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Redistribution to the less productive: parallel characterizations of the egalitarian Shapley and consensus values

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Abstract

In cooperative game theory with transferable utilities (TU games), there are two well-established ways of redistributing Shapley value payoffs: using egalitarian Shapley values, and using consensus values. We present parallel characterizations of these classes of solutions. Together with the (weaker) axioms that characterize the original Shapley value, those that specify the redistribution methods characterize the two classes of values. For the class of egalitarian Shapley values, we focus on redistributions in one-person unanimity games from two perspectives: allowing the worth of coalitions to vary, while keeping the player set fixed; and allowing the player set to change, while keeping the worth of coalitions fixed. This class of values is characterized by efficiency, the balanced contributions property for equal contributors, weak covariance, a proportionately decreasing redistribution in one-person unanimity games, desirability, and null players in unanimity games. For the class of consensus values, we concentrate on redistributions in \((n-1)\)-person unanimity games from the same two perspectives. This class of values is characterized by efficiency, the balanced contributions property for equal contributors to social surplus, complement weak covariance, a proportionately decreasing redistribution in \((n-1)\)-person unanimity games, desirability, and null players in unanimity games.

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Notes

  1. The null player out property requires that a deletion of a null player from a game does not affect the other players’ payoffs. The weak null player out property requires that a deletion of a null player from a game affects the other players’ payoffs in the same way.

  2. In addition to these three values, the equal allocation of nonseparable cost value has been well researched. This value corresponds to the dual value (Oishi et al. 2016) of the equal surplus division value, and is characterized by Moulin (1985), Hwang (2006) and Béal et al. (2017).

  3. More precisely, on page 147 of Joosten (1996), the \(\alpha\)-egalitarian Shapley value is originally defined as a one-point solution with respect to \(\alpha \in {\mathbb {R}}\), in other words, as an element in the set of the affine combinations of Sh and ED. Later, van den Brink et al. (2013) investigate the egalitarian Shapley values as a set of solutions by considering the set of convex combinations of Sh and ED. Ju et al. (2007) originally define the consensus value as a one-point solution of the average of Sh and ESD, and they generalize it as the \(\alpha\)-consensus value with respect to \(\alpha \in [0,1]\), in other words, an element in the set of the convex combinations of Sh and ESD. In this manuscript, we consider both the egalitarian Shapley values and the consensus values as the sets of solutions defined in the above. One justification for restricting our attention not on the sets of the affine combinations (i.e., \(\alpha \in {\mathbb {R}}\)) but on the sets of the convex combinations (i.e., \(\alpha \in [0,1]\)) lies in our focus on redistribution from more productive players to less productive ones. If \(\alpha >1\), the two values redistribute from the less productive players to the more productive ones. In addition, if \(\alpha <0\), the more productive players receive less than the less productive ones. Another justification is in terms of monotonicity of solutions, which increases a player’s payoff if some information of the player measured by v increases (for details, see Young 1985, Casajus and Huettner 2014b and Yokote and Funaki 2017). Young (1985) states “Monotonicity is a general principle of fair division ....” It should be noted that we also obtain axiomatizations of the affine combinations variations of the egalitarian Shapley and consensus values by omitting last two axioms from our axiomatizations (see Sect. 4).

  4. As another combination, Ferrières (2016, 2017) and Kongo (2018) investigate the set of convex combinations of ED and ESD on the class of games on a fixed player set. In addition, van den Brink and Funaki (2009) and van den Brink et al. (2016) investigate the same set on the class of games on a variable player set.

  5. Chun (1989) calls this axiom triviality.

  6. C is often defined as \(f(N,\alpha v+\beta )=\alpha f(N,v)+\beta\) for any \((N,v) \in G\), any \(\alpha \in {\mathbb {R}}\) and any \(\beta =(\beta _i)_{i \in N} \in {\mathbb {R}}^N\), where \((\alpha v+\beta )(S)=\alpha v(S)+ \sum _{i \in S}\beta _i\) for any \(S \subseteq N\). Our definition of C is weaker than this definition, because we consider only when \(\alpha =1\).

  7. This axiom is also known as local monotonicity (van den Brink et al. 2013; Malawski 2013) and fair treatment (Radzik and Driessen 2013).

  8. Yokote et al. (2019) introduce a weaker axiom than both BCEC and BCESS by requiring BC for players ij satisfying both \(v(N {\setminus } i)=v(N {\setminus } j)\) and \(v(i)=v(j)\). This axiom, together with E, linearity, and weak null player out (van den Brink and Funaki 2009), characterizes the affine combinations of Sh, ED, and ESD.

