The weighted-egalitarian Shapley values

Takaaki Abe; Satoshi Nakada

doi:10.1007/s00355-018-1143-3

Social Choice and Welfare

February 2019, Volume 52, Issue 2, pp 197–213 | Cite as

The weighted-egalitarian Shapley values

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Takaaki Abe
Satoshi Nakada

Original Paper

First Online: 24 July 2018

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Abstract

We propose a new class of allocation rules for cooperative games with transferable utility (TU-games), weighted-egalitarian Shapley values, where each rule in this class takes into account each player’s contributions and heterogeneity among players to determine each player’s allocation. We provide an axiomatic foundation for the rules with a given weight profile (i.e., exogenous weights) and the class of rules (i.e., endogenous weights). The axiomatization results illustrate the differences among our class of rules, the Shapley value, the egalitarian Shapley values, and the weighted Shapley values.

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Appendix A: Proof of Theorem 2

Let $\mathcal {G}^{c} \subseteq \mathcal {G}$ denote the set of games such that $v(N)=c$. Also, for any player $i\in N$ and $c\in \mathbb {R}$, let $\mathcal {G}^{c,i}$ denote the set of games v such that $v(N)=c$ and i is null player. Note that $\mathcal {G}^{c,i} \subseteq \mathcal {G}^{c}$ for all $i\in N$ and $c\in \mathbb {R}$. Let $\varDelta _i(v)=(v(S\cup \{i\})-v(S))_{S\subseteq N{\setminus } \{i\}} \in \mathbb {R}^{2^{(N-1)}}$ be a vector of marginal contributions of i in v. Therefore player $i\in N$ is a null player in v if $\varDelta _i(v)= \mathbf{0}$. Let $\varLambda ^i$ be the set of all vectors of marginal contribution of i: $\varLambda ^i=\{ \varDelta _i(v) | v\in \mathcal {G}\}$.

For each $x \in \mathbb {R}^N$, let $m_x \in \mathcal {G}$ be an additive game, $m_x(S)=\sum _{i \in S}x_i$ for all $S \subseteq N$. Let $\mathcal {G}^{add}$ be the set of additive games. Since there is a one-to-one correspondence between $x \in \mathbb {R}^N$ and an additive game $m_x$, we can identify $\mathcal {G}^{add}$ with $\mathbb {R}^N$. Abe and Nakada (2017) provide the following result, which will be useful later.

Theorem A.1

(Abe and Nakada 2017) Let $n \ne 2$. $f:\mathcal {G}^{add} \rightarrow \mathbb {R}^n$ satisfies (E), ($\mathrm {M}^{-}$), (NY), and (RIN) if and only if there exists some $\delta \in [0,1]$ and $w \in \mathcal {W}$ such that $f_i(x)=\delta \cdot x_i+(1-\delta ) \cdot w_i \cdot \sum _{l \in N}x_l$ for all $x \in \mathbb {R}^n$ and $i \in N$.

Now, we offer the proof of Theorem 2. It is clear that the rule satisfies all the axioms. We suppose that a rule $f: \mathcal {G}\rightarrow \mathbb {R}^N$ satisfies (E), ($\mathrm {M}^{-}$), (RIN), (WDMSP), and (NY).

Claim 1

For each $i \in N$, there exist functions $\phi _i: \varLambda ^i \times \mathbb {R}\rightarrow \mathbb {R}$ and $\alpha _i: \mathbb {R}\rightarrow \mathbb {R}$ such that $f_i(v)=\phi _i(\varDelta _i(v), v(N))+\alpha _i(v(N))$.

We first take any $c\in \mathbb {R}$. For any $i\in N$ and $v \in \mathcal {G}^c$, we have the following equation: for any $\bar{v}\in \mathcal {G}^c$ such that $\varDelta _i(v)=\varDelta _i(\bar{v})$,

$$\begin{aligned} f_i(v)\overset{\text {(M}^-)}{=}f_i(\bar{v})\ =:\alpha _i(c,\varDelta _i(v)). \end{aligned}$$

(A.1)

Specifically, we denote

$$\begin{aligned} \alpha _i(c)=\alpha _i(c,\mathbf{0}). \end{aligned}$$

(A.2)

Moreover, for any $i\in N$ and $v,v' \in \mathcal {G}^c$, we have

$$\begin{aligned}&f_i(v)-f_i(v') \overset{(A.1)}{=} \alpha _i(c,\varDelta _i(v))-\alpha _i(c,\varDelta _i(v'))\nonumber \\&\quad =: \phi _i(\varDelta _i(v),\varDelta _i(v'),c). \end{aligned}$$

