p^{\pi _{c}}(C_{k},i)|}+\alpha ^{\pi _{c}}_{i}+\gamma ^{\pi _{c}}_{i}, \ \ &{} \pi _{c}(i)=|p^{\pi _{c}}(N,C_{k})|+|C_{k}|, \end{array} \right. \end{aligned}$$
(3)
$$\begin{aligned} \alpha ^{\pi _{c}}_{i}= \sum _{r=|p^{\pi _{c}}(N,C_{k})|+1}^{\pi _{c}(i)-1} \frac{M^{\pi _{c}}_{\pi _{c}^{-1}(r)}(N)-M^{\pi _{c}}_{\pi _{c}^{-1}(r)}(C_{k})}{|p^{\pi _{c}}(N,C_{k})|+|C_{k}|-r}, \end{aligned}$$
$$\begin{aligned} \beta ^{\pi _{c}}_{i}= \sum _{z=\pi _{c}(i)+1}^{|p^{\pi _{c}}(N,C_{k})|+|C_{k}|} \frac{M^{\pi _{c}}_{\pi _{c}^{-1}(z)}(C_{k})}{|p^{\pi _{c}}(C_{k},\pi _{c}^{-1}(z))|}. \end{aligned}$$
$$\begin{aligned} \psi _{i}(N,v,{\mathcal {C}}):=\frac{1}{|\Pi _{N,{\mathcal {C}}}|}\sum _{\pi _{c}\in \Pi _{N,{\mathcal {C}}}}\psi _{i}^{\pi _{c}}(N,v,{\mathcal {C}}). \end{aligned}$$
(4)
Theorem 3
For each TU game with coalition structure \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), the procedural outcome given by Eq.(4), \(\psi (N,v,{\mathcal {C}})\) coincides with the two-step Shapley-solidarity value \(TSS(N,v,{\mathcal {C}})\).
The proof is mainly technical, and can be found in Appendix A.
5 Axiomatizations
As we can see, the two-step Shapley-solidarity value and the two-step Shapley value exactly differ in what principle is agreed to be used in bargaining on each union’s worth. Hence, we next propose axiomatizations to further indicate the precise similarities and differences between these two values.
5.1 Coalitional A-null player
To begin with, we proceed with recalling some axioms in coalition structure setting.
Axiom 1
Efficiency (E). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), \(\sum _{i\in N}\psi _{i}(N,v, {\mathcal {C}})=v(N)\).
Axiom 2
Additivity (A). For \((N,v,{\mathcal {C}}), (N,w,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), \(\psi _{i}(N,v+w,{\mathcal {C}})=\psi _{i}(N,v,{\mathcal {C}})+\psi _{i}(N,w, {\mathcal {C}})\).
Axiom 3
Coalitional symmetry (CS). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), and \(\{h,r\}\subseteq M\), if \(C_{h}\) and \(C_{r}\) are symmetric coalitions in \((N,v,{\mathcal {C}})\), then \(\sum _{i\in C_{h}}\psi _{i}(N,v,{\mathcal {C}})=\sum _{i\in C_{r}}\psi _{i}(N,v,{\mathcal {C}})\).
Axiom 4
Internal equity (IE). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), each \(k\in M\) and \(\{i,j\}\subseteq C_{k}\), if i and j are symmetric players in \((C_{k},v_{|C_{k}})\), then \(\psi _{i}(N,v,{\mathcal {C}})=\psi _{j}(N,v,{\mathcal {C}})\).
Efficiency, additivity and coalition symmetry are standard and most of coalitional values meet these requirements. Coalitional symmetry requires to treat symmetric unions equally. Kamijo (2009) introduces the internal equity axiom to consider the symmetric situation from the player’s level. It states that two players who are judged to be symmetric in the internal situation should be treated equally and thus receive equal payoff. The following coalitional null player axiom is proposed in Kamijo (2009), with which the two-step Shapley value is characterized.
Axiom 5
Coalitional null player (CNP). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), each \(k\in M\) and \(i\in C_{k}\), if i is a null player in (N, v), and k is a dummy player in \((M,v^{{\mathcal {C}}})\), then \(\psi _{i}(N,v,{\mathcal {C}})=0\).
In the statement of the coalitional null player, a null player gets nothing if the union she belongs to is a dummy player in quotient game. Otherwise, it is possible that a null player receives nonzero payoff. Hence, it is not necessarily the case that a zero payoff is given to all null players, which means the null player axiom (Owen 1977) needs not hold. Actually, identifying which kind of players is supposed to get zero payoff or how to deal with the payoff of a null player is one of the key issues in axiomatizations with additivity. This pops up in quite a number of papers which apply variants of the null player axiom to characterize values or coalitional values. For example, the \(\delta\)-reducing player proposed by van den Brink and Funaki (2015), the p-null player proposed by Béal et al. (2017) for a class of solidarity values, two types of null players proposed by Borkotokey et al. (2020) for a class of k-lateral Shapley values, the partial A-null player introduced by Hu and Li (2018) for the Shapley-solidarity value, and the \(\alpha\)-indemnificatory null player introduced by Zou et al. (2020) for the \(\alpha\)-egalitarian Owen value, just to name a few.