  9. Yokote et al. (2018) invoke the name of “weak covariance” for the axiom in which the supposition part in the current form does not exist. The current form is weaker than C in the exact sense, as proven in Lemma 1 in Appendix A.0. By adding this supposition part, the above discussion of redistribution will be considered only for the solution that satisfies NG, however, NG is derived from the combination of other axioms we use together with \(\mathbf{{C}}^-\) (for details, see Remark 1 in Appendix A.0.). The authors deeply appreciate an anonymous referee for pointing out the insufficiency of the formulation of this axiom.

  10. This idea is advocated in the population monotonicity axiom (see Thomson 1983). In the context of bargaining games, page 71 of Moulin (1991) explains this axiom as follows: “Intuitively, if a new guest shows up to share the same cake, every other guest should tighten his belt.”

  11. While \({\bf{PDR}}^1\) considers games with two or more players, \({\bf{PDR}}^{n-1}\) considers those with three or more players. This slight difference comes from the difference between ED and ESD. Together with other axioms, \({\bf{PDR}}^1\) characterizes the combinations of Sh and ED, and \({\bf{PDR}}^{n-1}\) characterizes those of Sh and ESD (for further detail, see Sect. 4). Because Sh is different from ED, but coincides with ESD in games with two players, \({\bf{PDR}}^1\) has to treat games with two players, whereas \({\bf{PDR}}^{n-1}\) does not.

  12. As an alternative to NU, a weaker variation of NPE, which requires that null players receive nonnegative payoffs if the worth of the grand coalition is nonnegative and if the game is superadditive, i.e., for any \(S,T \subseteq N\) with \(S \cap T=\emptyset\), it holds that \(v(S)+v(T) \le v(S \cup T)\), also works in our results. The authors deeply appreciate an anonymous referee for letting us know this alternative.

  13. See the discussion on page 117, after Corollary 1, in Yokote et al. (2018).

  14. f is linear if, for any \((N,v),(N,w) \in G\) and \(\lambda \in {\mathbb {R}}\), \(f(N,v+\lambda w)=f(N,v)+\lambda f(N,w)\), where, for any \(S \subseteq N\), \((v+\lambda w)(S)= v(S)+\lambda w(S)\).

  15. The case for \(n=2\) is untractable with general arguments starting from Lemma 4 below. This is because, if \(n=2\), it holds that \(Sh=ESD=\alpha _2 Sh+(1-\alpha _2)ESD\) for any \(\alpha _2 \in {\mathbb {R}}\), which means that the coefficient \(\alpha _2\) is indeterminate. We also note that \({\bf{PDR}}^{n-1}\) requires at least three players.

  16. Lemma 3 in Yokote et al. (2019) obtains the same result using a weaker axiom than BCESS.

  17. If \(v(i)+v(N {\setminus } i)=v(j)+v(N {\setminus } j)\), \(\lambda _{ij}=0\), and simply applying BCESS to (Nv) suffices to obtain (12).

  18. The linear independence of the Eqs. (12)–(14) is visualized as follows: Let \(N=\{1,2,3\}\) and \(i=1\). Then, (12)–(14) yield the following equation:

    $$\begin{aligned}\left( \begin{array}{ccc} 1 &{} -1 &{} 0 \\ 1 &{} 0 &{} -1 \\ 1 &{} 1 &{} 1 \end{array}\right) \left( \begin{array}{c} f_{1}(N,v) \\ f_{2}(N,v) \\ f_{3}(N,v) \end{array}\right) =\left( \begin{array}{c} f_{1}(N \backslash 2, v)-f_{2}\left( N \backslash 1, v+\lambda _{1, 2} u_{N \backslash 1}\right) +\lambda _{1, 2} \frac{\alpha _{3}}{2} \\ f_{1}(N \backslash 3, v)-f_{3}\left( N \backslash 1, v+\lambda _{1, 3} u_{N \backslash 1}\right) +\lambda _{1, 3} \frac{\alpha _{3}}{2} \\ v(N) \end{array}\right) .\end{aligned}$$

    Note that the matrix on the left-hand side is linearly independent.

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Acknowledgements

The authors are grateful to an associate editor and anonymous referees for their comments on the previous version of our paper. This work was supported by JSPS KAKENHI grant numbers 17H02503 and 20K13458, and Waseda University Grants for Special Research Projects (Grant Number 2019C-015).

