Axiomatizations of the proportional Shapley value

Abstract

We present new axiomatic characterizations of the proportional Shapley value, a weighted TU-value with the worths of the singletons as weights. The presented characterizations are proportional counterparts to the famous characterizations of the Shapley value by Shapley (Contributions to the theory of games, vol. 2. Princeton University Press, Princeton, pp 307–317, 1953b) and Young (Cost allocation: methods, principles, applications. North Holland Publishing Co, 1985a). We introduce two new axioms, called proportionality and player splitting, respectively. Each of them makes a main difference between the proportional Shapley value and the Shapley value. If the stand-alone worths are plausible weights, the proportional Shapley value is a convincing alternative to the Shapley value, for example in cost allocation. Especially, the player splitting property, which states that players’ payoffs do not change if another player splits into two new players who have the same impact to the game as the original player, justifies the use of the proportional Shapley value in many economic situations.

Appendix

1.1 Additional lemmas and a remark, used in the proofs

Remark 4

We can consider the collection of all TU-games $v \in \mathcal {G}^N, N\in \mathcal {N},$ as a vector space $\mathbb {R}^{2^N-1}$. Each game v is represented by a vector $\overrightarrow{v} \in \mathbb {R}^{2^N-1}$, where the entries in the $2^{|N|-1}$ coordinates of the $2^{|N|-1}$ coalitions $S\subseteq N,\, S\ne \emptyset ,$ get the worth v(S) of the respective coalition S. Hence, there exists for every game v a vector $\overrightarrow{\varDelta _v} \in \mathbb {R}^{2^{N}-1}$, which corresponds to the vector $\overrightarrow{v}$, where the entries of the coordinates get the dividends of the respective coalitions. By Eq. (1), we obtain with $v, v_1, v_2 \in \mathcal {G}^N,$ and for all $S \subseteq N$:

$$\begin{aligned}&\overrightarrow{\varDelta _v}&=\overrightarrow{\varDelta _{v_1}}+\overrightarrow{\varDelta _{v_2}}\\ \Leftrightarrow&\varDelta _v(S)&= \varDelta _{v_1}(S) +\varDelta _{v_2}(S)\\ \Leftrightarrow&v(S)-\sum _{R\subsetneq S}\varDelta _v(R)&={v_1}(S)-\sum _{R\subsetneq S}\varDelta _{v_1}(R)+{v_2}(S)-\sum _{R\subsetneq S}\varDelta _{v_2}(R)\\ \Leftrightarrow&v&= v_1 +v_2. \end{aligned}$$

Lemma 4

Equation (4) in WCSE can be replaced equivalently by the following:

$$\begin{aligned} \varDelta _{v}(S)={\left\{ \begin{array}{ll}\varDelta _w(R)+c,\text { if } S=R, \\ \varDelta _w(S), \text { otherwise. }\end{array}\right. } \end{aligned}$$

Proof

Let the notation and the preconditions as in WCSE. By (2), if $v(S)=w(S)$ for all $S\nsupseteq R$, we have $\varDelta _{v}(S)=\varDelta _w(S)$ for all such S and vice versa. Hence, by (1), $v(R)=w(R)+c$ is equivalent to $\varDelta _{v}(R)=\varDelta _w(R)+c$. By induction on the size $s:=|S|$, we now show that $v(S)= w(S)+c\;\Leftrightarrow \;\varDelta _{v}(S)=\varDelta _w(S)$ for all proper supersets $S \supsetneq R$.

Initialization: Let $S\supsetneq R$ and $s=|R|+1$. R is the only proper subset of S where there is a difference of the related dividends in both coalition functions and we obtain the following:

$$\begin{aligned}\begin{array}{rcrcl} v(S)=w(S)+c&{}\underset{(1)}{\Leftrightarrow }&{}v(S)-\displaystyle {\sum _{T\subsetneq S}\varDelta _v(T)}&{}=&{}w(S)+c-\displaystyle {\sum _{\begin{array}{c} T\subsetneq S,\\ T \ne R \end{array}}\varDelta _w(T)}-(\varDelta _w(R)+c)\\ &{}\underset{(1)}{\Leftrightarrow }&{}\varDelta _{v}(S)&{} =&{}\varDelta _w(S). \end{array} \end{aligned}$$

Induction step: Assume that equivalence holds for $s'=s-1$, $|R|+1\le s'\le n-1$ (IH). Then, by (IH), R is again the only proper subset of S with not equal related dividends in v and w. By (1), we get  $v(S)= w(S)+c\;\Leftrightarrow \;\varDelta _{v}(S)=\varDelta _w(S)$  as before and Lemma 4 is shown. $\square$

Lemma 5

(Casajus and Huettner 2008). If $i \in N$ and $v,w \in \mathcal {G}^N$, then ${\text {MC}}_i^v(S)={\text {MC}}_i^w(S)$ for all $S \subseteq N\backslash \{i\}$ iff $\varDelta _v(S \cup \{i\}) = \varDelta _w(S \cup \{i\})$ for all $S \subseteq N\backslash \{i\}$.