In line with these works, we introduce the coalitional A-null player axiom for cooperative games with coalition structure. It states that if a player is an A-null player in the subgame with a player set consisting of her union members, and the union she belongs to is a dummy player in the quotient game, then this player should obtain zero payoff.
Axiom 6
Coalitional A-null player (CANP). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), each \(k\in M\) and \(i\in C_{k}\), if i is an A-null player in \((C_{k},v_{|C_{k}})\), and k is a dummy player in \((M,v^{{\mathcal {C}}})\), then \(\psi _{i}(N,v,{\mathcal {C}})=0\).
With the aid of this axiom, we obtain an axiomatization of the two-step Shapley-solidarity value. Before we give the formal axiomatization, some definitions and lemmas are needed. For \({\mathcal {C}}\in {\mathcal {C}}_{N}\), we firstly define a family of TU games \(\{(N,{\tilde{u}}_{T})\}_{T\in 2^{N}\setminus \emptyset }\) with respect to \({\mathcal {C}}\) as follows.
Given \({\mathcal {C}}=\{C_{1},C_{2},\ldots ,C_{m}\}\), for any \(T\subseteq C_{k}\), \(k\in M\),and if \(T\nsubseteq C_{k}\), \(\forall k\in M\),Note that \((N,{\tilde{u}}_{T})\) is an ordinary unanimity game when \(T\nsubseteq C_{k}\). Next, we show that the family of \(\{(N,{\tilde{u}}_{T})\}_{T\in 2^{N}\setminus \emptyset }\) forms a basis of \({\mathcal {G}}^{N}\).
$$\begin{aligned} {\tilde{u}}_{T}(S)=\left\{ \begin{array}{ll} \left( {\begin{array}{c}s\\ t\end{array}}\right) ^{-1}\cdot \left( {\begin{array}{c}|C_{k}|\\ t\end{array}}\right) , \ \ &{} T\subseteq S \subseteq C_{k}; \\ 1, \ \ &{} T\subseteq S \nsubseteq C_{k}; \\ 0, \ \ &{} T \nsubseteq S, \end{array} \right. \end{aligned}$$
$$\begin{aligned} {\tilde{u}}_{T}(S)=\left\{ \begin{array}{ll} 1, \ \ &{} T\subseteq S; \\ 0, \ \ &{} T\nsubseteq S. \end{array} \right. \end{aligned}$$
Lemma 4
For each \({\mathcal {C}}\in {\mathcal {C}}_{N}\), the family of TU games \(\{(N,{\tilde{u}}_{T})\}_{T\in 2^{N}\setminus \emptyset }\) is a basis of the linear space \({\mathcal {G}}^{N}\).
Proof
It is well-known that \({\mathcal {G}}^{N}\) is a \((2^{n}-1)\)-dimensional linear space. Similar to the spirit of the proof of Lemma 2.2 in Nowak and Radzik (1994), we just show that TU games \(\{(N,{\tilde{u}}_{T})\}_{T\in 2^{N}\setminus \emptyset }\) consist of a set of \(2^{n}-1\) independent vectors in \({\mathcal {G}}^{N}\). To that end, let \(S_{1},S_{2},\cdots ,S_{2^{n}-1}\) be a fixed sequence containing all non-empty set of N such that \(n=|S_{1}|\ge |S_{2}|\ge \cdots \ge |S_{2^{n}-1}|\). Moreover, define a \((2^{n}-1)\times (2^{n}-1)\) matrix \(A=[a_{i,j}]\) whose entries are given byNotice that A is a triangular matrix and its diagonal entries equal \(\left( {\begin{array}{c}|C_{k}|\\ t\end{array}}\right)\) if \(T\subsetneq C_{k}\) \((k\in M)\) and 1 otherwise. Hence, we see that \(\det (A)=\prod \limits _{k=1}^{m}\prod \limits _{\emptyset \ne T \subsetneqq C_{k}}\left( {\begin{array}{c}|C_{k}|\\ t\end{array}}\right) \ne 0\). It follows that vectors \(\{{\tilde{u}}_{T}\}_{T\in 2^{N}\setminus \emptyset }\) are independent, and thus, \(\{(N,{\tilde{u}}_{T})\}_{T\in 2^{N}{\setminus } \emptyset }\) forms a basis of \({\mathcal {G}}^{N}\). This holds for all coalition structures \({\mathcal {C}}\). \(\square\)
$$\begin{aligned} a_{i,j}={\tilde{u}}_{S_{i}}(S_{j}),\; i,j=1,2,\dots ,2^{n}-1. \end{aligned}$$
Then, we have the following, main theorem.
Theorem 5
A coalitional value \(\psi\) on \(\mathcal{CG}\mathcal{}^{N}\) satisfies efficiency, additivity, coalitional symmetry, internal equity, and coalitional A-null player if and only if \(\psi (N,v,{\mathcal {C}})\) is the two-step Shapley-solidarity value.