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Correspondence to Takumi Kongo.

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The original online version of this article was revised: In sub-section 3.3, the terms “one-person” and “(n-1)-person” were incorrectly updated by mistake in production process. Now, they have been corrected.

Appendix

Appendix

A.0. A modification of \(\hbox {C}^-\) in Yokote et al. (2018)

As we have discussed in footnote 9, we add the supposition “Suppose \(f(N,{\mathbf {0}})=(0,\dots ,0) \in {\mathbb {R}}^N\)” to the definition of C in Yokote et al. (2018). This modification makes C weaker than C.

Lemma 1

C is weaker than C.

Proof

Let f satisfy C. We prove that f satisfies C. Suppose \(f(N, {\mathbf {0}})=(0,\dots ,0)\). Fix \(i\in N\). Then,

$$\begin{aligned} f_j(N,u_i)&=f_j(N,{\mathbf {0}}+ u_i) \nonumber \\&={\left\{ \begin{array}{ll} f_j(N,{\mathbf {0}})+1 &{} \text { if } j=i, \\ f_j(N,{\mathbf {0}}) &{} \text { if } j\ne i. \end{array}\right. }\nonumber \\&={\left\{ \begin{array}{ll} 1 &{} \text { if } j=i, \\ 0 &{} \text { if } j\ne i, \end{array}\right. } \end{aligned}$$
(1)

where the second equality follows from C and the third equality follows from \(f(N, {\mathbf {0}})=(0,\dots ,0)\). It follows that, for any \((N, v) \in G\) and \(\lambda \in {\mathbb {R}}\),

$$\begin{aligned} f_j\left( N, v+\lambda u_{i}\right)&={\left\{ \begin{array}{ll}f_j(N,v)+\lambda &{} \text { if } j=i, \\ f_j(N,v) &{} \text { if } j\ne i, \end{array}\right. } \\&=f_j(N,v)+\lambda f_j(N, u_i) \text { for all } j\in N, \end{aligned}$$

where the first equality follows from C and the second equality follows from (1). \(\square\)

Furthermore, this modification does not matter for results in Yokote et al. (2018). This is because Yokote et al. (2018) use \(\mathbf{C}^-\) together with E and BCEC. If we consider null games only, BCEC (and BCESS also) is equivalent to BC. Then, because E and BC characterize the Shapley value (Myerson 1980) and because the Shapley value satisfies NG, we obtain NG which guarantees the supposition part we add to the new definition of \(\mathbf{C}^-\).

Remark 1

Let f satisfy E and BCEC (or BCESS). Then, f satisfies NG.

A.1. Proof of Theorem 1

\(\alpha Sh+(1-\alpha )ED\) clearly satisfies \({\bf{PDR}}^{1}\), for any \(\alpha \in {\mathbb {R}}\). It also satisfies D and NU if \(\alpha \in [0,1]\). Together with Remark 1 above, Yokote et al. (2018, Theorem 1) show that f satisfies E, BCEC, and \(\mathbf{C}^-\) if and only if there exists a sequence \({\mathbf {r}}\in \bigl \{\{r_{\ell }\}_{{\ell }=1}^{{\bar{n}}} : r_{\ell } \in {\mathbb {R}} \text { for all } {\ell } \in \{1, \dots , {\bar{n}}\}\bigr \}\), such that \(f_i(N,v)=\frac{v(N)-v^{\mathbf {r}}(N)}{n}+Sh_i(N, v^{\mathbf {r}}) \text { for all } (N,v)\in G\) and \(i \in N\), where \(v^{\mathbf {r}}(S)=r_s v(S) \text { for all } S\subseteq N, S\ne \emptyset\), and any \((N,v) \in G\). Furthermore, the sequence \({\mathbf {r}}\) is recursively determined, as follows:Footnote 13

  • \(1-r_1=2f_j(\{i,j\},u_i)\).

  • For \(\ell \in \{2, \dots , {\bar{n}}-1\}\), by choosing \(N \subseteq {\mathcal {U}}\) with \(n=\ell\), \(i,j\in N\) with \(i\ne j\), and \(k\in {\mathcal {U}}\backslash N\):

    $$\begin{aligned} 1-r_{{\ell }}=\sum _{m=1}^{{\ell }}(1-r_m)-\sum _{m=1}^{{\ell }-1}(1-r_m)={\ell }({\ell }+1)f_j(N\cup k, u_i)-{\ell }({\ell }-1)f_j(N, u_i). \end{aligned}$$
  • \(r_{{\bar{n}}}\) is an arbitrary value.

