The class of ASN-position values

Abstract

This paper introduces a class of cooperative allocation rules for TU-games with a network structure that account for an agent’s centrality in the network. In contrast to existing centrality measures, this approach analyzes the consequences of tie failure (rather than node failure). Though not directly applied to the centrality issue, tie failures are the idea underlying the Position value (the Shapley value of the arc-game). In contrast, we allow for a whole class of allocation rules as a basis since the Shapley approach might be unreasonable in political applications, especially in networks with incompatibilities. Requiring additivity (A), equal treatment of equals (Symmetry S), and irrelevance of unproductive agents (nullplayer irrelevance N) for the underlying basis, we define the class of ASN-position values and provide axiomatic characterizations. To emphasize the crucial role of links, we refine this class to multiplicative ASN-position values and provide a monotonicity result with respect to links. We apply our approach to the case of the 2001 state parliament elections in Hamburg, Germany to use our allocation rule as a power index and further provide an example where we use our approach as a centrality measure to identify top key nodes. In both applications, the position value taking the Banzhaf value as a basis turns out to be more convincing than the Shapley-based approach.

Appendices

Appendix A: Proofs

Proof of Lemma 1: If $Y\in \mathcal {Y}$ satisfies S, the corresponding Y-Position value $\pi ^Y$ satisfies DEG.

Proof

Consider any (N, v, g) which is link anonymous, that is, there exists a function $f:\{0,1,\ldots ,|g|\}\longrightarrow {\varvec{\mathbb {R}}}$ such that $v^N(g')=f(|g'|)$ for all $g'\subseteq g$.Then, for any two links $\lambda ,\tilde{\lambda }\in g$, we have

$$\begin{aligned} v^N(g'\cup \lambda )= & {} f(|g'\cup \lambda |)=f(|g'|+1)=f(|g'\cup \tilde{\lambda }|)\\= & {} v^N(g'\cup \tilde{\lambda })\quad \forall \,g'\subseteq g{\setminus }\{\lambda ,\tilde{\lambda }\} \end{aligned}$$

that is, all $\lambda ,\tilde{\lambda }\in g$ are (pairwise) symmetric in $(g,v^N)$. Therefore, for any allocation rule $Y\in \mathcal {Y}$ which satisfies S, we have $Y_{\lambda }(g,v^N)=Y_{\tilde{\lambda }}(g,v^N)=:\alpha \in {\varvec{\mathbb {R}}}$ (constant due to S). Hence, for the corresponding Y-Position value we have for any $i\in N$

$$\begin{aligned} \pi ^Y_i(N,v,g)=\sum \limits _{\lambda \in g_i}\frac{1}{2}Y_{\lambda }(g,v^N)=\alpha \frac{|g_i|}{2} \end{aligned}$$

which proves DEG. $\square$

Proof of Lemma 2: If $Y\in \mathcal {Y}$ satisfies NI, the corresponding Y-Position value $\pi ^Y$ satisfies SLP.

Proof

Consider any (N, v, g). For all $\tilde{\lambda }\in g$ which are superfluous in (N, v, g), that is, $v^N(g'\cup \tilde{\lambda })=v^N(g')$ for all $g'\subseteq g{\setminus }\{\tilde{\lambda }\}$, we have that $\tilde{\lambda }$ is a Nullplayer in $(g,v^N)$. By Y satisfying NI=NPO+N we have for all $i\in N$

$$\begin{aligned} \pi _i^Y(N,v,g)&=\sum \limits _{\lambda \in g_i{\setminus }\tilde{\lambda }}\frac{1}{2}Y_{\lambda }(g,v^N)+{\left\{ \begin{array}{ll}\frac{1}{2}Y_{\tilde{\lambda }}(g,v^N),&{} \text {if }\tilde{\lambda }\in g_i\\ 0,&{} \text {otherwise }\end{array}\right. }\\&\mathop {=}\limits ^{\text {NPO}}\sum \limits _{\lambda \in g_i{\setminus }\tilde{\lambda }}\frac{1}{2}Y_{\lambda }(g{\setminus }\tilde{\lambda },v^N)+{\left\{ \begin{array}{ll}\frac{1}{2}Y_{\tilde{\lambda }}(g,v^N),&{} \text {if }\tilde{\lambda }\in g_i\\ 0,&{} \text {otherwise }\end{array}\right. }\\&\mathop {=}\limits ^{\text {N}}\sum \limits _{\lambda \in g_i{\setminus }\tilde{\lambda }}\frac{1}{2}Y_{\lambda }(g{\setminus }\tilde{\lambda },v^N)=\pi _i^Y(N,v,g{\setminus }\tilde{\lambda }) \end{aligned}$$

which proves SLP. $\square$

Proof of Theorem 1: If an allocation rule for TU games with a network structure satisfies A, DEG, SLP and Y-CLP, $Y\in ASN$, it is uniquely determined on cycle-free networks and it is given by the ASN-Position value $\pi ^Y\in \Pi (ASN)$ corresponding to Y.