(A.3)

Hence, for any $i\in N$ and $v \in \mathcal {G}^c$, we obtain the following equation: for any $v' \in \mathcal {G}^{c,i}$,

$$\begin{aligned} \phi _i(\varDelta _i(v),\varDelta _i(v'),c) \overset{(A.3)}{=} f_i(v)-f_i(v') \overset{(A.1)}{=} f_i(v)-\alpha _i(c). \end{aligned}$$

(A.4)

Note that $f_i(v)-\alpha _i(c)$ is independent from $v' \in \mathcal {G}^{c,i}$. For any $i\in N$ and $v \in \mathcal {G}^c$ let

$$\begin{aligned} \phi _i(\varDelta _i(v),c):=f_i(v)-\alpha _i(c). \end{aligned}$$

(A.5)

Hence, for any $i\in N$ and $v \in \mathcal {G}$, we obtain

$$\begin{aligned} f_i(v)\overset{(A.5)}{=}\phi _i(\varDelta _i(v),v(N))+\alpha _i(v(N)). \end{aligned}$$

(A.6)

This completes Claim 1.

Claim 2

For any $i\in N$, and $c \in \mathbb {R}$, the function $\phi _i(\cdot , c): \varLambda ^i \rightarrow \mathbb {R}$ satisfies (M) within $\mathcal {G}^c$: for any $v,v'\in \mathcal {G}^c$, if $v(S\cup \{i\})-v(S)\ge v'(S\cup \{i\})-v'(S)$ for any $S\subset N{\setminus } \{i\}$, then $\phi (\varDelta _i(v), c)\ge \phi (\varDelta _i(v'), c)$. Moreover, $\phi _i(\mathbf{0},c)=0$.

Let $c=v(N)=v'(N)$. We have

$$\begin{aligned} \phi (\varDelta _i(v),c)- \phi (\varDelta _i(v'),c)&= \phi (\varDelta _i(v),c) + \alpha _i(c)- (\phi (\varDelta _i(v'),v'(N))+\alpha _i(c))\\&\overset{\text {C1}}{=} f_i(v) - f_i(v')\\&\overset{\text {(M}^-)}{\ge } 0. \end{aligned}$$

Moreover, for any $c\in \mathbb {R}$,

$$\begin{aligned} \phi _i(\mathbf{0},c) \overset{(A.5),(A.4)}{=}\phi _i(\mathbf{0},\mathbf{0},c) \overset{(A.3),(A.1)}{=}\alpha _i(c,\mathbf{0})-\alpha _i(c,\mathbf{0})=0. \end{aligned}$$

(A.7)

Claim 3

The function $\phi $ is symmetric: for any $v \in \mathcal {G}$ and $i, j \in N$, if i, j is symmetric in v, then $\phi _i(\varDelta _i(v), v(N))=\phi _j(\varDelta _j(v), v(N))$.

For any $i,j\in N$ and $v \in \mathcal {G}$ such that $v(S \cup \{i\})-v(S) = v(S \cup \{j\})-v(S)$ for all $S \subseteq N{\setminus }\{i,j\}$, let $v'=v(N)u_{N \backslash \{i,j\}}$. Then, we have

$$\begin{aligned} \phi _i(\varDelta _i(v), v(N)) \overset{(A.6)}{=} f_i(v)-f_i(v') \overset{{\mathrm{(WDMSP)}}}{=}f_j(v)-f_j(v') \overset{(A.6)}{=} \phi _j(\varDelta _j(v), v(N)). \end{aligned}$$

(A.8)

This completes Claim 3.

Claim 4

The function $\phi $ satisfies $\delta $-efficiency ($\delta $-E): there is a $\delta \in [0,1]$ such that

$\sum _{i \in N}\phi _i(\varDelta _i(v), v(N))=\delta v(N)$ for any $v \in \mathcal {G}$.