1.2 Proofs

1.2.1 Proof of Lemma 1

Let $i,j \in N$ and $v \in \mathcal {G}^N$. If $S= \emptyset$, we have $v(S\cup \{k\})=v(S)+v(\{k\})$. By induction on the size $s:=|S|$ of all coalitions $S \subseteq N\backslash \{i,j\},\,S \ne \emptyset ,$ we show that

$$\begin{aligned} v(S\cup \{k\})=v(S)+v(\{k\})\quad \Leftrightarrow \quad \varDelta _v(S \cup \{k\})=0. \end{aligned}$$

Initialization: Let $s=1$. For $k \in \{i,j\}$ we have the following:

$$\begin{aligned}{}\begin{array}[b]{crcl} &{}v(S\cup \{k\}) &{}=&{}v(S)+v(\{k\})\\ \underset{(1)}{\Leftrightarrow }&{} \varDelta _v(S\cup \{k\})+\varDelta _v(S)+\varDelta _v(\{k\})&{}=&{}\varDelta _v(S)+\varDelta _v(\{k\})\\ \Leftrightarrow &{}\varDelta _v(S\cup \{k\})&{}=&{}0. \end{array} \end{aligned}$$

Induction step: Assume that equivalence and equality in the first and last line of the system above hold for all coalitions $S'$ with $s'\ge 1$ (IH), and let $s=s'+1$ and $k \in \{i,j\}$. We get the following:

$$\begin{aligned}\begin{array}{crcl} &{}v(S\cup \{k\}) &{}=&{}v(S)+v(\{k\})\\ \underset{(1)}{\Leftrightarrow }&{} \varDelta _v(S\cup \{k\})+\displaystyle {\sum _{R\subsetneq (S\cup \{k\})}\varDelta _v(R)}&{}=&{}\displaystyle {\sum _{R\subseteq S}\varDelta _v(R)}+\varDelta _v(\{k\})\\ \underset{(IH)}{\Leftrightarrow }&{}\varDelta _v(S\cup \{k\})+\varDelta _v(\{k\})+\displaystyle {\sum _{R\subseteq S}\varDelta _v(R)}&{}=&{}\displaystyle {\sum _{R\subseteq S}\varDelta _v(R)}+\varDelta _v(\{k\})\\ \Leftrightarrow &{}\varDelta _v(S\cup \{k\})&{}=&{}0. \end{array} \end{aligned}$$

Thus, the equivalence is shown. $\square$

1.2.2 Proof of Theorem 1

I. Existence: By Béal et al. (2018), ${\text {Sh}}^p$ satisfies E and D.

$\bullet$P: Let $v \in \mathcal {G}_{0}^N$ and $i,j \in N$, such that i and j are weakly dependent in v. We have the following:

$$\begin{aligned}{}\begin{array}[b]{rcl} {\text {Sh}}^{p}_i(v)&{} = &{}\displaystyle {\sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ v(\{i\})}{\sum _{k \in S}v(\{k\})}\varDelta _v(S)}\underset{Lem. 1}{=}v(\{i\})+ \sum _{\begin{array}{c} S\subseteq N,\\ {\{i,j\} \subseteq S} \end{array}} \frac{ v(\{i\})}{\sum _{k \in S}v(\{k\})}\varDelta _v(S)\\ &{}=&{}\displaystyle {\frac{v(\{i\})}{v(\{j\})}v(\{j\})+\frac{v(\{i\})}{v(\{j\})} \sum _{\begin{array}{c} S\subseteq N,\\ {\{i,j\} \subseteq S} \end{array}} \frac{ v(\{j\})}{\sum _{k \in S}v(\{k\})}\varDelta _v(S)}\\ &{}\underset{Lem.1}{=}&{}\displaystyle {\frac{v(\{i\})}{v(\{j\})} \sum _{\begin{array}{c} S\subseteq N,\\ {S\ni j} \end{array}} \frac{ v(\{j\})}{\sum _{k \in S}v(\{k\})}\varDelta _v(S)}=\frac{v(\{i\})}{v(\{j\})}{\text {Sh}}^{p}_j(v). \end{array} \end{aligned}$$