Proof
Existence. Firstly, we show that the two-step Shapley-solidarity value satisfies the above five axioms. Efficiency and additivity are trivial due to the definition of the two-step Shapley-solidarity value. Coalition symmetry and internal equity can be easily verified since both Shapley value and solidarity value satisfy symmetry. It also turns out to be true that the two-step Shapley-solidarity value satisfies coalitional A-null player axiom, because the solidarity value satisfies the A-null player axiom and the Shapley value assigns a dummy player his stand-alone worth.
Uniqueness. Let \(\psi\) be a coalitional value over \(\mathcal{CG}\mathcal{}^{N}\) which satisfies the five axioms. Lemma 4 immediately implies that, given \({\mathcal {C}}\in {\mathcal {C}}_{N}\), for each \((N,v)\in {\mathcal {G}}^{N}\), there exists \(\{\lambda _{T}\; |\;\lambda _{T}\in {\mathbb {R}}, T\in 2^{N}{\setminus } \emptyset \}\) such that \(v=\sum _{T\in 2^{N}{\setminus } \emptyset }\lambda _{T}{\tilde{u}}_{T}\). According to additivity, it is now sufficient to prove that, for each TU game \((N,\lambda _{T}{\tilde{u}}_{T},{\mathcal {C}})\), \(\psi (N,\lambda _{T}{\tilde{u}}_{T},{\mathcal {C}})\) is uniquely determined by efficiency, coalitional symmetry, internal equity and the coalitional A-null player axiom.
For \({\mathcal {C}}\in {\mathcal {C}}_{N}\) and each \(T\in 2^{N}{\setminus } \emptyset\), denote \(D=\{k\in M\;\vert \;C_{k}\cap T\ne \emptyset \}\). Note that the corresponding quotient game \((M,(\lambda _{T}{\tilde{u}}_{T})^{{\mathcal {C}}})\) for \((N,\lambda _{T}{\tilde{u}}_{T},{\mathcal {C}})\) is equivalent to the unanimity game \((M,\lambda _{T}u_{D})\) since, for each \(T\in 2^{N}{\setminus } \emptyset\), there isfor all \(Q\subseteq M\). Hence, each \(k\notin D\) is a null player in \((M,(\lambda _{T}{\tilde{u}}_{T})^{{\mathcal {C}}})\). Moreover, for each \(k\notin D\), the subgame \((C_{k},(\lambda _{T}{\tilde{u}}_{T})_{|C_{k}})\) is a null game, namely \((\lambda _{T}{\tilde{u}}_{T})_{|C_{k}}(S)=0\) for all \(S\subseteq C_{k}\). By the coalitional A-null player axiom, we have \(\psi _{i}(N,\lambda _{T}{\tilde{u}}_{T},{\mathcal {C}})=0\) for each \(i\in C_{k}\) \((k\notin D)\). For \(k\in D\), there is \(\sum _{i\in C_{k}}\psi _{i}(N,\lambda _{T}{\tilde{u}}_{T},{\mathcal {C}})=\frac{\lambda _{T}}{d}\), which derives from efficiency and coalitional symmetry.
$$\begin{aligned} (\lambda _{T}{\tilde{u}}_{T})^{{\mathcal {C}}}(Q)= \lambda _{T}{\tilde{u}}_{T}(\cup _{k\in Q}C_{k})= \left\{ \begin{array}{ll} \lambda _{T}, \ \ &{} T \subseteq \cup _{k\in Q}C_{k}; \\ 0, \ \ &{} \text{ otherwise }, \end{array} \right. \end{aligned}$$
Now, let us focus on the internal distribution of the payoff that one union obtains from their collective bargaining. For each \(T\in 2^{N}{\setminus } \emptyset\), there is the corresponding D, and we consider the following two cases.
(\(i\))
\(d=1\). Let \(D=\{k\}\), notice that each player \(i\in C_{k}{\setminus } T\) is an A-null player in \((C_{k},(\lambda _{T}{\tilde{u}}_{T})_{|C_{k}})\) since, for each coalition \(S\subseteq C_k\) satisfying \(T\subseteq S\) and \(i\in S\), Besides, k is a dummy player in the quotient game \((M,(\lambda _{T}{\tilde{u}}_{T})^{{\mathcal {C}}})\). By the coalitional A-null player axiom, we have \(\psi _{i}(N,\lambda _{T}{\tilde{u}}_{T},{\mathcal {C}})=0\) for each \(i\in C_{k}\setminus T\). Furthermore, the symmetry of any two players \(i,j\in T\) in subgame \((C_{k},(\lambda _{T}{\tilde{u}}_{T})_{|C_{k}})\) immediately implies \(\psi _{i}(N,\lambda _{T}{\tilde{u}}_{T},{\mathcal {C}})=\frac{\lambda _{T}}{t}\).