Let ${\tilde{f}}: \mathcal {G}^{add} \rightarrow \mathbb {R}$ be the restriction of f on $\mathcal {G}^{add}$. Then, by Theorem A.1, for each $m_x \in \mathcal {G}^{add}$, we have

$$\begin{aligned} {\tilde{f}}_i(m_x)= & {} \phi _i(\varDelta _i(m_x), \sum _{l \in N}x_l)+\alpha _i\left( \sum _{l \in N}x_l\right) \nonumber \\= & {} \delta \cdot x_i+(1-\delta ) \cdot w_i \cdot \sum _{l \in N}x_l. \end{aligned}$$

(A.9)

for some $\delta \in [0,1]$ and $w \in \mathcal {W}$. In particular, for $x_i=0$, (A.7) implies $\alpha _i(\sum _{l \in N}x_l)$$= (1-\delta ) \cdot w_i \cdot \sum _{l \in N}x_l$. Hence, from Claim 1, it follows that for each $v \in \mathcal {G}$,

$$\begin{aligned} f_i(v)=\phi _i(\varDelta _i(v), v(N))+(1-\delta ) \cdot w_i \cdot v(N). \end{aligned}$$

(A.10)

Since f satisfies (E), $\sum _{i \in N}f_i(v)=\sum _{i \in N}\phi _i(\varDelta _i(v), v(N))+(1-\delta ) v(N)=v(N)$, which implies that $\phi $ satisfies the following property:

$$\begin{aligned} (\delta -\text {E}): \sum _{i \in N}\phi _i(\varDelta _i(v), v(N))=\delta v(N). \end{aligned}$$

This completes Claim 4.

Claim 5

There is a $\delta \in [0,1]$ and a $w \in \mathcal {W}$ such that $f_i(x)=\delta \cdot Sh_i(v)+(1-\delta ) \cdot w_i \cdot v(N)$ for any $v \in \mathcal {G}$.

Fixing $c \in \mathbb {R}$, we write $\phi _i^c(v)=\phi _i(\varDelta _i(v), c)$ for each $v \in \mathcal {G}^c$. The function $\phi ^c(v): \mathcal {G}^c \rightarrow \mathbb {R}^N$ satisfies ($\delta $-E), (M) and (SYM) within $\mathcal {G}^c$ by Claims 2, 3 and 4. Hence, by the same argument of Theorem 1, we have

$$\begin{aligned} \phi _i^c(v)=\delta Sh_i(v). \end{aligned}$$

Since $c \in \mathbb {R}$ is arbitrarily chosen, for any $v\in \mathcal {G}$, we have

$$\begin{aligned} \phi _i(\varDelta _i(v), v(N))=\phi _i^{v(N)}(v)=\delta Sh_i(v) \end{aligned}$$

(A.11)

Finally, by (A.10) and (A.11), we obtain $f_i(x)=\delta \cdot Sh_i(v)+(1-\delta ) \cdot w_i \cdot v(N)$, which completes the proof. $\square $

Appendix B: Proof of Theorem 3

We first show that (E$^*$), (M$^{-*}$), (SYM$^{*}$) characterizes the egalitarian Shapley value when we fix $w=(\frac{1}{n},\ldots ,\frac{1}{n})$, which can be useful in later.

Lemma 1

Suppose that $\mathcal {W}=\{(\frac{1}{n},\ldots ,\frac{1}{n})\}$ and $n \ne 2$. Then, an allocation rule $f: \mathcal {G}\times \mathcal {W}\rightarrow \mathbb {R}^N$ satisfies (E$^*$), (M$^{-*}$), (SYM$^{*}$) if and only if it is an egalitarian-Shapley value.

Proof

This follows from the axioms and arguments in Casajus and Huettner (2014) if $w=(\frac{1}{n},\ldots ,\frac{1}{n})$. $\square $

Now, we offer the proof of Theorem 3. It is clear that the rule satisfies all the axioms. We suppose that a rule $f: \mathcal {G}\rightarrow \mathbb {R}^N$ satisfies (E$^*$), ($\mathrm {M}^{-*}$), (RIN$^*$), (SYM$^{*}$), and (FEC$^*$). Claim 1 can be thought of as an analog of that of Theorem 2. The differences lie in Claims 2, 3 and 4. In this proof, we first specify the form of the Shapley value, while we first specify the weighted division in Theorem 2.

Claim 1

For each $i \in N$, there exists functions $\phi _i(v): \varLambda ^i \times \mathbb {R}$$ \rightarrow \mathbb {R}$ and $\alpha _i: \mathcal {W} \times \mathbb {R} \rightarrow \mathbb {R}$ such that $f_i(v,w)=\phi _i(\varDelta _i(v), v(N))+\alpha _i(w, v(N))$.