By \({\bf{PDR}}^1\), \((\ell +1)f_j(N \cup k,u_i)=\ell f_j(N,u_i)\), for \(\ell \in \{2,3,\dots , {\bar{n}}-1\}\) and, thus, \(1-r_{\ell }=(\ell -(\ell -1))\ell f_j(N,u_i)=\ell f_j(N,u_i)=2f_j(\{i,j\},u_i)\). By letting \(\alpha =1-2f_j(\{i,j\},u_i)\), \(\alpha \in {\mathbb {R}}\) satisfies \(r_{\ell }=r_{\ell +1}=\alpha\), for any \(1 \le \ell \le {\bar{n}}-1\). By the linearityFootnote 14 of Sh, \(f_i(N,v)=(1-\alpha )\frac{v(N)}{n}+Sh_i(N,\alpha v)=\alpha Sh_i(N,v)+(1-\alpha )ED_i(N,v)\).

It suffices to show that D and NU imply \(\alpha \in [0,1]\). We prove this by contradiction. Consider \((\{i,j\},u_i) \in G\). Now, \(f_i(\{i,j\},u_i)=\alpha + \frac{1-\alpha }{2}\) and \(f_j(\{i,j\},u_i)=\frac{1-\alpha }{2}\). Suppose \(\alpha >1\). Then, \(f_j(\{i,j\},u_i)<0\), which violates NU. Suppose \(\alpha < 0\). Then, \(f_i(\{i,j\},u_i)<f_j(\{i,j\},u_i)\), which violates D. Therefore, \(\alpha \in [0,1]\).

A.2. Proof of Theorem 2

First, we prepare for the proof. In the proof, we use the following two axioms. Given \((N,v) \in G\), \(i \in N\) and \(j \in N {\setminus } i\) are symmetric in (Nv) if, for any \(S \subseteq N {\setminus } \{i,j\}\), \(v(S \cup i)=v(S \cup j)\).

Symmetry (S)::

For any \((N,v) \in G\), if \(i \in N\) and \(j \in N {\setminus } i\) are symmetric in (Nv), then \(f_i(N,v)=f_j(N,v)\).

Anonymity (A)::

For any \((N,v) \in G\) and \(\pi : N \rightarrow {\mathbb {N}}\), \(f_i(N,v)=f_{\pi (i)}(\pi (N),\pi v)\), where \(\pi v(S)=v(\pi ^{-1}(S))\), for any \(S \subseteq \pi (N)\).

We divide the proof of Theorem 2 into 11 lemmas and remarks. The role of each lemma/remark is summarized as follows. Lemma 2 shows the existence of the value satisfying the six axioms. Remark 2 and Lemma 3 show the uniqueness of the value for \(n=1\) and \(n=2\), respectively. Remark 3 shows A of the value for \(n=2\). Remark 4 shows S of the value for \(n=t\) if the value satisfies A, for \(n=t-1\). Lemmas 4 to 6 together show the uniqueness of the value for \(n=3\) , with respect to a coefficient \(\alpha _3 \in {\mathbb {R}}\). Remark 5 shows A of the value for \(n=3\). Lemma 7 shows the uniqueness of the value for \(n=t+1\) , with respect to a coefficient \(\alpha _{t+1} \in {\mathbb {R}}\), if the value is unique for \(n=t\) \((\ge 3)\) , with respect to a coefficient \(\alpha _{t} \in {\mathbb {R}}\). Lemma 8 shows the coincidence of the coefficients for any game. Lastly, we narrow the range of coefficients to [0, 1].

In the following, the proofs of remarks are apparent, and hence, we omit them. Further, each of R3, R4, L5, \(\dots\) denote Remark 3, Remark 4, Lemma 5, \(\dots\), respectively.

Lemma 2

For any \(\alpha \in {\mathbb {R}}\), \(\alpha Sh+(1-\alpha )ESD\) satisfies E, BCESS, CC and PDR\(^{n-1}\). Furthermore, for any \(\alpha \in [0,1]\), \(\alpha Sh+(1-\alpha )ESD\) satisfies D and NU.