$$\begin{aligned} (\lambda _{T}{\tilde{u}}_{T})_{|C_{k}}(S)&= \lambda _{T}\left( {\begin{array}{c}s\\ t\end{array}}\right) ^{-1}\cdot \left( {\begin{array}{c}|C_{k}|\\ t\end{array}}\right) \\&=\lambda _{T}\frac{t!(s-t)!}{s!}\cdot \frac{|C_{k}|!}{t!(|C_{k}|-t)!}\\&=\lambda _{T}\frac{1}{s}\cdot (s-t) \cdot \frac{t!(s-t-1)!}{(s-1)!}\frac{|C_{k}|!}{t!(|C_{k}|-t)!}\\&= \frac{1}{s}\sum _{j\in S}(\lambda _{T}{\tilde{u}}_{T})_{|C_{k}}(S\setminus j). \end{aligned}$$
(\(ii\))
\(d\ge 2\). For each \(C_{k} \;(k\in D)\) and \(\{i,j\} \subseteq C_{k}\), we have \((\lambda _{T}{\tilde{u}}_{T})_{|C_{k}}(S\cup i)=(\lambda _{T}{\tilde{u}}_{T})_{|C_{k}}(S\cup j)\) for each \(S\subseteq C_{k}\setminus \{i,j\}\). Thus, by internal equity, there is \(\psi _{i}(N,\lambda _{T}{\tilde{u}}_{T},{\mathcal {C}})=\frac{\lambda _{T}}{d\cdot |C_{k}|}\) for each \(i\in C_{k}\; (k\in D)\).
Hence, it is clear that \(\psi (N,\lambda _{T}{\tilde{u}}_{T},{\mathcal {C}})\) is unique, which completes the proof. \(\square\)
5.2 Quasi-balanced contributions for grand coalition
This section provides two other axiomatizations which are related to the principle of balanced contributions. The balanced contributions property was firstly proposed by Myerson (1980) to characterize the Shapley value. It requires that any two players must have the same impacts on mutual payoff when one of them departs from the game. Subsequently, Xu et al. (2016) introduce quasi-balanced contributions to verify that the solidarity value is the unique efficient value satisfying this property.
Axiom 7
Quasi-balanced contributions (QBC). For each \((N,v)\in {\mathcal {G}}^{N}\) and \(\{i,j\}\subseteq N\),
$$\begin{aligned} \varphi _{i}(N,v)-\varphi _{i}(N\setminus j,v)+\frac{1}{n}v(N\setminus j)=\varphi _{j}(N,v)-\varphi _{j}(N\setminus i,v)+\frac{1}{n}v(N\setminus i). \end{aligned}$$
Moreover, there are counterparts for the coalitional structure setting to describe the mutual influence of two unions and two players within the same union, namely coalitional balanced contributions and intracoalitional balanced contributions as introduced by Calvo et al. (1996).
Axiom 8
Intracoalitional balanced contributions (IBC). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), and \(i,j\in C_{h}\in {\mathcal {C}}\) with \(i\ne j\),
$$\begin{aligned} \psi _{i}(N,v,{\mathcal {C}})-\psi _{i}(N\setminus j,v_{|N\setminus j},{\mathcal {C}}_{|N\setminus j})= \psi _{j}(N,v,{\mathcal {C}})-\psi _{j}(N\setminus i,v_{|N\setminus i},{\mathcal {C}}_{|N\setminus i}). \end{aligned}$$
Axiom 9
Coalitional balanced contributions (CBC). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), and each \(C_{h}, C_{r}\in C\) with \(r\ne h\),
$$\begin{aligned} \begin{aligned}&\sum _{i\in C_{h}}\psi _{i}(N,v,{\mathcal {C}})-\sum _{i\in C_{h}}\psi _{i}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}}) \\&\quad =\sum _{i\in C_{r}}\psi _{i}(N,v,{\mathcal {C}})-\sum _{i\in C_{r}}\psi _{i}(N\setminus C_{h},v_{|N\setminus C_{h}},{\mathcal {C}}_{|N\setminus C_{h}}). \end{aligned} \end{aligned}$$
Intracoalitional balanced contributions means that given two players in the same union, the amounts that both players gain or lose when the other leaves the game should be equal. Correspondingly, from the perspective of unions, there is the coalitional balanced contributions property. These two axioms together with efficiency give rise to the Owen value (Calvo et al. 1996). Calvo and Gutiérrez (2010) also prove that the two-step Shapley value can be characterized with the CBC axiom, in which there are two other axioms being involved, called population solidarity within unions and coherence.
Axiom 10
Population solidarity within unions (PSU). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), each \(C_{h}, C_{r}\in {\mathcal {C}}\) with \(r\ne h\), and each \(\{i,j,k\}\subseteq N\) with \(\{i,j\}\subseteq C_{h}\) and \(k\in C_{r}\),
$$\begin{aligned} \psi _{i}(N,v,{\mathcal {C}})-\psi _{i}(N\setminus k,v_{|N\setminus k},{\mathcal {C}}_{|N\setminus k})=\psi _{j}(N,v,{\mathcal {C}})-\psi _{j}(N\setminus k,v_{|N\setminus k},{\mathcal {C}}_{|N\setminus k}). \end{aligned}$$
Axiom 11
Coherence (C). For each \((N,v)\in {\mathcal {G}}^{N}\), \(\psi (N,v,{\mathcal {C}}^{N})=\psi (N,v,{\mathcal {C}}^{n})\).
Population solidarity within unions states that players in the same union follow the solidarity principle in such a way that all members in the union experience the same gains or losses when the game changes due to addition or deletion of players outside the union. Coherence means that it is indistinguishable between games in which all players belong to one union and when all of them act as singletons. The following theorem is due to Calvo and Gutiérrez (2010).