$\bullet$WA: Let $v,w \in \mathcal {G}_{0}^N$ with $w(\{i\})=c\cdot v(\{i\})$ for all $i \in N,\, c>0$. We have the following:

$$\begin{aligned}{}\begin{array}[b]{rcl} {\text {Sh}}_i^p(v)+ {\text {Sh}}_i^p(w)&{} \underset{(3)}{=}&{} \displaystyle {\sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ v(\{i\})}{\sum _{j \in S}v(\{j\})}\varDelta _v(S)+\sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ w(\{i\})}{\sum _{j \in S}w(\{j\})}\varDelta _w(S)} \\ &{}=&{}\displaystyle {\sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ v(\{i\})}{\sum _{j \in S}v(\{j\})}\varDelta _v(S)+\sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ c\cdot v(\{i\})}{\sum _{j \in S}c\cdot v(\{j\})}\varDelta _w(S)} \\ &{}=&{}\displaystyle {\sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ v(\{i\})}{\sum _{j \in S}v(\{j\})}\big [\varDelta _v(S)+\varDelta _w(S)\big ]} \\ &{}=&{} \displaystyle {\sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ (1+c)v(\{i\})}{\sum _{j \in S}(1+c)v(\{j\})}\big [\varDelta _v(S)+\varDelta _w(S)\big ]} \\ &{}=&{}\displaystyle {\sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ v(\{i\})+w(\{i\})}{\sum _{j \in S}\big [v(\{j\})+w(\{j\})\big ]}\varDelta _{v+w}(S)}\\[1ex] &{}=&{} {\text {Sh}}_i^p(v+w). \end{array} \end{aligned}$$

II. Uniqueness: Let $N\in \mathcal {N}, \,n:=|N|$, $v \in \mathcal {G}^{N}_{0},$ and $\varphi$ a TU-value which satisfies all axioms of Theorem 1. To prove uniqueness, we will show that $\varphi$ equals ${\text {Sh}}^p$.

For $n=1$, $\varphi$ equals ${\text {Sh}}^p$ by E.

Let now $n\ge 2$. For each coalition $S \subseteq N,\,S\ne \emptyset ,$ we define, corresponding to Remark 4, a TU-game $v_S \in \mathcal {G}^{N}_{0}$ through a vector $\overrightarrow{v_S}\in \mathbb {R}^{2^N-1}$ by assigning the coordinates of the related vector $\overrightarrow{\varDelta _{v_S}}\in \mathbb {R}^{2^N-1}$ in the entry of a coalition $R \subseteq N,\, R \ne \emptyset ,$ the dividend:

$$\begin{aligned} \varDelta _{v_S}(R):= {\left\{ \begin{array}{ll}\dfrac{v(\{j\})}{2^n-1}, \text { if } R=\{j\} \text { for all } j\in N, \\ \varDelta _{v}(S), \text { if } R=S, \,|S| \ge 2,\\ 0, \,\text {otherwise.} \end{array}\right. } \end{aligned}$$

Thus, each vector $\overrightarrow{v_S}\in \mathbb {R}^{2^N-1}$ gets in the coordinates of coalitions $R\subseteq N,\,R \ne \emptyset ,$ the entry:

$$\begin{aligned} v_S(R)= {\left\{ \begin{array}{ll} \varDelta _v(S)+\sum \nolimits _{j\in R}\dfrac{v(\{j\})}{2^n-1}, \text { if } R\supseteq S,\,|S| \ge 2,\\ \sum \nolimits _{j \in R}\dfrac{v(\{j\})}{2^n-1},\,\text {otherwise.} \end{array}\right. } \end{aligned}$$

(5)

We have $\displaystyle {\overrightarrow{\varDelta _{v}}=\sum _{\begin{array}{c} S \subseteq N, \\ S\ne \emptyset \; \end{array}}\overrightarrow{\varDelta _{v_S}}}$, and so, by Remark 4, $\displaystyle {v=\sum _{\begin{array}{c} S \subseteq N, \\ S\ne \emptyset \; \end{array}}v_S.}$

By D, we obtain the following:

$$\begin{aligned} \varphi _i(v_S)={\left\{ \begin{array}{ll}v_S(\{i\})=\dfrac{v(\{i\})}{2^n-1} \text { for all } i \in N \text { and }|S|= 1, \text { and} \\ v_S(\{i\})=\dfrac{v(\{i\})}{2^n-1} \text { for all } i \in N,\, i \notin S,\,|S|\ge 2. \end{array}\right. } \end{aligned}$$

(6)