Proof

E , D and NU are obvious. CC is from the linearity of Sh and ESD. For \({\bf{PDR}}^{n-1}\), if \(n \ge 3\), \(\alpha Sh_i(N,u_{N {\setminus } i})+(1-\alpha )ESD_i(N,u_{N {\setminus } i})=\frac{1-\alpha }{n}\). For BCESS, for any \((N,v) \in G\) and \(i,j \in N\),

$$\begin{aligned} ESD_i(N,v)-ESD_j(N,v)=v(i)-v(j) \end{aligned}$$
(2)

and

$$\begin{aligned}&ESD_i(N {\setminus } j,v)-ESD_j(N {\setminus } i,v)\nonumber \\&\quad =\frac{v(N {\setminus } j)-\sum _{k \in N {\setminus } j}v(k)}{n-1}+v(i)-\frac{v(N {\setminus } i)-\sum _{k \in N {\setminus } i}v(k)}{n-1}-v(j) \nonumber \\&\quad =\frac{v(N {\setminus } j)-v(N {\setminus } i)}{n-1}+\frac{n-2}{n-1}(v(i)-v(j)). \end{aligned}$$
(3)

Here, \(v(i)+v(N {\setminus } i)=v(j)+v(N {\setminus } j)\) implies that (2)=(3). Because Sh satisfies BC, and Sh and ESD both satisfy linearity, the desired result is obtained. \(\square\)

Remark 2

Let f satisfy E. Then, for any game \((N,v) \in G\), with \(n=1\), \(f_i(N,v)=v(i)\) (i.e., f is uniquely determined).

Lemma 3

Footnote 15Let f satisfy E and BCESS. For any game \((N,v) \in G\), with \(n=2\), \(f_i(N,v)=\frac{v(N)+v(i)-v(j)}{2}\) (i.e., f is uniquely determined).

Proof

\(n=2\) implies that \(N {\setminus } j=i\) and \(N {\setminus } j=i\), for \(i,j \in N\). Hence, BCESS and Remark 2 together imply that \(f_i(N,v)-f_j(N,v)=v(i)-v(j)\). Further, E implies that \(f_i(N,v)+f_j(N,v)=v(N)\). These together imply the desired result. \(\square\)

Remark 3

Let f satisfy E and BCESS. Then, f satisfies A on the class of games with two players.

Remark 4

Footnote 16 Let f satisfy E and BCESS. Let \(t \in {\mathbb {N}}\), with \(t \ge 3\). If f satisfies A on the class of games with \(t-1\) players, then f satisfies S on the class of games with t players.

From here, we prepare for the induction with respect to the number of players by showing the uniqueness of the value with respect to a real number \(\alpha _3\) for games with three players. The induction is shown in Lemma 7.

Lemma 4

Let \({|N|}=3\), and let f satisfy E, BCESS, and CC. Then, there exists \(\alpha _N\), such that \(f_i(N,u_{N {\setminus } i})=\frac{1-\alpha _N}{ {3}}\), for any \(i \in N\).

Proof

Players \(k,\ell \in N {\setminus } i\) are symmetric in \((N,u_{N {\setminus } i})\). Hence,

$$\begin{aligned} f_k(N,u_{N {\setminus } i})-f_{\ell }(N,u_{N {\setminus } i}) \overset{\mathbf { {R3,R4}}}{=}0. \end{aligned}$$

For any \(k \ne i\), let

$$\begin{aligned} \alpha _{N,i}= {6}f_k(N,u_{N {\setminus } i})- {2} \iff f_k(N,u_{N {\setminus } i})=\frac{ {2}+\alpha _{N,i}}{ {6}}. \end{aligned}$$
(4)

Then,

$$\begin{aligned} f_i(N,u_{N {\setminus } i})\overset{\mathrm {\mathbf{E}}}{=}u_{N{\setminus } i}(N)- {2}f_k(N,u_{N {\setminus } i})\overset{{\mathbf {(4)}}}{=}\frac{1-\alpha _{N,i}}{ {3}}. \end{aligned}$$
(5)

Similarly, consider the game \((N,u_{N {\setminus } j}) \in G\). For any \(\ell \ne j\), by letting

$$\begin{aligned} \alpha _{N,j} = {6}f_{\ell }(N,u_{N {\setminus } j})- {2} \iff f_{\ell }(N,u_{N {\setminus } j})=\frac{ {2}+\alpha _{N,j}}{ {6}}, \end{aligned}$$
(6)

we obtain that

$$\begin{aligned} f_j(N,u_{N {\setminus } j})\overset{\mathrm {\mathbf{E}}}{=}u_{N{\setminus } j}(N)- {2}f_{\ell }(N,u_{N {\setminus } j})\overset{{\mathbf {(6)}}}{=}\frac{1-\alpha _{N,j}}{ {3}}. \end{aligned}$$
(7)