Theorem 6
(Calvo and Gutiérrez 2010) The two-step Shapley value is the only value that satisfies efficiency, coalitional balanced contributions, population solidarity within unions and coherence.
Besides, they also introduce the axiom of null coalition which requires null coalitions should get nothing.
Axiom 12
Null coalition (NC). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), and \(k\in M\), if k is a null coalition, then \(\sum _{i\in C_{k}}\psi _{i}(N,v,{\mathcal {C}})=0\).
Then the following theorem holds.
Theorem 7
(Calvo and Gutiérrez 2010) A coalitional value \(\psi\) satisfies efficiency, additivity, coalitional symmetry, null coalitional axiom, population solidarity within unions and coherence if and only if \(\psi (N,v,{\mathcal {C}})=TSh(N,v,{\mathcal {C}})\).
Both Theorems 6 and 7 invoke the coherence axiom which is violated by the two-step Shapley-solidarity value. Next, we will show if we replace the coherence in the above two theorems with a coalitional version of the quasi-balanced contributions for TU games with coalition structures, called the quasi-balanced contributions for the grand coalition, we can get corresponding axiomatizations of the two-step Shapley-solidarity value. First, we formulate the mentioned axiom.
Axiom 13
Quasi-balanced contributions for the grand coalition (QCGC). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\) with \(|{\mathcal {C}}|=1\), and \(i,j\in C_{k}\in {\mathcal {C}}\),
$$\begin{aligned} \begin{aligned}&\psi _{i}(N,v,{\mathcal {C}})-\psi _{i}(N\setminus j,v_{|N\setminus j},{\mathcal {C}}_{|N\setminus j})+\frac{1}{n}v(N\setminus j)\\&\quad =\psi _{j}(N,v,{\mathcal {C}})-\psi _{j}(N\setminus i,v_{|N\setminus i},{\mathcal {C}}_{|N\setminus i})+\frac{1}{n}v(N\setminus i). \end{aligned} \end{aligned}$$
Note that this axiom has exactly the same requirement as the condition for the solidarity value. As we know, the solidarity value can be characterized by quasi-balanced contributions and efficiency. Hence, quasi-balanced contributions for the grand coalition is the corresponding feature for TU games with coalition structure in which the coalition structure is just one union.
Theorem 8
A coalitional value \(\psi\) satisfies efficiency, additivity, coalitional symmetry, null coalitional axiom, population solidarity within unions and quasi-balanced contributions for the grand coalition if and only if \(\psi (N,v,{\mathcal {C}})=TSS(N,v,{\mathcal {C}})\).
Proof
The proof follows the same spirit as the proof of Theorem 4 in Calvo and Gutiérrez (2010) (Theorem 7 above). For clarity, we here restate it in order to highlight the difference.
Existence. It is straightforward to verify that the two-step Shapley-solidarity value satisfies efficiency, additivity, coalitional symmetry, null coalition axiom and the population solidarity within unions. As for the quasi-balanced contributions for the grand coalition, if \({\mathcal {C}}=\{C_1\}=\{N\}\), then \(M=\{1\}\) and \(Sh_{1}(M,v^{{\mathcal {C}}})=v(N)\), and there is \(TSS_{i}(N,v,{\mathcal {C}})=Sol_{i}(N,v)\) for each \(i\in N\). Hence, the two-step Shapley-solidarity value satisfies the quasi-balanced contributions for the grand coalition because the solidarity value satisfies the quasi-balanced contributions.
Uniqueness. Let \(\psi\) be a coalitional value satisfying the above six axioms. Given \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), define value \(\phi\) on \({\mathcal {G}}^{M}\) by, for each \(k\in M\), \(\phi _{k}(M,v^{{\mathcal {C}}})=\sum _{i\in C_{k}}\psi _{i}(N,v,{\mathcal {C}})\).
It turns out that the value \(\phi\) is well-defined by efficiency, additivity, coalitional symmetry and null coalition axiom, and there is \(\phi _{k}(M,v^{{\mathcal {C}}})=Sh_{k}(M,v^{{\mathcal {C}}})\). Thus, when \({\mathcal {C}}={\mathcal {C}}^{n}\), \(\psi _{i}(N,v,{\mathcal {C}}^{n})=\phi _{i}(M,v^{{\mathcal {C}}^{n}})=Sh_{i}(N,v)=TSS_{i}(N,v,{\mathcal {C}}^{n})\). On the other hand, when \({\mathcal {C}}={\mathcal {C}}^{N}\), because \(\psi\) satisfies efficiency and quasi-balanced contributions for the grand coalition, then by Theorem 4.2 in Xu et al. (2016), we can obtain \(\psi _{i}(N,v,{\mathcal {C}}^{N})=Sol_{i}(N,v)=TSS_{i}(N,v,{\mathcal {C}}^{N})\).