By Lemma 1, all players $i \in S,\,|S|\ge 2,$ are pairwise weakly dependent in $v_S$. We get for an arbitrary $i \in S,\,|S|\ge 2,$ and by $v_S(N)\underset{\begin{array}{c} (5) \end{array}}{=}\varDelta _v(S)+\sum _{j\in N}\dfrac{v(\{j\})}{2^n-1}$:

$$\begin{aligned}{}\begin{array}[b]{rcl} \sum \nolimits _{j \in S}\varphi _j(v_S)&{}\underset{(\mathbf P )}{=}&{} \displaystyle {\sum _{j \in S}\frac{v_S(\{j\})}{v_S(\{i\})}\varphi _i(v_S)}= \sum _{j \in S}\frac{v(\{j\})}{v(\{i\})}\varphi _i(v_S)\\ &{}\underset{\begin{array}{c} (\mathbf E ) \end{array}}{=}&v_S(N)- \displaystyle {\sum _{j \in N\backslash S}\varphi _j(v_S)} \underset{\begin{array}{c} (6) \end{array}}{=}\varDelta _v(S)+\sum _{j\in S}\dfrac{v(\{j\})}{2^n-1} \\ \Leftrightarrow \;\;\varphi _i(v_S)&{}=&{} \displaystyle {\frac{v(\{i\})}{\sum _{j \in S}v(\{j\})}\varDelta _v(S)}+ \frac{v(\{i\})}{2^n-1}. \end{array} \end{aligned}$$

(7)

Therefore, we have by (3), (6), and (7) for all $S \subseteq N,\,S\ne \emptyset :$

$$\begin{aligned} \varphi _i(v_S)={\text {Sh}}^p_i(v_S) \text { for all } i\in N. \end{aligned}$$

${\text {Sh}}^p$ and $\varphi$ satisfy WA. It follows:

$$\begin{aligned} \varphi _i(v)={\text {Sh}}_i^p(v) \text { for all }i \in N, \end{aligned}$$

and uniqueness is shown. $\square$

1.2.3 Proof of Proposition 1

$\Rightarrow$: We show that WM implies WCSE: Let v and w two TU-games satisfying the hypotheses of WCSE, i.e. for a coalition $R \subseteq N, \,|R|\ge 2,$$c \in \mathbb {R},$ we have the following:

$$\begin{aligned} v(S)={\left\{ \begin{array}{ll}w(S)+c, \text { if }S\supseteq R,\\ w(S),\text { if }S\nsupseteq R.\end{array}\right. } \end{aligned}$$

Let $\varphi$ be a TU-value which obeys WM. By Lemma 4, we have

$$\begin{aligned} \varDelta _{v}(S)={\left\{ \begin{array}{ll}\varDelta _w(R)+c,\text { if } S=R, \\ \varDelta _w(S), \text { otherwise.}\end{array}\right. } \end{aligned}$$

Thus, we have $\varDelta _v(S \cup \{i\}) = \varDelta _w(S \cup \{i\})$ for all $i \in N\backslash R$ and $S \subseteq N\backslash \{i\}$. It follows from Lemma 5 that ${\text {MC}}_i^v(S)={\text {MC}}_i^w(S)$ for all $S \subseteq N\backslash \{i\}$. Therefore, we can use WM and get  $\varphi _i(v)= \varphi _i(w)\, \text { for all } i \in N\backslash R$ and WCSE is satisfied.

$\Leftarrow$: we show that WCSE implies WM: let $N \in \mathcal {N},\,i \in N,\,v,w \in \mathcal {G}^N$ two coalition functions satisfying the hypothesis of WM, i.e., ${\text {MC}}_i^v(S)={\text {MC}}_i^w(S)$ for all $S \subseteq N\backslash \{i\}$ and $w(\{k\})= v(\{k\})$ for all $k \in N$ and $\varphi$ a value satisfying WCSE. Then, by Lemma 5, we have $\varDelta _v(T) = \varDelta _w(T)$ for all $T \subseteq N,\,T\ni i$. Let $\mathcal {R}=\{R_j\subseteq N:\varDelta _v(R_j)\ne \varDelta _w(R_j) \}$ an indexed set of all subsets of N with different dividends in v and w, $1 \le j \le |\mathcal {R}|$. We inductively define a sequence of coalition functions $w_j,\,0\le j \le |\mathcal {R}|,$ by $w_j:=w$ if $j=0$, and, if $1\le j \le |\mathcal {R}|:$

$$\begin{aligned} \varDelta _{w_{j}}(S):={\left\{ \begin{array}{ll}\varDelta _{w_{j-1}}(R_j)+\big [\varDelta _v(R_j)-\varDelta _{w_{j-1}}(R_j)\big ],\text { if } S=R_j, \\ \varDelta _{w_{j-1}}(S), \text { if }S\subseteq N, \,S\ne R_j.\end{array}\right. } \end{aligned}$$