Proof

Existence: Since $\pi ^Y\in \Pi (ASN)$, that is, the corresponding Y satisfies A, S, NI, we know that $\pi ^Y$ satisfies A, DEG and SLP by Corollary 1, Lemmas 1 and 2, respective ly. Y-CLP follows by Corollary 2.

Uniqueness: We follow the idea of the proof for the (Shapley-) Position value of Borm et al. (1992). Let Y in ASN and let $W\in \mathcal {Y}_G$ satisfy A, DEG, SLP and Y-CLP. Consider any TU-game with a network structure (N, v, g) such that g is cycle free. The unanimity games $\{u_T\}_{T\in 2^N{\setminus } \{\emptyset \}}$ with $u_T(K)=1$, if $T\subseteq K$ and 0, otherwise, build a basis of V. Hence, every $v\in V$ can be written as

$$\begin{aligned} v=\sum \limits _{T\in 2^N{\setminus } \{\emptyset \}}\mu _T(v)u_T \end{aligned}$$

where the coefficients $\{\mu _T\}_{T\in 2^N{\setminus } \{\emptyset \}}$ are called the Harsanyi dividends of v. Therefore, by A, it is sufficient to show that $W(N,\beta u_T,g)$ is uniquely determined for all $\beta \in {\varvec{\mathbb {R}}}$ and $T\in 2^N$ such that $|T|\ge 2$. Let such $\beta , T$ be arbitrary but fixed.

Case 1: $\not \exists$ $C\in \mathcal {C}(N,g)$ such that $T\subseteq C$.

That is, there exists $i,j\in T$ being unconnected in g and hence, $\beta u_T^N(g')=0$ for all $g'\subseteq g$. Therefore, every $\lambda \in g$ is superfluous, and hence, by SLP, we have $W(N,\beta u_T,g)=W(N,\beta u_T,g{\setminus }\lambda _1)=W(N,\beta u_T,g{\setminus }\{\lambda _1,\lambda _2\})=\cdots =W(N,\beta u_T,\emptyset )$.

Trivially, the game $(N,\beta u_t,\emptyset )$ is link anonymous and hence, by DEG, there exists a constant $\alpha \in {\varvec{\mathbb {R}}}$ such that

$$\begin{aligned} W_i(N,\beta u_T,g)=W_i(N,\beta u_t,\emptyset )=\alpha \cdot C^d_i(\emptyset )=0\,\forall \, i\in N\quad \text {(uniquely determined)} \end{aligned}$$

Case 2: $\exists$ $C\in \mathcal {C}(N,g)$ such that $T\subseteq C$.

Consider the (unique) connected hull (cf. Owen 1986) of T, H(T), given by

$$\begin{aligned} H(T):=\bigcap \big \{S|T\subseteq S\subseteq C\text { such that }g|_{S}\text { is connected subgraph}\big \} \end{aligned}$$

As g is cycle-free, H(T) is the minimal set of nodes that are essential to connect T. Note that cycle-freeness is essential here: if there is more than one path connecting T, the intersection is empty on the disjoint parts of the connecting paths. We have

$$\begin{aligned} \beta u_T^N(g')={\left\{ \begin{array}{ll}\beta ,&{}\text { if }g|_{H(T)}\subseteq g'\\ 0,&{}\text { otherwise}\end{array}\right. } \end{aligned}$$

All links $\lambda \notin g|_{H(T)}$ are superfluous in $(N,\beta u_T,g)$, hence, by SLP,$W(N,\beta u_T,g)=W(N,\beta u_T,g|_{H(T)})$.