We first take any $c \in \mathbb {R}$. For any $i \in N$, $v \in \mathcal {G}$ and $w \in \mathcal {W}$, we have the following equation: for any ${\bar{v}} \in \mathcal {G}^c$ such that $\varDelta _i(v)=\varDelta _i({\bar{v}})$,

$$\begin{aligned} f_i(v, w)\overset{(\text {M}^{-*})}{=}f_i({\bar{v}},w)=: \alpha _i(w, c, \varDelta _i(v)). \end{aligned}$$

(B.1)

Specifically, we denote

$$\begin{aligned} \alpha _i(w, c)=\alpha _i(w, c, \mathbf 0 ). \end{aligned}$$

(B.2)

By (FEC$^*$), for any $c\in \mathbb {R}$ and $i\in N$, there is a function $\phi ^c_i:\mathcal {G}^c \rightarrow \mathbb {R}$ such that

$$\begin{aligned} \phi ^c_i(v)&=f_i(v,w)-f_i(cu_{N{\setminus } \{i\}},w)\\&\quad \overset{(B.2)}{=}f_i(v,w)-\alpha _i(w, c). \end{aligned}$$

By (B.1), we know that $\phi ^c_i(v)=\phi ^c_i({\bar{v}})$ if $v(N)={\bar{v}}(N)=c$ and $\varDelta _i(v)=\varDelta _i({\bar{v}})$. Hence, we can define $\phi _i(\varDelta _i(v), c): \varLambda ^i \times \mathbb {R} \rightarrow \mathbb {R}$ as $\phi _i(\varDelta _i(v), c)=:\phi ^c_i(v)$. Therefore, for any $i \in N$, $v \in \mathcal {G}$ and $w \in \mathcal {W}$, we obtain $f_i(v,w)=\phi _i(\varDelta _i(v), v(N))$$+\alpha _i(w, v(N))$. This completes Claim 1.

Claim 2

For any $v \in \mathcal {G}$, there exists a $\delta \in [0,1]$ and $d^{v(N)}_i \in \mathbb {R}$ such that $\phi _i(\varDelta _i(v), v(N))=\delta Sh_i(v)+d^{v(N)}_i$.

Let $w^*=(1/n,\ldots ,1/n) \in \mathcal {W}$, i.e., the equal weight. For any $c \in \mathbb {R}$ and any $v \in \mathcal {G}^c$, by Claim 1, we have

$$\begin{aligned} f_i(v, w^*)=\phi _i(\varDelta _i(v), v(N))+\alpha _i(w^*, v(N)), \end{aligned}$$

(B.3)

and, by Lemma 1, there exists $\delta \in [0,1]$ such that

$$\begin{aligned} f_i(v, w^*)=\delta Sh_i(v)+(1-\delta )\frac{1}{n}c. \end{aligned}$$

(B.4)

Note that $\delta $ does not depend on $c \in \mathbb {R}$. For any $v'\in \mathcal {G}^{c,i}$, we have

$$\begin{aligned}&\phi _i(\varDelta _i(v'), c)+\alpha _i(w^*, c) \overset{(B.3)}{=} f_i(v', w^*) \overset{(B.4)}{=} \delta Sh_i(v')+(1-\delta )\frac{1}{n}c\nonumber \\&\quad =(1-\delta )\frac{1}{n}c. \end{aligned}$$

(B.5)

Note that player i is a null player in game $v'\in \mathcal {G}^{c,i}$. Hence, for any $v',v''\in \mathcal {G}^{c,i}$, we have $\phi _i(\varDelta _i(v'), c)+\alpha _i(w^*, c)\overset{(B.5)}{=}(1-\delta )\frac{1}{n}c\overset{(B.5)}{=}\phi _i(\varDelta _i(v''), c)+\alpha _i(w^*, c)$ and, so, denote $d^c_i:=\phi _i(\varDelta _i(v'), c)=\phi _i(\varDelta _i(v''), c)$. We obtain

$$\begin{aligned} \alpha _i(w^*, c)\overset{(B.5), d^c_i}{=}(1-\delta )\frac{1}{n}c-d^c_i. \end{aligned}$$

(B.6)

Therefore, for every $v \in \mathcal {G}^c$, we must have

$$\begin{aligned} \phi _i(\varDelta _i(v), c) \overset{(B.3)(B.4)(B.6)}{=} \delta Sh_i(v)+d^c_i. \end{aligned}$$

(B.7)

Since $c \in \mathbb {R}$ is arbitrary chosen, we obtain $\phi _i(\varDelta _i(v'), v(N))=\delta Sh_i(v)+d^{v(N)}_i$ for all $v \in \mathcal {G}$.