Then, we have $w_{|\mathcal {R}|}=v$ and, by Lemma 4 and WCSE, we get $\varphi _i(w_{j})=\varphi _i(w_{j-1})$ for all $j,\, 1 \le j \le |\mathcal {R}|,$ and therefore, $\varphi _i(v)=\varphi _i(w)$ and WM is satisfied. $\square$

1.2.4 Proof of Theorem 2

I. Existence: by Theorem 1, ${\text {Sh}}^p$ satisfies E and P.

$\bullet$WCSE: By Lemma 4, we have for v and a coalition R from WCSE:

$$\begin{aligned} \varDelta _{v}(S)={\left\{ \begin{array}{ll}\varDelta _w(R)+c,\text { if } S=R, \\ \varDelta _w(S), \text { otherwise}.\end{array}\right. } \end{aligned}$$

Thus, we obtain for all $i\in N\backslash R$ by the following:

$$\begin{aligned} {\text {Sh}}^{p}_i(v) = \sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ v(\{i\})}{\sum _{j \in S}v(\{j\})}\varDelta _v(S)=\sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i} \end{array}} \frac{ w(\{i\})}{\sum _{j \in S}w(\{j\})}\varDelta _w(S)={\text {Sh}}^{p}_i(w). \end{aligned}$$

II. Uniqueness: Let $N\in \mathcal {N}, \,n:=|N|$, $v \in \mathcal {G}^{N}_{0}$, and $\varphi$ a TU-value which satisfies all axioms of Theorem 2. We will show that $\varphi$ satisfies Eq. (3).

For $n=1$, Eq. (3) is satisfied by E.

Let $n\ge 2$. We use an induction on the size $r:=|\{R \subseteq N: R \text { is active in } v$ and $|R|\ge 2\}|$.

Initialization: Let $r=0$. By Lemma 1, all players $i,j \in N$ are pairwise weakly dependent in v. We get for an arbitrary $i \in N$:

$$\begin{aligned} \sum _{j \in N}\varphi _j(v)\underset{(\mathbf P )}{=} \sum _{j \in N}\frac{v(\{j\})}{v(\{i\})}\varphi _i(v)\underset{(\mathbf E )}{=}v(N). \end{aligned}$$

With $v(N)= \sum _{j \in N}v(\{j\})$ it follows that $\varphi _i(v) =v(\{i\})$ and Eq. (3) holds to $\varphi$ if r = 0.

Induction step: Assume that Eq. (3) holds to $\varphi$ if $r\ge 0,\,r$ arbitrary (IH), and let exactly $r+1$ coalitions $Q_k \subseteq N, \,|Q_k|\ge 2,\, 1\le k\le r+1,$ active in v. Let Q be the intersection of all such coalitions $Q_k$:

$$\begin{aligned} Q=\bigcap _{1\le k\le r+1}Q_k. \end{aligned}$$

We distinguish two cases: (a) $i \in N\backslash Q$ and (b) $i \in Q$.

(a) Each player $i \in N\backslash Q$ is a member of at most r active coalitions $Q_k,\, |Q_k|\ge 2,$ and v gets at least one active coalition $R_i,\,|R_i|\ge 2$, $i \notin R_i$. Hence, there exists a coalition function $w_i\in \mathcal {G}^{N}_{0}$, where all coalitions get the same dividend in $w_i$ as in v, except coalition $R_i$ which gets the dividend $\varDelta _{w_i}(R_i)=0,$ and there is a scalar $c \in \mathbb {R},\,c\ne 0,$ with the following:

$$\begin{aligned} \varDelta _{v}(S)={\left\{ \begin{array}{ll}\varDelta _{w_i}(R_i)+c,\text { if } S=R_i, \\ \varDelta _{w_i}(S), \text { otherwise. }\end{array}\right. } \end{aligned}$$

By Lemma 4 and WCSE, we get $\varphi _i(v)=\varphi _i(w_i) \text { with } i \in N\backslash R_i$ and because there exists for all $i \in N\backslash Q$ a such $R_i$, we get $\varphi _i(v)=\varphi _i(w_i) \text { for all } i \in N\backslash Q$. All coalition functions $w_i$ get at most r active coalitions with at least two players and Eq. (3) follows by (IH). Thus, we have the following:

$$\begin{aligned} \varphi _i(v)={\text {Sh}}_i^p(v) \text { for all }i \in N\backslash Q. \end{aligned}$$

(8)