The game $(N,\beta u_T,g|_{H(T)})$ is link anonymous (all links have the same number of swings, namely, one) with

$$\begin{aligned} f:\{0,1,\ldots ,|g|_{H(T)}|\}\longrightarrow {\varvec{\mathbb {R}}}, f(x):={\left\{ \begin{array}{ll} \beta ,&{}\text { if }x=|g|_{H(T)}|\\ 0,&{}\text { otherwise} \end{array}\right. } \end{aligned}$$

hence, by DEG, there exists a constant $\alpha \in {\varvec{\mathbb {R}}}$ such that

$$\begin{aligned} W_i(N,\beta u_T,g)\mathop {=}\limits ^{\text {SLP}}W_i(N,\beta u_T, g|_{H(T)})\mathop {=}\limits ^{\text {DEG}}\alpha \cdot C^d_i(g|_{H(T)}) \end{aligned}$$

(1)

It directly follows that

$$\begin{aligned} W_i(N,\beta u_T,g)=0\quad \forall \, i\in N{\setminus } H(T) \end{aligned}$$

(2)

By (2) and Y-CLP, we have

$$\begin{aligned} \sum \limits _{i\in H(T)}W_i(N,\beta u_T,g)&\mathop {=}\limits ^{(2)}\sum \limits _{i\in C}W_i(N,\beta u_T,g) \mathop {=}\limits ^{\text {Y-CLP}}\sum \limits _{\lambda \in g|_C}Y_{\lambda }(g,\beta u_T^N). \end{aligned}$$

Note that all $\lambda \in g{\setminus } g|_{H(T)}=g|_{N{\setminus } H(T)}$ are Nullplayers in $(g,\beta u_T^N)$ which yields and $Y_{\lambda }(g,\beta u_T^N)=0$ since Y satisfies N. Furthermore, all $\lambda ,\tilde{\lambda }\in g|_{H(T)}$ are pairwise symmetric in $(g,\beta u_T^N)$ and since Y satisfies S, there exist $B^Y_T\in {\varvec{\mathbb {R}}}$ such that

$$\begin{aligned} Y_{\lambda }(g,\beta u_T^N)={\left\{ \begin{array}{ll} B^Y_T,&{}\text {if }\lambda \in g|_{H(T)}\\ 0,&{}\lambda \notin g|_{H(T)} \end{array}\right. }\Rightarrow \sum \limits _{\lambda \in g|_C}Y_{\lambda }(g,\beta u_T^N)=B_T^Y\cdot |g|_{H(T)}| \end{aligned}$$

where $B^Y_T\in {\varvec{\mathbb {R}}}$ is uniquely determined by $Y, \beta$ and T due to single-valuedness of Y and independent on g by NPO.

On the other hand, by (1), we have

$$\begin{aligned} \sum \limits _{i\in C}W_i(N,\beta u_T,g)=\alpha \sum \limits _{i\in C}C^d_i(g|_{H(T)})=\alpha \cdot 2\cdot |g|_{H(T)}| \end{aligned}$$

Combining this, we get

$$\begin{aligned} \alpha \cdot 2\cdot |g|_{H(T)}|=B_T^Y\cdot p|g|_{H(T)}|\Leftrightarrow \alpha =\frac{B_T^Y}{2} \end{aligned}$$

and hence

$$\begin{aligned} W_i(N,\beta u_T,g)=\frac{B_T^Y}{2}C^d_i(g|_{H(T)})=\frac{B_T^Y}{2}|g_i|_{H(T)}| \end{aligned}$$

which is uniquely determined, given $Y\in ASN$. $\square$

Proof of Lemma 3: If $Y\in ASN$, the corresponding Y-Position value $\pi ^Y$ satisfies BLC.

Proof

Let (N, v, g) be a TU game with a network structure. Following Slikker (2005a), there exists a unique linear combination of unanimity games (on link sets) representing the link game $v^N$, that is $v^N$ exists $\{\beta ^{v}_{g'}\}_{g'\subseteq g}$ such that

$$\begin{aligned} v^N=\sum \limits _{g'\subseteq g}\beta ^v_{g'}u_{g'}. \end{aligned}$$

Note that the dividends $\beta ^v$ do not depend on the choice set g (cf. Slikker 2005a) and that $\beta ^v_{\emptyset }=0$ for zero-normalized v.

Now let $Y\in ASN$ and consider $(g,\beta ^v_{g'}u_{g'})$ with $g'\subseteq g$ arbitrary but fixed. All $\lambda \in g{\setminus } g'$ are Nullplayers in this game which yields $Y_\lambda (g,\beta ^v_{g'}u_{g'})=0$ for all $\lambda \in g{\setminus } g'$ by N and $Y_\lambda (g,\beta ^v_{g'}u_{g'})=Y_\lambda (g',\beta ^v_{g'}u_{g'})$ for all $\lambda \in g'$ by NPO. All $\lambda ,\tilde{\lambda }\in g'$ are (pairwise) symmetric which yields $Y_\lambda (g,\beta ^v_{g'}u_{g'})=Y_\lambda (g',\beta ^v_{g'}u_{g'})=B_{g'}^{v,Y}$ by S where $B_{g'}^{v,Y}\in \mathbb {R}$ is unique by single-valuedness and does not depend on the choice set g (NPO).