Now, consider a game \((N,u_{N {\setminus } i}+u_{N {\setminus } j}) \in G\). Because i and j are symmetric in \((N,u_{N {\setminus } i}+u_{N {\setminus } j})\),

$$\begin{aligned} f_i(N,u_{N {\setminus } i}+u_{N {\setminus } j})-f_j(N,u_{N {\setminus } i}+u_{N {\setminus } j})\overset{{\mathbf {{R3,R4}}}}{=}0. \end{aligned}$$
(8)

At the same time,

$$\begin{aligned}&f_i(N,u_{N {\setminus } i}+u_{N {\setminus } j})-f_j(N,u_{N {\setminus } i}+u_{N {\setminus } j})\nonumber \\&\quad \overset{\mathbf {{CC}^-}}{=}f_i(N,u_{N {\setminus } i})+f_i(N,u_{N {\setminus } j})-f_j(N,u_{N {\setminus } i})-f_j(N,u_{N {\setminus } j})\nonumber \\&\quad \overset{\mathbf {(5),(6),(4),(7)}}{=}\frac{-\alpha _{N,i}+\alpha _{N,j}}{{2}}. \end{aligned}$$
(9)

By (8) and (9), \(\alpha _{N,i}=\alpha _{N,j}\), for any \(i,j \in N\). By letting \(\alpha _N\) denote the equal value, (5) establishes the desired result. \(\square\)

Lemma 5

Let \({|N|}=3\) and let f satisfy E, BCESS, CC, and PDR\(^{n-1}\). For any \((N,u_{N {\setminus } i}),(M,u_{M {\setminus } j}) \in G\), with \(i \in N\), \(j \in M\), and \(|N|=|M|= {3}\), there exists \(\alpha _{ {3}} \in {\mathbb {R}}\), such that \(f_i(N,u_{N {\setminus } i})=f_j(M,u_{ {M} {\setminus } j})=\frac{1-\alpha _{ {3}}}{ {3}}\).

Proof

By Lemma 4,

$$\begin{aligned} \alpha _N\overset{\mathrm {(5)}}{=} 1- {3}f_i(N,u_{N {\setminus } i})\overset{\mathrm {\mathbf{PDR}^{n-1}}}{=} 1- {3}f_j(M,u_{ {M} {\setminus } j})\overset{\mathrm {(5)}}{=}\alpha _{M}. \end{aligned}$$

Thus, \(\alpha _N=\alpha _M=\alpha _{ {3}}\). \(\square\)

Lemma 6

Let \({|N|}=3\), and let f satisfy E, BCESS, CC, and \({\bf{PDR}}^{n-1}\). For any game with three players, there exists \(\alpha _{ {3}} \in {\mathbb {R}}\), such that \(f=\alpha _{ {3}} Sh+(1-\alpha _{ {3}})ESD\).

Proof

By Lemma 2, \(f=\alpha _{ {3}} Sh+(1-\alpha _{ {3}})ESD\) satisfies the four axioms with respect to \(\alpha _{ {3}} \in {\mathbb {R}}\).

In the following, we show the uniqueness of f with respect to \(\alpha _{ {3}}\). Given \((N,v) \in G\), with \(|N|={ {3}}\), let \(i,j \in N\), and let \(\lambda _{i,j}=v(j)+v(N {\setminus } j)-v(i)-v(N {\setminus } i)\). Consider the game \((N,v+\lambda _{i,j}u_{N {\setminus } i})\).Footnote 17 In this game,

  • \((v+\lambda _{i,j}u_{N {\setminus } i})(i)=v(i)\),

  • \((v+\lambda _{i,j}u_{N {\setminus } i})(N {\setminus } i)=v(j)+v(N {\setminus } j)-v(i)\),

  • \((v+\lambda _{i,j}u_{N {\setminus } i})(j)=v(j)\), and

  • \((v+\lambda _{i,j}u_{N {\setminus } i})(N {\setminus } j)=v(N {\setminus } j)\).