(b) If $Q=\{i\}$, we get, by E of $\varphi$ and ${\text {Sh}}^p$ and case (a), $\varphi _i(v)= {\text {Sh}}^p_i(v)$. If $|Q|\ge 2$, each player $j \in Q$ is a member of all $r+1$ active coalitions $Q_k \subseteq N, \,|Q_k|\ge 2,\, 1\le k\le r+1,$ and therefore, by Lemma 1, all players $j \in Q$ are weakly dependent. By P and E of $\varphi$ and ${\text {Sh}}^p$, we get for an arbitrary $i \in Q$ the following:

$$\begin{aligned} \sum _{j \in Q}\varphi _j(v)&\underset{(\mathbf P )}{=} \sum _{j \in Q}\frac{v(\{j\})}{v(\{i\})}\varphi _i(v) \underset{\begin{array}{c} (\mathbf E )\\ (8) \end{array}}{=}v(N)- \sum _{j \in N\backslash Q}{\text {Sh}}^p_j(v)\underset{\begin{array}{c} (\mathbf E ) \end{array}}{=}\sum _{j \in Q}{\text {Sh}}^p_j(v)\\&\underset{(\mathbf P )}{=} \sum _{j \in Q}\frac{v(\{j\})}{v(\{i\})}{\text {Sh}}^p_i(v) \;\;\Leftrightarrow \;\;\varphi _i(v)={\text {Sh}}_i^p(v) \end{aligned}$$

and together with I. the proof is complete. $\square$

1.2.5 Proof of Proposition 2

Let $(N,v) \in \mathcal {G}_{0}^N,\,j \in N,$ and $(N^j,v^j) \in \mathcal {G}_0^{N^j}$ a corresponding split player game to (N, v). We point out that we have for all $S \subseteq N\backslash \{j\},\, S \ne \emptyset ,$$\varDelta _{v^j}(S)=\varDelta _{v}(S),$$\varDelta _{v^j}(S\cup \{k,l\})=\varDelta _v(S\cup \{j\})$, and $\varDelta _{v^j}(S\cup \{k\})=\varDelta _{v^j}(S\cup \{\ell \})=0$. Then, we get for all $i \in N\backslash \{j\}$ the following:

$$\begin{aligned} {\text {Sh}}^{p}_i(N,v) =&\sum _{\begin{array}{c} R\subseteq N,\\ {R\ni i} \end{array}} \frac{ v(\{i\})}{\sum _{m \in R}v(\{m\})}\varDelta _v(R)\\ =&\sum _{\begin{array}{c} S\subseteq N\backslash \{j\},\\ {S\ni i} \end{array}} \frac{ v(\{i\})}{\sum _{m \in S}v(\{m\})}\varDelta _v(S)\\&+\sum _{\begin{array}{c} S\subseteq N\backslash \{j\},\\ {S\ni i} \end{array}} \frac{ v(\{i\})}{\sum _{m \in S\cup \{j\}}v(\{m\})}\varDelta _v(S\cup \{j\}) \\ =&\sum _{\begin{array}{c} S\subseteq N^j\backslash \{k,\ell \},\\ {S \ni i} \end{array}} \frac{ v^j(\{i\})}{\sum _{m \in S}v^j(\{m\})}\varDelta _{v^j}(S)\\&+\sum _{\begin{array}{c} \begin{array}{c} S\subseteq N^j\backslash \{k,\ell \},\\ {S\ni i} \end{array} \end{array}} \frac{ v^j(\{i\})}{\sum _{m \in S\cup \{k,\ell \}}v^j(\{m\})}\varDelta _{v^j}(S\cup \{k,\ell \})\\ =&\;\sum _{\begin{array}{c} R\subseteq N^j,\\ {R\ni i} \end{array}} \frac{ v^j(\{i\})}{\sum _{m \in R}v^j(\{m\})}\varDelta _{v^j}(R)\\ =&\;{\text {Sh}}^{p}_i(N^j,v^j). \end{aligned}$$

$\square$

1.2.6 Proof of Lemma 2

Let $N=\{1,2,..., n\},\, |N|\ge 2$, $v \in \mathcal {G}_{0}^N$, $\varphi$ a TU-value which satisfies E and PS for all $v \in \mathcal {G}_{0}^N$, and w.l.o.g., player 1 and player 2 be symmetric in v. If we split player 1 according to PS into two new players, player $n+1$ and player $n+2$, $N^1= \{2,3,...,n, n+1, n+2\}$, we have the following:

$$\begin{aligned} \varphi _{2}(N^1,v^1)= \varphi _{2}(N,v), \end{aligned}$$

(9)

and, if we split player 2 according to PS into the same players as before, player $n+1$ and player $n+2$, instead, $N^2= \{1,3, 4,...,n, n+1, n+2\}$, we have the following:

$$\begin{aligned} \varphi _{1}(N^2,v^2)= \varphi _{1}(N,v), \end{aligned}$$

(10)

where we choose $v^2(\{n+1\}):=v^1(\{n+1\})$ and $v^2(\{n+2\}):=v^1(\{n+2\})$.