By A we obtain

$$\begin{aligned} Y_\lambda (g,v^N)=\sum \limits _{g'\subseteq g}Y_\lambda (g,\beta ^v_{g'}u_{g'})=\sum \limits _{\begin{array}{c} g'\subseteq g:\\ \lambda \in g' \end{array}}B_{g'}^{v,Y}\quad \forall \,\lambda \in g. \end{aligned}$$

For the corresponding Y-Position value we hence have

$$\begin{aligned} \pi ^{Y}_i(N,v,g)&=\sum \limits _{\lambda \in g_i}\frac{1}{2}\sum \limits _{\begin{array}{c} g'\subseteq g:\\ \lambda \in g' \end{array}}B_{g'}^{v,Y}=\sum \limits _{\begin{array}{c} g'\subseteq g,\\ g'\ne \emptyset \end{array}}\sum \limits _{\lambda \in g_i'}\frac{1}{2}B_{g'}^{v,Y}=\sum \limits _{\begin{array}{c} g'\subseteq g,\\ g'\ne \emptyset \end{array}}B_{g'}^{v,Y}\frac{|g_i'|}{2} \end{aligned}$$

and therefore (using that $B_{g'}^{v,Y}$ does not depend on g) we obtain

which proves BLC. $\square$

Proof of Theorem 2: If an allocation rule for TU games with a network structure satisfies BLC and Y-CLP, $Y\in ASN$, it is uniquely determined for general networks and it is given by the ASN-Position value $\pi ^Y\in \Pi (ASN)$ corresponding to Y.

Proof

Existence: Since $\pi ^Y\in \Pi (ASN)$, that is, the corresponding Y is an ASN-value, we know that $\pi ^Y$ satisfies BLC by Lemma 3. Y-CLP follows by Corollary 2. Uniqueness: Suppose V, W being two allocation rules for network structures satisfying BLC and Y-CLP, $Y\in ASN$. We proceed by induction over |g|.

Induction basis [IB]: For $|g|=0$, that is, $g=\emptyset$, we have that $\mathcal {C}(N,g)=\{i\}_{i\in N}$ and hence, by Y-CLP, we have

$$\begin{aligned} V_i(N,v,g)&=\sum \limits _{i\in C}V_i(N,v,g)= \sum \limits _{\lambda \in \emptyset }Y_{\lambda }(g,v^N)=0\\&=\sum \limits _{i\in C}W_i(N,v,g)=W_i(N,v,g). \end{aligned}$$

Now suppose that $V(N,v,g)=W(N,v,g)$ for all g such that $|g|=k, k\ge 0$ (induction hypothesis [IH]).

Consider g such that $|g|=k+1$. As $k+1\ge 1$, there exists $i,j\in N, i\ne j$ such that $j\in C_i(N,g)$ (for all l with $|C_l(N,g)|=1$, we have $V_l(N,v,g)=W_l(N,v,g)$ by Y-CLP). By BLC, we have for all $i,j\in C_i(N,g)$:

$$\begin{aligned} \sum \limits _{\lambda \in g_j}V_i(N,v,g)-\sum \limits _{\lambda \in g_i}V_j(N,v,g)&\mathop {=}\limits ^{\text {BLC}}\sum \limits _{\lambda \in g_j}V_i(N,v,g{\setminus }\lambda )-\sum \limits _{\lambda \in g_i}V_j(N,v,g{\setminus }\lambda )\\&\mathop {=}\limits ^{\text {[IH]}}\sum \limits _{\lambda \in g_j}W_i(N,v,g{\setminus }\lambda )-\sum \limits _{\lambda \in g_i}W_j(N,v,g{\setminus }\lambda )\\&\mathop {=}\limits ^{\text {BLC}}\sum \limits _{\lambda \in g_j}W_i(N,v,g)-\sum \limits _{\lambda \in g_i}W_j(N,v,g) \end{aligned}$$

and hence

$$\begin{aligned} |g_j|V_i(N,v,g)-|g_i|V_j(N,v,g)=|g_j|W_i(N,v,g)-|g_i|W_j(N,v,g) \end{aligned}$$