Now we focus on the cases when the coalition structure is not trivial. Assume that \(|{\mathcal {C}}|\ge 2\), for each \(\{h,r\}\subseteq M\), each \(i\in C_{h}\) and \(k\in C_{r}\), according to population solidarity within unions, there exists \(\gamma _{h}\in {\mathbb {R}}\) such that \(\psi _{i}(N,v,{\mathcal {C}})-\psi _{i}(N{\setminus } k,v_{|N{\setminus } k},{\mathcal {C}}_{|N{\setminus } k})=\gamma _{h}\), and hence, for each \(i\in C_{h}\),Using the population solidarity within unions repeatedly until only \(C_{h}\) is in the game, we obtainfor each \(i\in C_{h} \in {\mathcal {C}}\). Hence, there is \(\psi (N,v,{\mathcal {C}})=TSS(N,v,{\mathcal {C}})\) for each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), which completes the proof. \(\square\)
$$\begin{aligned} \psi _{i}(N,v,{\mathcal {C}})=\psi _{i}(N\setminus k,v_{|N\setminus k},{\mathcal {C}}_{|N\setminus k})+\frac{1}{|C_{h}|}[Sh_{h}(M,v^{{\mathcal {C}}}) -Sh_{h}(M,v^{{\mathcal {C}}_{|N\setminus k}})]. \end{aligned}$$
$$\begin{aligned} \psi _{i}(N,v,{\mathcal {C}})&= \psi _{i}(C_{h},v_{|C_{h}},{\mathcal {C}}_{|C_{h}})+\frac{1}{|C_{h}|}[Sh_{h}(M,v^{{\mathcal {C}}}) -Sh_{h}(\{h\},v^{{\mathcal {C}}_{|C_{h}}})] \\&= Sol_{i}(C_{h},v_{|C_{h}})+ \frac{1}{|C_{h}|}[Sh_{h}(M,v^{{\mathcal {C}}}) -v(C_{h})]\\&= TSS_{i}(N,v,{\mathcal {C}}) \end{aligned}$$
Theorem 9
A coalitional value \(\psi\) satisfies efficiency, coalitional balanced contributions, population solidarity within unions and quasi-balanced contributions for the grand coalition if and only if \(\psi (N,v,{\mathcal {C}})=TSS(N,v,{\mathcal {C}})\).
Proof
Existence. It is left to show the two-step Shapley-solidarity value satisfies the coalitional balanced contribution. By definition, for each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\) and each \(C_{h}, C_{r}\in C\) with \(r\ne h\), \(\sum _{i\in C_{h}}TSS_{i}(N,v,{\mathcal {C}})=Sh_h(M,v^{{\mathcal {C}}})\) and \(\sum _{i\in C_{r}}TSS_{i}(N,v,{\mathcal {C}})=Sh_r(M,v^{{\mathcal {C}}})\). Hence, the coalitional balanced contributions of the two-step Shapley-solidarity value immediately follows from the balanced contributions of the Shapley value (Myerson 1980).
Uniqueness. Let \(\psi\) be a coalitional value satisfying the above four axioms. We show \(\psi (N,v,{\mathcal {C}})=TSS(N,v,{\mathcal {C}})\) for all \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^N\) by induction on \(|{\mathcal {C}}|\).
Let \(|{\mathcal {C}}|=1\). This means that the coalition structure is trivial and \({\mathcal {C}}={\mathcal {C}}^N\). Given a TU game with coalition structure \((N,v,{\mathcal {C}}^N)\in \mathcal{CG}\mathcal{}^N\), quasi-balanced contributions for the grand coalition together with efficiency implies \(\psi _{i}(N,v,{\mathcal {C}}^N)=Sol_i(N,v)\) for all \(i\in N\). Hence, we have \(\psi (N,v,{\mathcal {C}})=TSS(N,v,{\mathcal {C}})\) for \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^N\) with \(|{\mathcal {C}}|=1\).
Now, assume \(\psi (N,v,{\mathcal {C}})=TSS(N,v,{\mathcal {C}})\) holds for all TU games with coalition structure \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^N\) when \(|{\mathcal {C}}|\le m\), we prove \(\psi (N,v,{\mathcal {C}})=TSS(N,v,{\mathcal {C}})\) can also be established for \((N,v,{\mathcal {C}})\) with \(|{\mathcal {C}}|=m+1\).
Let \((N,v,{\mathcal {C}})\) be a TU game with coalition structure where \(|{\mathcal {C}}|=m+1\). Since both \(\psi\) and TSS satisfy CBC, we haveandMoreover, according to the induction hypothesis, we haveThe above three equations yieldfor all \(C_{h},C_{r}\in {\mathcal {C}}\). Then, fixing h in the left part in the above equation and summing over \(r\in M\) of the right, we haveCombining with efficiency, we getfor all \(h\in M\).