Thus, \((v+\lambda _{i,j}u_{N {\setminus } i})(i)+(v+\lambda _{i,j}u_{N {\setminus } i})(N {\setminus } i)=(v+\lambda _{i,j}u_{N {\setminus } i})(j)+(v+\lambda _{i,j}u_{N {\setminus } i})(N {\setminus } j)\). BCESS implies that

$$\begin{aligned} f_i(N,v+\lambda _{i,j}u_{N {\setminus } i})-f_j(N,v+\lambda _{i,j}u_{N {\setminus } i})=f_i(N {\setminus } j,v)-f_j(N {\setminus } i,v+\lambda _{i,j}u_{N {\setminus } i}). \end{aligned}$$
(10)

Now,

$$\begin{aligned}&f_i(N,v+\lambda _{i,j}u_{N {\setminus } i})-f_j(N,v+\lambda _{i,j}u_{N {\setminus } i})\nonumber \\&\quad \overset{\mathbf {CC}^-}{=}f_i(N,v)+\lambda _{i,j}f_i(N,u_{N {\setminus } i})-f_j(N,v)-\lambda _{i,j}f_j(N,u_{N {\setminus } i})\nonumber \\&\quad \overset{\mathbf { {L5}},(5),(4)}{=}f_i(N,v)+\lambda _{i,j}\frac{1-\alpha _{ {3}}}{ {3}}-f_j(N,v)-\lambda _{i,j}\frac{ {2}+\alpha _{ {3}}}{ {6}}. \end{aligned}$$
(11)

Then,

$$\begin{aligned} f_i(N,v)-f_j(N,v)\overset{\mathbf {(10),(11)}}{=}f_i(N {\setminus } j,v)-f_j(N {\setminus } i,v+\lambda _{i,j}u_{N {\setminus } i})+\lambda _{i,j}\frac{\alpha _{ {3}}}{ {2}}. \end{aligned}$$
(12)

By Lemma 3, \(f_i(N {\setminus } j,v)\) and \(f_j(N {\setminus } i,v+\lambda _{i,j}u_{N {\setminus } i})\) are uniquely determined. Hence, \(f_i(N,v)-f_j(N,v)\) is uniquely determined with respect to \(\alpha _3\).

Taking the above argument for i and any \(k \ne i,j\) we obtain that

$$\begin{aligned} f_i(N,v)-f_k(N,v)=f_i(N {\setminus } k,v)-f_k(N {\setminus } i,v+\lambda _{i,k}u_{N {\setminus } i})+\lambda _{i,k}\frac{\alpha _{ {3}}}{ {2}}. \end{aligned}$$
(13)

Furthermore, by E

$$\begin{aligned} \sum _{\ell \in N}f_{\ell }(N,v)=v(N). \end{aligned}$$
(14)

(12), (13), and (14) are linearly independent of each other.Footnote 18 Thus, given \(\alpha _{ {3}} \in {\mathbb {R}}\), they constitute mutually independent linear equations with three variables, with a unique solution equal to \(\alpha _{ {3}} Sh+(1-\alpha _{ {3}}) ESD\). \(\square\)

Remark 5

Let f satisfy E, BCESS, CC, and \({\bf{PDR}}^{n-1}\). f satisfies A on the class of games with three players.

Now, we can complete the induction.

Lemma 7

Let f satisfy E, BCESS, CC, and \({\bf{PDR}}^{n-1}\), and let \(t \ge 3\).(i) If there exists \(\alpha _t \in {\mathbb {R}}\), such that \(f=\alpha _t Sh+(1-\alpha _t)ESD\) for any game with t players, then there exists \(\alpha _{t+1} \in {\mathbb {R}}\), such that \(f=\alpha _{t+1} Sh+(1-\alpha _{t+1})ESD\) for any game with \(t+1\) players. (ii) f satisfies S for any games with \(t+2\) players.

Proof

(i) From Lemmas 4–6, there exists \(\alpha _{ {3}} \in {\mathbb {R}}\), such that \(f=\alpha _{ {3}} Sh+(1-\alpha _{ {3}})ESD\) for any game with three players. Now, suppose that there exists \(\alpha _t \in {\mathbb {R}}\), such that \(f=\alpha _t Sh+(1-\alpha _t)ESD\) for any game with t players, and consider any game (Nv) with \(t+1\) players, where \(t \ge 3\). We divide the proof into three steps.