(3)

Summing up (3) over all $j\in C$ yields:

$$\begin{aligned}&\displaystyle \sum \limits _{j\in C}\left[ |g_j|V_i(N,v,g)-|g_i|V_j(N,v,g)\right] =\sum \limits _{j\in C}\left[ |g_j|W_i(N,v,g)-|g_i|W_j(N,v,g)\right] \\&\displaystyle \Leftrightarrow V_i(N,v,g)\underbrace{\sum \limits _{j\in C}|g_j|}_{\begin{array}{c} :=A(j)>0 \\ \text {by }|C_i|\ge 1 \end{array}}-|g_i|\sum \limits _{j\in C}V_j(N,v,g) =W_i(N,v,g)\underbrace{\sum \limits _{j\in C}|g_j|}_{\begin{array}{c} :=A(j)>0 \\ \text {by }|C_i|\ge 1 \end{array}}-|g_i|\sum \limits _{j\in C}W_j(N,v,g)\\&\displaystyle \mathop {\Rightarrow }\limits ^{\text {Y-CLP}} A(j)\cdot V_i(N,v,g)-|g_i|\sum \limits _{\lambda \in g|_C}Y_{\lambda }(g,v^N) =A(j)\cdot W_i(N,v,g)-|g_i|\sum \limits _{\lambda \in g|_C}Y_{\lambda }(g,v^N)\\&\displaystyle \Leftrightarrow V_i(N,v,g)=W_i(N,v,g)\quad \forall \, i\in N \end{aligned}$$

which finishes the proof. $\square$

Proof of Theorem 3: Any multiplicative Y-Position value $\pi ^Y\in \Pi (ASN_m)$ can be written as

$$\begin{aligned} \pi _i^Y(N,v,g)=\sum \limits _{\lambda \in g_i}\sum \limits _{g'\subseteq g{\setminus }\lambda }\frac{w^Y_{|g|,|g'|}}{2} \left( v^N(g'\cup \lambda )-v^N(g')\right) \end{aligned}$$

where the weighting scheme $\left\{ w^Y_{|g|,|g'|}\right\} _{|g'|=0,\ldots ,|g|-1}$ satisfies

• 1. $w^Y_{|g|,|g'|}=Y_{\lambda }(g,1_{g'\cup \lambda })$ for any $\lambda \in g$ and $g'\subseteq g{\setminus }\lambda$

• 2. $w^Y_{|g|,|g'|}+w^Y_{|g|,|g'|+1}=w^Y_{|g|-1,|g'|}\quad \forall \,|g'|=0,\ldots ,|g|-1$

Proof

Let $Y\in ASN, K\subseteq N$ be arbitrary but fixed and consider the game $(N,1_K)$. All players in K as well as all players in $N{\setminus } K$ are (pairwise) symmetric in this game, hence, by S, there must exist $w^Y(N,K)$ and $\tilde{w}^Y(N,K)$ (uniquely determined by single-valuedness) such that we have

$$\begin{aligned} Y_i(N,1_K)={\left\{ \begin{array}{ll} w^Y(N,K),&{}i\in K\\ \tilde{w}^Y(N,K),&{}i\notin K \end{array}\right. } \end{aligned}$$

(4)

for all $i\in N$. Let $i\in N$ be arbitrary but fixed and, for all $K\subseteq N{\setminus }\{i\}$, consider the game $(N,1_{K\cup \{i\}}+1_K)$. One can easily check that i is a Nullplayer in this game and using N and A we observe

$$\begin{aligned} \tilde{w}^Y(N,K)=-w^Y(N,K\cup \{i\})\text { for all }i\in N\text { and all }K\subseteq N{\setminus }\{i\}. \end{aligned}$$

(5)

Now, let $i,j\in N, j\ne i$ be arbitrary but fixed and, for all $K\subseteq N{\setminus }\{i,j\}$, consider the game $(N,1_{K\cup \{i\}}+1_{K\cup \{j\}})$. One can easily check that i and j are symmetric in this game and using S, A and (5) we observe

$$\begin{aligned} w^Y(N,K\cup \{i\})=w^Y(N,K\cup \{j\})\text { for all }i,j\in N,i\ne j,\text { and all }K\subseteq N{\setminus }\{i,j\}. \end{aligned}$$

(6)