$$\begin{aligned} \begin{aligned}&\sum _{i\in C_{h}}\psi _{i}(N,v,{\mathcal {C}})-\sum _{i\in C_{r}}\psi _{i}(N,v,{\mathcal {C}})\\&\quad =\sum _{i\in C_{h}}\psi _{i}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}})-\sum _{i\in C_{r}}\psi _{i}(N\setminus C_{h},v_{|N\setminus C_{h}},{\mathcal {C}}_{|N\setminus C_{h}}), \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned}&\sum _{i\in C_{h}}TSS_{i}(N,v,{\mathcal {C}})-\sum _{i\in C_{r}}TSS_{i}(N,v,{\mathcal {C}})\\&\quad =\sum _{i\in C_{h}}TSS_{i}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}})-\sum _{i\in C_{r}}TSS_{i}(N\setminus C_{h},v_{|N\setminus C_{h}},{\mathcal {C}}_{|N\setminus C_{h}}). \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned}&\sum _{i\in C_{h}}\psi _{i}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}})-\sum _{i\in C_{r}}\psi _{i}(N\setminus C_{h},v_{|N\setminus C_{h}},{\mathcal {C}}_{|N\setminus C_{h}})\\&\quad =\sum _{i\in C_{h}}TSS_{i}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}})-\sum _{i\in C_{r}}TSS_{i}(N\setminus C_{h},v_{|N\setminus C_{h}},{\mathcal {C}}_{|N\setminus C_{h}}). \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned}&\sum _{i\in C_{h}}\psi _{i}(N,v,{\mathcal {C}})-\sum _{i\in C_{h}}TSS_{i}(N,v,{\mathcal {C}})=\sum _{i\in C_{r}}\psi _{i}(N,v,{\mathcal {C}})-\sum _{i\in C_{r}}TSS_{i}(N,v,{\mathcal {C}}), \end{aligned} \end{aligned}$$
$$\begin{aligned}&|M |\left( \sum _{i\in C_{h}}\psi _{i}(N,v,{\mathcal {C}})-\sum _{i\in C_{h}}TSS_{i}(N,v,{\mathcal {C}})\right) \\&\quad =\sum _{r\in M}\sum _{i\in C_{r}}\psi _{i}(N,v,{\mathcal {C}})-\sum _{r\in M}\sum _{i\in C_{r}}TSS_{i}(N,v,{\mathcal {C}}). \end{aligned}$$
$$\begin{aligned} \sum _{i\in C_{h}}\psi _{i}(N,v,{\mathcal {C}})=\sum _{i\in C_{h}}TSS_{i}(N,v,{\mathcal {C}}), \end{aligned}$$
(5)
Then, it remains to show \(\psi _{i}(N,v,{\mathcal {C}})=TSS_{i}(N,v,{\mathcal {C}})\) for all \(i\in C_h\in {\mathcal {C}}\). This can be obtained by induction on \(|C_h|\). Given \(C_h\in {\mathcal {C}}\) with \(|C_h|=1\), Eq.(5) yields \(\psi _{i}(N,v,{\mathcal {C}})=TSS_{i}(N,v,{\mathcal {C}})\) for \(\{i\}=C_h\). We now assume \(|C_h|\ge 2\). For each \(C_h,C_r\in {\mathcal {C}}\), and \(\{i,j\}\subseteq C_{h}\), by repeatedly using PSU on \(\psi\) and TSS until the players within union \(C_r\) are ruled out, we haveandAgain, by induction hypothesis, we haveHence, there isThen, fixing i and summing over \(j\in C_h\), we obtainBy Eq.(5), we conclude that \(\psi _{i}(N,v,{\mathcal {C}})=TSS_{i}(N,v,{\mathcal {C}})\) for all \(i\in C_h\in {\mathcal {C}}\), which completes the proof. \(\square\)
$$\begin{aligned} \psi _{i}(N,v,{\mathcal {C}})-\psi _{j}(N,v,{\mathcal {C}})=\psi _{i}(N\setminus C_{r},v_{|{N\setminus C_{r}}},{\mathcal {C}}_{|{N\setminus C_{r}}})-\psi _{j}(N\setminus C_{r},v_{|{N\setminus C_{r}}},{\mathcal {C}}_{|{N\setminus C_{r}}}), \end{aligned}$$
$$\begin{aligned}&TSS_{i}(N,v,{\mathcal {C}})-TSS_{j}(N,v,{\mathcal {C}})\\&\quad = TSS_{i}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}})-TSS_{j}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}}). \end{aligned}$$
$$\begin{aligned} \psi _{i}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}})= & {} TSS_{i}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}}),\\ \psi _{j}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}})= & {} TSS_{j}(N\setminus C_{r},v_{|N\setminus C_{r}},{\mathcal {C}}_{|N\setminus C_{r}}). \end{aligned}$$
$$\begin{aligned} \psi _{i}(N,v,{\mathcal {C}})-TSS_{i}(N,v,{\mathcal {C}})=\psi _{j}(N,v,{\mathcal {C}})-TSS_{j}(N,v,{\mathcal {C}}). \end{aligned}$$
$$\begin{aligned} |C_h|(\psi _{i}(N,v,{\mathcal {C}})-TSS_{i}(N,v,{\mathcal {C}}))=\sum _{j\in C_h}(\psi _{j}(N,v,{\mathcal {C}})-TSS_{j}(N,v,{\mathcal {C}})). \end{aligned}$$
6 Conclusion
The two-step Shapley-solidarity value is in our opinion a conceptually simple value for cooperative games with coalition structure that captures the solidarity concept within unions. The given axiomatizations exactly pinpoint this and show similarities and also the subtle difference when compared to the two-step Shapley value as defined by Kamijo (2009). The given Example 1 also highlights the difference to the two closest relatives, but of course, other examples can be constructed to show opposite effects, too. It is an interesting question for further research to find subclasses of games to turn the “empirical” observations of Example 1 into a firm theorem. In this context, observe that for anonymous games where \(v(S)=|S|\), and a specific class of simple games, namely when \(v(S)=1\) for all \(|S|\ge 2\) and \(v(S)=0\) otherwise, all three values that we defined in Table 1 are identical. Moreover, for additive games where \(v(S)=\sum _{i\in S}v(i)\), the two-step Shapley value is given by the stand-alone worth of the players, so \(TSh_i(N,v)=v(i)\), and the Shapley-solidarity value and two-step Shapley-solidarity value are identical, but different from the former.