  • Step 1 (Corresponds to Lemma 4): There exists \(\alpha _N\) such that \(f_i(N,u_{N {\setminus } i})=\frac{1-\alpha _N}{t+1}\) for any \(i \in N\). \(\square\)

Proof

Let \(i,j \in N\). Players in \(N {\setminus } i\) and \(N {\setminus } j\) are symmetric in \((N ,u_{N {\setminus } i})\) and \((N,u_{N {\setminus } j})\), respectively. Hence, by letting \(\alpha _{N,i}=t(t+1)f_k(N,u_{N {\setminus } i})-t\) and \(\alpha _{N.j}=t(t+1)f_k(N,u_{N {\setminus } j})-t\) with \(k \in N {\setminus } \{i,j\}\), E implies that \(f_i(N,u_{N {\setminus } i})=\frac{1-\alpha _{N,i}}{t+1}\) and that \(f_j(N,u_{N {\setminus } j})=\frac{1-\alpha _{N,j}}{t+1}\). Similar to the proof of Lemma 4, by considering a game \((N,u_{N {\setminus } i}+u_{N {\setminus } j})\), S and CC together imply \(\alpha _{N,i}=\alpha _{N,j}\), and we let \(\alpha _N=\alpha _{N,i}\). \(\square\)

  • Step 2 (Corresponds to Lemma 5): There exists \(\alpha _{t+1}\) such that \(f_i(N,u_{N {\setminus } i})=f_j(M,u_{M {\setminus } j})=\frac{1-\alpha _{t+1}}{t+1}\) for any player set N and M with \(|N|=|M|=t+1\).

Proof

Similar to Lemma 5, Step 1 and \(\mathbf{PDR} ^{n-1}\) together imply the desired result. \(\square\)

  • Step 3 (Corresponds to Lemma 6): \(f=\alpha _{t+1}Sh+(1-\alpha _{t+1})ESD\) for any game with \(t+1\) players.

Proof

Similar to Lemma 6, given a game (Nv) with \(t+1\) players, given \(i \in N\) and any \(j \in N {\setminus } i\), we consider a new game \((N,v+\lambda _{i,j}u_{N {\setminus } i})\) with \(\lambda =v(j)+v(N {\setminus } j)-v(i)-v(N {\setminus } i),\) and apply BCESS to it. Then, by inductive hypothesis, \(\mathbf {CC}^-\) and Steps 1 and 2, \(f_i(N,v)-f_j(N,v)\) is uniquely determined with respect to \(\alpha _{t+1}\). E implies \(\sum _{\ell \in N}f_{\ell }(N,v)=v(N)\). Hence, with respect to \(\alpha _{t+1}\), f is uniquely determined for games with \(t+1\) players. \(\square\)

(ii) Apparent from (i) and Remark 4.

By (i) and (ii), an induction with respect to \(t \ge 3\) implies the desired result. \(\square\)

Lemma 8

Let f satisfy E, BCESS, CC, and \({\bf{PDR}}^{n-1}\). Then, there exists \(\alpha \in {\mathbb {R}}\), such that \(f=\alpha Sh+(1-\alpha )ESD\).

Proof

Lemma 7 and \({\bf{PDR}}^{n-1}\) together imply that \(\alpha _t=\alpha _{t+1}\), for any \(t \ge 3\). For any game with one or two players, \(Sh=ESD\). Therefore, the desired result is obtained. \(\square\)

Lastly, it suffices to show that D and NU imply \(\alpha \in [0,1]\). We prove the fact by contradiction. Consider \((\{i,j,k\},u_{\{i,j\}}) \in G\). Now, \(f_i(\{i,j,k\},u_{\{i,j\}})=\frac{\alpha }{2} + \frac{1-\alpha }{3}\) and \(f_k(\{i,j,k\},u_{\{i,j\}})=\frac{1-\alpha }{3}\). Suppose \(\alpha >1\). Then, \(f_k(\{i,j,k\},u_{\{i,j\}})<0\), which violates NU. Suppose \(\alpha < 0\). Then, \(f_i(\{i,j,k\},u_{\{i,j\}})<f_k(\{i,j,k\},u_{\{i,j\}})\), which violates D.

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Yokote, K., Kongo, T. & Funaki, Y. Redistribution to the less productive: parallel characterizations of the egalitarian Shapley and consensus values. Theory Decis 91, 81–98 (2021). https://doi.org/10.1007/s11238-020-09781-1

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Keywords

  • Redistribution
  • TU game
  • Axiomatization
  • Egalitarian Shapley value
  • Consensus value