In the same manner, we split now in the game $(N^1, v^1)$ player 2 into two new players, player $n+3$ and player $n+4$, and analogous in the game $(N^2, v^2)$ player 1 into the same players as before, player $n+3$ and player $n+4$, and choose ${v^{2}}^1(\{n+3\}):={v^{1}}^2(\{n+3\})$ and ${v^{2}}^1(\{n+4\}):={v^{1}}^2(\{n+4\})$. We have ${N^{1}}^2={N^{2}}^1=\{3,4,..., n,n+1,n+2,n+3,n+4\}$ and ${v^{1}}^2={v^{2}}^1$ and get by E, according to Remark 1:

$$\begin{aligned} \varphi _{n+3}\Big ({{N^1}^2,{v^1}^2}\Big )+\varphi _{n+4}\Big ({{N^1}^2,{v^1}^2}\Big )&= \varphi _{2}(N^1,v^1)\underset{(9)}{=}\varphi _{2}(N,v),\\ \varphi _{n+3}\Big ({{N^{2}}^1,{v^2}^1}\Big )+\varphi _{n+4}\Big ({{N^{2}}^1,{v^2}^1}\Big )&= \varphi _{1}(N^2,v^2)\underset{(10)}{=}\varphi _{1}(N,v). \end{aligned}$$

Hence, we have $\varphi _{1}(N,v)= \varphi _{2}(N,v)$ and S is shown. $\square$

1.2.7 Proof of Lemma 3

Let $N \in \mathcal {N},\, |N|\ge 2$, $v \in \mathcal {G}_{0_{\mathbb {Q}}}^N$ a TU-game, and, w.l.o.g., players $i,j \in N$, such that i and j are weakly dependent in v. Furthermore, let $\varphi$ a TU-value which satisfies E and PS for all $v \in \mathcal {G}_{0_{\mathbb {Q}}}^N$, and therefore, by Lemma 2, also S. Due to $v(\{i\}),v(\{j\}) \in \mathbb {Q}\backslash \{0\}$, the worths of the singletons $v(\{k\}),\,k \in \{i,j\}$, can be written as a fraction. We distinguish two cases: (a) $v(\{k\})>0$ and (b) $v(\{k\})<0$.

(a) We have the following:

$$\begin{aligned} v(\{k\})=\;\dfrac{p_k}{q_k} \;\text { with }\; p_k,q_k \in \mathbb {N}. \end{aligned}$$

We choose a main denominator q of these two fractions by $q:=q_iq_j$. With $z_i:= p_iq_j$ and $z_j:= p_jq_i$, we get the following:

$$\begin{aligned} v(\{i\})= \;\frac{z_i}{q}\;\text { and }\;v(\{j\})= \;\frac{z_j}{q}. \end{aligned}$$

(11)

Now, we define a player set $N'$ and a coalition function $v'$ by “splitting” each player $k \in \{i,j\}$ into $z_k$ players $k_{1}$ to $k_{z_k}$, such that we have $N'=(N\backslash \{i,j\})\cup \{i_m: 1\le m \le z_i\}\cup \{j_m: 1\le m \le z_j\}$. Each player $k_{m}\in N'\backslash (N\backslash \{i,j\}),\,\, 1\le m \le z_k,$ gets a singleton worth $v'(\{k_{m}\})=\frac{1}{q} \text { for } k \in \{i,j\}$, synonymous with

$$\begin{aligned} v'(\{\ell \})= \frac{1}{q} \text { for all } \ell \in N'\backslash (N\backslash \{i,j\}), \end{aligned}$$

where $|N'\backslash (N\backslash \{i,j\})|=z_i +z_j$ and $v(\{k\})= \sum _{1\le m \le z_k}v'(\{k_{m}\}),\,k \in \{i,j\}$. We define $v'(R'):=v(R)$ for all $R'= R\backslash \{i,j\} \cup N'\backslash (N\backslash \{i,j\}),\, R \subseteq N, \{i,j\}\subseteq R,$ and $v'(S):= v(S)$ for all $S\subseteq N'$ with $S\subseteq N$. All other coalitions $T\subseteq N'$ are defined as not active in $v'$.

