Abstract
This paper is devoted to a class of Social-Shapley values for cooperative games with graph structure. The Social-Shapley value compromises the utilitarianism of the Shapley value and the egalitarianism of the Solidarity value, in which the sociality is reflected by the Solidarity value. Through defining the corresponding properties in graph-restricted games, the paper axiomatically characterizes the Social-Shapley value when the coefficient is given exogenously. Moreover, we axiomatize the class of all possible Social-Shapley values in the graph-restricted games.
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Acknowledgement
This research has been supported by the National Natural Science Foundation of China (Grant No. 71571143), the Science and Technology Research and Development Program in Shaanxi Province of China (Grant Nos. 2017GY-095, 2017JM5147).
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Appendix
Appendix
1.1 Proof of Lemma 3
Given \(i\in N\), let \(\langle v,L\rangle =\mathcal {G}^0_N(i)\bigcap \mathcal {G}^1_N(i)\), then for each coalition \(S\in C^L_{N \backslash \{i\}}\), we get \(v^L(S\cup \{i\})=v^(S)\) and \(\sum _{k\in S\cup \{i\}}[v^{L}(S\cup \{i\})-v^L(S\cup \{i\}\backslash \{k\})]=0\). Thus, the following equation holds,
In this equation, let \(S=\{j\}\), where \(j\ne i\), we get \(v^L(\{i,j\})=v^L(\{i\})\). By \(\langle v,L\rangle \in \mathcal {G}^0_N(i)\), we deduce \(v^L(\{i\})=0\) and \(v^L(\{i,j\})=v^L(\{j\})\). Therefore, \(v^L(\{j\})=v^L(\{i,j\})=v^L(\{i\})=0\).
Using mathematical induction on the cardinality of S, we can verify that \(v^L(S)=v^L(S\cup \{i\})=0\) exists for each \(S\in C^L_{N\backslash \{i\}}\), which means \(\langle v,L\rangle \) is the null game, hence \(\mathcal {G}^0_N(i)\bigcap \mathcal {G}^1_N(i)=\emptyset \).
Note that the dimension of the space \(\mathcal {G}_N(i)\) equals \(2^n-2\). According to \(\mathcal {G}^0_N(i)\subset \mathcal {G}_N(i)\), \(\mathcal {G}^1_N(i)\subset \mathcal {G}_N(i)\), it remains to prove that the dimensions of the linear space \(\mathcal {G}^0_N(i)\) and \(\mathcal {G}^1_N(i)\) are at least \(2^{n-1}-1\), respectively. Define two classes of games as
where \(\langle w_T,L\rangle \) is defined as in Sect. 3 and
Clearly, both \(\mathcal {B}^0_N(i)\) and \(\mathcal {B}^1_N(i)\) are composed of a set of linearly independent games.
Besides, since \(\mathcal {B}^0_N(i)\subset \mathcal {G}^0_N(i)\), \(\mathcal {B}^1_N(i)\subset \mathcal {G}^1_N(i)\) and the cardinalities of \(\mathcal {B}^0_N(i)\) and \(\mathcal {B}^1_N(i)\) are respectively \(2^{n-1}-1\), we can deduce the cardinalities of \(\mathcal {G}^0_N(i)\) and \(\mathcal {G}^1_N(i)\) are respectively \(2^{n-1}-1\). \(\square \)
1.2 Proof of Theorem 2
It is easy to check that \(\varPhi ^\lambda (v,L)\) satisfies component efficiency, additivity, symmetry, rationality, proportionality of Shapley value and proportionality of Solidarity value, the uniqueness will be proved as follow:
Step 1. We will prove for each \(i\in N\), there exists \(\lambda _i\in \mathbb {R}\), so that the following equality establish, \(\langle v,L\rangle \in \mathcal {G}^L_N\),
Choose any game \(\langle w,L\rangle \in \mathcal {B}^1_N(i)\subset \mathcal {G}^1_N(i)\) such that \(Sh_i(w,L)>0\), let \(\lambda _i=\frac{\varPhi _i(w,L)}{Sh_i(w,L)}\), by proportionality of Shapley value, for all \(\langle v,L\rangle \in \mathcal {G}^1_N(i)\), we have
Similarly, choose any game \(\langle u,L\rangle \in \mathcal {B}^0_N(i)\subset \mathcal {G}^0_N(i)\) such that \(Sol_i(u,L)>0\), let \(\gamma _i=\frac{\varPhi _i(u,L)}{Sol_i(u,L)}\), combining proportionality of Solidarity value, for each \(\langle v,L\rangle \in \mathcal {G}^0_N(i)\), the following equality holds:
According to the definitions of \(\mathcal {G}^0_N(i)\) and \(\mathcal {G}^1_N(i)\), given \(\langle v,L\rangle \in \mathcal {G}^0_N(i)\), we have
Considering game \(\langle w,L\rangle \in \mathcal {G}^L_N\), where
by component efficiency and symmetry, it follows \(\varPhi _i(w,L)=1\), \(i\in N\). Similarly, it follows that
In the Eq. (4), let \(\langle w,L\rangle =\langle v,L\rangle \), we get \(\gamma _i=1-\lambda _i\). Consequently, the Eq. (3) holds for any \(\langle v,L\rangle \in \mathcal {G}_N(i)\).