For any $K,K'\subseteq N$ with $|K|=|K'|$, we can build a sequence$\{K=K_0,K_1,\ldots , K_{l}=K'\}$ where every set can be obtained from its predecessor by replacing exactly one player. Using (6) we hence obtain that the weights only depend on sizes of coalitions (cf. also Ruiz et al. (1998), proof of Lemma 9). We obtain

$$\begin{aligned} Y_i(N,1_K)=\left\{ \begin{array}{lll} w^Y(|N|,|K|),&{}i\in K\\ -w^Y(|N|,|K\cup \{i\}|),&{}i\notin K\end{array}\right. \quad \text { for all }K\subseteq N\text { with }|K|=k. \end{aligned}$$

(7)

Now let $Y\in ASN_m$ and (N, v) be any TU-game. Note that any $v\in V$ can be written as $v=\sum _{K\subseteq N}v(K)\cdot 1_K.$ Using (7) and that Y is not only additive but also multiplicative (m), we have

$$\begin{aligned} Y_i(N,v)\mathop {=}\limits ^{\mathbf{A }}\sum \limits _{K\subseteq N}Y_i(N,v(K)\cdot 1_K)&\mathop {=}\limits ^{\mathbf{m }}\sum \limits _{K\subseteq N}v(K)\cdot Y_i(N,1_K)\nonumber \\&=\sum \limits _{K\subseteq N:i\in K}v(K)\cdot w^Y(|N|,|K|)\nonumber \\&\quad +\sum \limits _{K\subseteq N:i\notin K}v(K)\cdot [-w^Y(|N|,|K\cup \{i\}|)]\nonumber \\&=\sum \limits _{K\!\subseteq \! N{\setminus }\{i\}}w^Y(|N|,|K\cup \{i\}|)[v(K\cup \{i\})\!-\!v(K)]. \end{aligned}$$

(8)

Now let (N, v, g) be any TU-game with a network structure. Using (8), we obtain

$$\begin{aligned} \pi _i^Y(N,v,g)&=\sum \limits _{\lambda \in g_i}\frac{1}{2}\sum \limits _{g'\subseteq g{\setminus }\lambda }w^Y_{|g|,|g'|}\left( v^N(g'\cup \lambda )-v^N(g')\right) \end{aligned}$$

where $w^Y_{|g|,|g'|}=Y_{\lambda }(g,1_{g'\cup \lambda })$ for any $\lambda \in g$, and $g'\subseteq g{\setminus }\lambda$.

To show the second part of the Theorem, we will make use of NPO. Let $\lambda ,\lambda '\in g$ with $\lambda '\ne \lambda$ being a Nullplayer in $(g,v^N)$. Following the same steps as in Derks and Haller (1999) we obtain

$$\begin{aligned} w^Y_{|g|,|g'|}+w^Y_{|g|,|g'|+1}=w^Y_{|g|-1,|g'|}\,\forall \,|g'|=0,\ldots ,|g|-1 \end{aligned}$$

(9)

$\square$

Proof of Theorem 4: We will make use of the following expression:

Lemma 6

The weights from Theorem 3 satisfy

$$\begin{aligned} w^Y_{m,k}=\sum \limits _{l=0}^{n-m}\left( {\begin{array}{c}n-m\\ l\end{array}}\right) w^Y_{n,k+l} \quad \text { for all }n\ge m>k. \end{aligned}$$

The proof follows by induction over n (starting at $n=m$).

Proof of Theorem 4.1: Any multiplicative ASN-Position value $\pi ^Y\in \Pi (ASN_m)$ satisfies LMON for all TU-games with a network structure such that the corresponding link-game is link-convex.

Proof

Let $Y\in ASN_m$ and let (N, v, g) be any TU-game with a network structure such that the corresponding link-game $v^N$ is link-convex. By Theorem 3, for any $i\in N$ we have

$$\begin{aligned} \pi _i^Y(N,v,g)&=\sum \limits _{\lambda \in g_i}\frac{1}{2}\sum \limits _{g'\subseteq g{\setminus }\lambda }w^Y_{|g|,|g'|}\left( v^N(g'\cup \lambda )-v^N(g')\right) \\&=\frac{1}{2}\sum \limits _{\lambda \in g_i}\sum \limits _{\begin{array}{c} g'\subseteq g:\\ \lambda \in g' \end{array}}w^Y_{|g|,|g'|-1}\left( v^N(g')-v^N(g'{\setminus }\lambda )\right) \\&=\frac{1}{2}\sum \limits _{\begin{array}{c} g'\subseteq g:\\ g'\ne \emptyset \end{array}}w^Y_{|g|,|g'|-1}\sum \limits _{\lambda \in g'_i}\left( v^N(g')-v^N(g'{\setminus }\lambda )\right) \end{aligned}$$