There is another coalitional value which is akin to our proposed value and the two-step Shapley value. It can be obtained by using the solidarity value for both, the induced internal game and the quotient game in Kamijo’s two step approach. Formally, it is given by, for each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\) and \(i\in C_{k}\in {\mathcal {C}}\),Let us call it the two-step solidarity value.
$$\begin{aligned} TS_{i}(N,v,{\mathcal {C}})=Sol_{i}(C_{k},v_{|C_{k}})+\frac{Sol_{k}(M,v^{{\mathcal {C}}})-v(C_{k})}{|C_{k}|}. \end{aligned}$$
By definition, this value supports a higher degree of solidarity among unions compared to the other two values, due to the fact that the revenue distribution among unions is by the solidarity value instead of the Shapley value. When applying the two-step solidarity value to the game in Example 1, one obtains the payoff vector \(TS(N,v,{\mathcal {C}})=(\frac{L}{12},\frac{L}{12},\frac{L}{12}+\frac{1}{4},\frac{3L}{4}-\frac{1}{4})\). Although this value leads to the same difference within players of union \(C_{I}\) as the two-step Shapley-solidarity value does, it has a smaller difference of \(\frac{L-1}{2}\) between the two unions, compared to the difference \(L-1\) in the two-step Shapley-solidarity value.
This difference can be confirmed from the axiomatic perspective as well. With a modification of the axiom of CANP, we obtain an axiom based on A-dummy players. Here, a player \(i\in N\) is called an A-dummy player if \(\frac{1}{s}\sum _{j\in S}(v(S)-v(S{\setminus } j))=v({i})\) for all \(S\subseteq N\) with \(i\in S\).
Axiom 14
Coalitional A-dummy player (CADP). For each \((N,v,{\mathcal {C}})\in \mathcal{CG}\mathcal{}^{N}\), each \(k\in M\) and \(i\in C_{k}\), if i is an A-null player in \((C_{k},v_{|C_{k}})\), and k is an A-dummy player in \((M,v^{{\mathcal {C}}})\), then \(\psi _{i}(N,v,{\mathcal {C}})=0\) and \(\sum _{i\in C_k}\psi _{i}(N,v,{\mathcal {C}})=v(C_k)\).
CADP replaces the condition for dummy players of CANP with A-dummy players, and requires the total payoff for a union which is an A-dummy player in the quotient game being equal to the union’s worth. Then, the two-step solidarity value can be axiomatized by E, A, CS, IE and CADP. The proof is similar to the proof of Theorem 5, except that we work with yet another basis for the set of TU games, namely \(\{(N,{\tilde{w}}_{T})\}_{T\subseteq 2^{N}\setminus \emptyset }\), with respect to coalition structure \({\mathcal {C}}\), where \({\tilde{w}}_{T}\) is defined by, for any \(T\subseteq C_{k}\), \(k\in M\),andfor all \(T\nsubseteq C_{k}\) and all \(k\in M\). Here, recall that \({\mathcal {C}}_{|S}\) is the restriction of \({\mathcal {C}}\) on the player set S for all \(S\subseteq N\), namely \({\mathcal {C}}_{|S}=\{C_{h}\cap S\, |\,C_{h}\in {\mathcal {C}} \; \text {and} \; C_{h}\cap S\ne \emptyset \}\).
$$\begin{aligned} {\tilde{w}}_{T}(S)=\left\{ \begin{array}{ll} \left( {\begin{array}{c}s\\ t\end{array}}\right) ^{-1}\cdot \left( {\begin{array}{c}|C_{k}|\\ t\end{array}}\right) , \ \ &{} T\subseteq S \subseteq C_{k}; \\ \frac{|{\mathcal {C}}_{|S}|+1}{2}, \ \ &{} T\subseteq S \nsubseteq C_{k}; \\ 0, \ \ &{} T \nsubseteq S, \end{array} \right. \end{aligned}$$
$$\begin{aligned} {\tilde{w}}_{T}(S)=\left\{ \begin{array}{ll} \left( {\begin{array}{c}|{\mathcal {C}}_{|S}|\\ |{\mathcal {C}}_{|T}|\end{array}}\right) ^{-1}, \ \ &{} T\subseteq S; \\ 0, \ \ &{} T\nsubseteq S, \end{array} \right. \end{aligned}$$
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 71671140, 72071159), and the China Scholarship Council (Grant No. 202006290073). We thank the anonymous reviewers for their helpful comments and the suggestion to also discuss the two-step solidarity value.
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