Now, defining a game \(\langle u,L\rangle \in \mathcal {G}^L_N\) as follow:
Obviously \(\langle u,L\rangle \notin \mathcal {G}_N(i)\).
Note that as linear space, the dimensions of \(\mathcal {G}^L_N\) and \(\mathcal {G}_N(i)\) are \(2^n-1\) and \(2^n-2\) respectively, hence, any \(\langle v,L\rangle \in \mathcal {G}^L_N\) can be represented as follow
where \(\langle v^0,L\rangle \in \mathcal {G}_N(i)\) and c is a constant coefficient. According to component efficiency, additivity, symmetry, it follows that
In addition, we have
Consequently, Eq. (3) holds for any \(\langle v,L\rangle \in \mathcal {G}^L_N\).
Step 2. We will prove that coefficient \(\lambda _i\) in Eq. (3) is independent of i. Assuming that \(n\ge 2\), for \(\langle v,L\rangle \in \mathcal {G}^L_N\), component efficiency implies:
By component efficiency of Solidarity value, this equation is equivalent to
By component efficiency of the Shapley and Solidarity value, we have
This means for \(\langle v,L\rangle \in \mathcal {G}^L_N\),
where \(\gamma _i=\lambda _i-\lambda _1\), \(i=2,3,\ldots ,n\).
Defining games \(\langle v_k,L\rangle \), \(k=2,3,\ldots ,n\), as follow
According to Eq. (5),
and
we get the linear system
The only solution of the above linear system is \(\gamma _2=\gamma _3=\cdots ,\gamma _n=0\).
Then for every \(i\in N\), it implies \(\lambda _i-\lambda _1=0\), \(i=2,3,\ldots , n\). Now we have proved that coefficient \(\lambda _i\) is independent of i and \(\lambda _i=\lambda _1=\lambda \), \(i=2,3,\ldots , n\).
Step 3. The coefficient \(\lambda \) will be proved belonging to interval [0, 1] in this step.
Assuming that \(n\ge 2\), for any player \(i\in N\), considering the following unanimity games:
By applying the property of rationality to \(\langle u_{N\backslash \{i\}},L\rangle \), we obtain \(\varPhi _i(u_{N\backslash \{i\}},L)\ge 0\). In addition, combining the following equations:
we have
Evidently, we deduce that \(\lambda \le 1\).
Considering the game \(\langle \bar{w},L\rangle \), which is defined as follow,
Applying the property of rationality to \(\langle \bar{w},L\rangle \), we get \(\varPhi _i(\bar{w},L)\ge 0\). By the conditions:
we have
which implies \(\lambda \ge 0\). Then for \(n\ge 2\), we have proven the conclusion. When \(n=1\), the theorem is obviously established. \(\square \)
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Yang, H., Sun, H., Xu, G. (2019). A Class of Social-Shapley Values of Cooperative Games with Graph Structure. In: Li, DF. (eds) Game Theory. EAGT 2019. Communications in Computer and Information Science, vol 1082. Springer, Singapore. https://doi.org/10.1007/978-981-15-0657-4_3
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DOI: https://doi.org/10.1007/978-981-15-0657-4_3
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