(*)

Now consider any $\tilde{g}\subseteq g$. We can rewrite any $g'\subseteq g$ as $g'=h_1\cup h_2$ with $h_1\subseteq \tilde{g}$ and $h_2\subseteq g{\setminus }\tilde{g}$. Note that we either have $h_1\ne \emptyset$ or $h_2\ne \emptyset$. From ($*$) and using link convexity of $v^N$ we obtain

$$\begin{aligned} \pi _i^Y(N,v,g)&=\frac{1}{2}\sum \limits _{h_1\subseteq \tilde{g}}\sum \limits _{h_2\subseteq g{\setminus }\tilde{g}}w^Y_{|g|,|h_1\cup h_2|-1}\sum \limits _{\lambda \in (h_1\cup h_2)_i}\\&\quad \times \left( v^N(h_1\cup h_2)-v^N((h_1\cup h_2){\setminus }\lambda )\right) \\&\ge \frac{1}{2}\sum \limits _{h_1\subseteq \tilde{g}}\sum \limits _{h_2\subseteq g{\setminus }\tilde{g}}w^Y_{|g|,|h_1\cup h_2|-1}\sum \limits _{\lambda \in (h_1)_i}\left( v^N(h_1)-v^N(h_1{\setminus }\lambda )\right) \\&=\frac{1}{2}\sum \limits _{h_1\subseteq \tilde{g}}\sum \limits _{\lambda \in (h_1)_i}\left( v^N(h_1)-v^N(h_1{\setminus }\lambda )\right) \sum \limits _{h_2\subseteq g{\setminus }\tilde{g}}\underbrace{w^Y_{|g|,|h_1|+|h_2|-1}}_{\begin{array}{c} \text {only depends on}\\ \text {sizes of link-sets} \end{array}}\\&=\frac{1}{2}\sum \limits _{h_1\subseteq \tilde{g}}\sum \limits _{\lambda \in (h_1)_i}\left( v^N(h_1)-v^N(h_1{\setminus }\lambda )\right) \\&\quad \times \sum \limits _{k=0}^{|g|-|\tilde{g}|}\left( {\begin{array}{c}|g|-|\tilde{g}|\\ k\end{array}}\right) w^Y_{|g|,(|h_1|-1)+k}\\&{\mathop {=}\limits ^{\text {Lemma }6}}\frac{1}{2}\sum \limits _{h_1\subseteq \tilde{g}}w^Y_{|\tilde{g}|,|h_1|-1}\sum \limits _{\lambda \in (h_1)_i}\left( v^N(h_1)-v^N(h_1{\setminus }\lambda )\right) \\&\mathop {=}\limits ^{(*)}\pi _i^Y(N,v,\tilde{g}) \end{aligned}$$

$\square$

Proof of Theorem 4.2: Any multiplicative ASN-Position value $\pi ^Y\in \Pi (ASN_m)$ satisfies Component Decomposability CD:

$$\begin{aligned} \pi _i^Y(N,v,g)=\pi _i^Y(C_i(N,g),v|_{C_i(N,g)},g|_{C_i(N,g)}). \end{aligned}$$

Proof

Since $g_i\subseteq (g|_{C_i(N,g)})_i$, it is sufficient to show that any ASN value Y satisfies

$$\begin{aligned} Y_{\lambda }(g,v^N)=Y_{\lambda }(g|_C,v^N|_{C})\quad \forall \, \lambda \in g|_C,\, C\in \mathcal {C}(N,g)\text { and all }v\in V^0. \end{aligned}$$

Let $Y\in ASN_m, (N,v)$ be a standard TU-game and let g be a network. Consider $C\in \mathcal {C}(N,g)$. For any $\lambda \in g|_C$, marginal contributions are not effected by connections outside C due to the form of $v^N$. Using this and following the same decomposition method as in the proof of Theorem 4.1, we obtain