Claim 3

$\alpha _i(w, v(N))=(1-\delta )\cdot w_iv(N)-d^{v(N)}_i$ for each $w \in \mathcal {W}$.

Consider any $w \in \mathcal {W}$ and player $k^* \in N$ such that $k^* \in \mathrm{argmin}_{i \in N, w_i>0} w_i$. Note that $k^*$ is well-defined because $\sum _{i \in N}w_i=1$ and $w_i \ge 0$ for any $i \in N$. By Claim 2, for any player $i \ne k^*$ and any $c\in \mathbb {R}$,

$$\begin{aligned} f_k(cu_{\{i\}},w)= \left\{ \begin{array}{ll} \delta c+d^{c}_k+\alpha _k(w, c) &{} \quad \mathrm{if}~~ k=i,\\ d^{c}_k+\psi ^{c}_k(w) &{}\quad \mathrm{otherwise}.\\ \end{array} \right. \end{aligned}$$

Hence, we have

$$\begin{aligned} \sum _{k \in N}(\alpha _k(w, c)+d^{c}_k)\overset{(\text {E}^*)}{=}(1-\delta )c. \end{aligned}$$

(B.8)

Moreover, for any $i\ne k^*$, j$(j\ne i,\ j\ne k^*)$ and, by considering a game $cu_{\{j\}}$, we have

$$\begin{aligned} \alpha _i(w, c)+d^{c}_i \overset{(\text {RIN}^*)}{=}\frac{w_i}{w_{k^*}}(\alpha _{k*}(w, c)+d^{c}_{k^*}), \end{aligned}$$

(B.9)

because i and $k^*$ are null players in $cu_{\{j\}}$. Therefore, for any $i\in N$, we have

$$\begin{aligned}&\alpha _{i}(w, c)+d^{c}_i-(1-\delta ) w_ic\\&\quad \overset{(B.9)}{=} w_i\cdot \left[ \frac{1}{w_{k^*}}(\alpha _{k*}(w, c)+d^{c}_{k^*})-(1-\delta )c\right] \\&\quad \overset{(B.8)}{=} w_i\cdot \left[ \frac{1}{w_{k^*}}(\alpha _{k*}(w, c)+d^{c}_{k^*})-\sum _{k \in N}(\alpha _{k}(w, c)+d^{c}_k)\right] \\&\quad \overset{(B.9)}{=} \frac{w_i}{w_{k^*}}\cdot \left[ (\alpha _{k*}(w, c)+d^{c}_{k^*})-\sum _{k \in N}w_k(\alpha _{k*}(w, c)+d^{c}_{k^*})\right] \\&\quad \overset{\sum _{k}w_k=1}{=} \frac{w_i}{w_{k^*}}\cdot \Bigl [(\alpha _{k*}(w, c)+d^{c}_{k^*})-(\alpha _{k*}(w, c)+d^{c}_{k^*})\Bigr ]\\&\quad = 0. \end{aligned}$$

Since $c \in \mathbb {R}$ is arbitrary chosen, we obtain $\alpha _{i}(w, v(N))=(1-\delta )\cdot w_iv(N)-d^{v(N)}_i$ for all $v \in \mathcal {G}$.

Claim 4

For any $v \in \mathcal {G}$ and $w\in \mathcal {W}$, there exists a $\delta \in [0,1]$ such that $f_i(v,w)=\delta \cdot Sh_i(v) + (1-\delta )\cdot w_i v(N)$.

For any $v \in \mathcal {G}$ and $w\in \mathcal {W}$, we have

$$\begin{aligned} f_i(v,w)&\overset{\text {C1}}{=}\phi _i(\varDelta _i(v), v(N))+\alpha _i(w, v(N))\\&\quad \overset{\text {C2}}{=}\delta Sh_i(v)+d^{v(N)}_i +\alpha _i(w, v(N))\\&\quad \overset{\text {C3}}{=}\delta Sh_i(v)+d^{v(N)}_i +(1-\delta )\cdot w_i v(N)-d^{v(N)}_i\\&\quad =\delta Sh_i(v)+(1-\delta )\cdot w_iv(N). \end{aligned}$$

This completes the proof. $\square $