Applying PS (repeatedly) to $v,\; \varphi$ and the two players $i,j \in N$, we can get the coalition function $v'$ defined just before and, by Remark 1, we have the following:

$$\begin{aligned} \varphi _k(N,v) = \sum _{1\le m \le z_k}\varphi _{k_{m}}(N',v') \text { for } k \in \{i,j\}. \end{aligned}$$

(12)

All players $\ell \in N'\backslash (N\backslash \{i,j\})$ are symmetric in $v'$. Hence, it follows from S that

$$\begin{aligned} \varphi _{\ell }(N',v')=\,\frac{\varphi _i(N,v)+\varphi _j(N,v)}{z_i+z_j}\text { for all } \ell \in N'\backslash (N\backslash \{i,j\}). \end{aligned}$$

We get

$$\begin{aligned} \varphi _k(N,v)\underset{(12)}{=}\,\sum _{1\le m \le z_k}\varphi _{k_{m}}(N',v')=\frac{z_k}{z_i+z_j}\big [\varphi _i(N,v)+\varphi _j(N,v)\big ]\text { for } k \in \{i,j\}. \end{aligned}$$

It follows:

$$\begin{aligned} \varphi _i(N,v) = \frac{z_i}{z_j}\varphi _j(N,v)\underset{(11)}{=} \frac{v(\{i\})}{v(\{j\})}\varphi _j(N,v) \end{aligned}$$

and P is shown.

(b) We have the following:

$$\begin{aligned} v(\{k\})=\;\dfrac{p_k}{q_k} \;\text { with }\; (-p_k),q_k \in \mathbb {N}. \end{aligned}$$

We choose a main denominator q of these two fractions by $q:=-q_iq_j$. With $z_i:= -p_iq_j$ and $z_j:= -p_jq_i$, we get

$$\begin{aligned} v(\{i\})= \;\frac{z_i}{q}\;\text { and }\;v(\{j\})= \;\frac{z_j}{q}. \end{aligned}$$

The remaining part of the proof equals the related part in case (a). $\square$

1.3 Logical independence

Finally, we want to show the independence of the axioms used in the characterizations.

Remark 5

Let $v \in \mathcal {G}^N_0,\; N\in \mathcal {N}.$ The axioms in Theorem 1/Corollary 2 are logically independent:

• E: The TU-value $\varphi$ defined by

  $$\begin{aligned} \varphi _i(v)= v(\{i\})+2\cdot \sum _{\begin{array}{c} S\subseteq N,\\ {S\ni i,\, S\ne \{i\}} \end{array}} \frac{ v(\{i\})}{\sum _{j \in S}v(\{j\})}\varDelta _v(S)\;\text { for all } \,i \in N \end{aligned}$$

  satisfies D, P/PS, and WA but not E.

• D: The proportional rule$\pi$ (Moriarity 1975), given by

  $$\begin{aligned} \pi _i(v) = \frac{ v(\{i\})}{\sum _{j \in N}v(\{j\})}v(N)\;\text { for all } \,i \in N, \end{aligned}$$

  (13)

  satisfies E, P/PS, and WA but not D.

• P/PS: ${\text {Sh}}$ satisfies E, D, and WA but not P/PS.

• WA: The TU-value $\varphi$ defined for all $i \in N$ by

  $$\begin{aligned} \varphi _i(v)= {\left\{ \begin{array}{ll} v(\{i\}) , \text { if}~ i~ \text {is a dummy player,}\\ \dfrac{v(\{i\})}{\sum _{\begin{array}{c} j \in N,\\ j {\textit{ is no dummy}} \end{array}}v(\{j\})}\Big [v(N)-\sum \nolimits _{\begin{array}{c} j \in N,\\ j {\textit{ is a dummy}} \end{array}}v(\{j\})\Big ], \text { otherwise},\end{array}\right. } \end{aligned}$$

  satisfies E, D, and P/PS but not WA.

Remark 6

Let $v \in \mathcal {G}^N_0,\, N\in \mathcal {N}.$ The axioms in Theorem 2/Corollary 1 are logically independent:

• E: The TU-value $\varphi$ defined for all $i \in N$ by

  $$\begin{aligned} \varphi _i(v)= {\left\{ \begin{array}{ll}0, \text { if } |N|=1, \\ {\text {Sh}}^p_i(v), \text { otherwise,}\end{array}\right. } \end{aligned}$$

  satisfies P/PS and WCSE but not E.

• P/PS: ${\text {Sh}}$ satisfies E and WCSE but not P/PS.

• WCSE: The proportional rule $\pi$ [Eq. (13)] satisfies E and P/PS but not WCSE.