$$\begin{aligned} Y_{\lambda }(g,v^N)&=\sum \limits _{g'\subseteq g{\setminus }\lambda }w^Y_{|g|,|g'|}\left[ v^N((g'\cap g|_C)\cup \lambda )-v^N(g'\cap g|_C)\right] \\&=\sum \limits _{g'\subseteq g{\setminus }\lambda }w^Y_{|g|,|g'|}\underbrace{\left[ v^N(g'|_C\cup \lambda )-v^N(g'|_C)\right] }_{\begin{array}{c} \text {the same for all }g',g''\subseteq g{\setminus }\lambda \\ \text {such that } g'|_C=g''|_C \end{array}}\\&=\sum \limits _{\tilde{g}\subseteq (g|_C){\setminus }\lambda }\left[ v^N(\tilde{g}\cup \lambda )-v^N(\tilde{g})\right] \sum \limits _{h\subseteq g|_{N{\setminus } C}}w^Y_{|g|,|\tilde{g}|+|h|}\\&=\sum \limits _{\tilde{g}\subseteq (g|_C){\setminus }\lambda }\left[ v^N(\tilde{g}\cup \lambda )-v^N(\tilde{g})\right] \sum \limits _{k=0}^{|g|_{N{\setminus } C}|}\left( {\begin{array}{c}|g|_{N{\setminus } C}|\\ k\end{array}}\right) w^Y_{|g|,|\tilde{g}|+k}\\&\mathop {=}\limits ^{\begin{array}{c} |g|_{N{\setminus } C}|=|g|-|g|_C|\text { and}\\ \text {Lemma 6} \end{array}}\sum \limits _{\tilde{g}\subseteq (g|_C){\setminus }\lambda }w^Y_{|g|_C|,|\tilde{g}|}\left[ v^N(\tilde{g}\cup \lambda )- v^N(\tilde{g})\right] \\&=Y_{\lambda }(g|_C,v^N|_C) \end{aligned}$$

which finishes the proof. $\square$

Proof of Theorem 4.3: Any multiplicative ASN-Position value $\pi ^Y\in \Pi (ASN_m)$ can be written as

$$\begin{aligned} \pi _i^Y(N,v,g)=\sum \limits _{\lambda \in g_i}\sum \limits _{\begin{array}{c} g'\subseteq g:\\ \lambda \in g' \end{array}}\beta _{g'}^v\frac{w^Y_{|g'|,|g'|-1}}{2} \end{aligned}$$

where $\{\beta _{g'}^v\}_{g'\subseteq g}$ are the Harsanyi dividends representing $v^N$ and $w^Y$ is the weight defined in Theorem 3.

Proof

Let $Y\in ASN_m$ and (N, v, g) be any TU game with a network structure. Recall from the proof of Theorem 3, that the link game $v^N$ can be written as a (unique) linear combination of (link set) unanimity games where the corresponding scalars $\{\beta ^v_{g'}\}_{g'\subseteq g}$ are called Harsanyi dividends. Let $g'\subseteq g$ be arbitrary but fixed. By N, we have $Y_{\lambda }(g,u_{g'})=0$ for all $\lambda \notin g'$. Consider any $\lambda \in g'$. Note that any (link set) unanimity game can be written as

$$\begin{aligned} u_{g'}=\sum \limits _{\bar{g}\subseteq g{\setminus } g'}1_{g'\cup \bar{g}}. \end{aligned}$$

Using the weights from Theorem 3 and by Lemma 6, we obtain

$$\begin{aligned} Y_{\lambda }(g,u_{g'})= & {} \sum \limits _{\bar{g}\subseteq g{\setminus } g'}Y_{\lambda }(g,1_{g'\cup \bar{g}})\mathop {=}\limits ^{\lambda \in g'}\sum \limits _{\bar{g}\subseteq g{\setminus } g'}w^Y_{|g|,|g'|+|\bar{g}|-1}\\= & {} \sum \limits _{l=0}^{|g|-|g'|}\left( {\begin{array}{c}|g|-|g'|\\ l\end{array}}\right) w^Y_{|g|,|g'|-1+l}=w^Y_{|g'|,|g'|-1} \end{aligned}$$

which yields

$$\begin{aligned} \pi _i^Y(N,v,g)= & {} \sum \limits _{\lambda \in g_i}\frac{1}{2}Y_{\lambda }(g,v^N)=\sum \limits _{\lambda \in g_i}\frac{1}{2}\sum \limits _{\begin{array}{c} g'\subseteq g:\\ \lambda \in g' \end{array}}\beta _{g'}^vY_{\lambda }(g,u_{g'})\\= & {} \sum \limits _{\lambda \in g_i}\sum \limits _{\begin{array}{c} g'\subseteq g:\\ \lambda \in g' \end{array}}\beta _{g'}^v\frac{w^Y_{|g'|,|g'|-1}}{2} \end{aligned}$$

$\square$

Proof of Lemma 5: The two characterizing axioms of the ASN-Position values are independent.

Proof

To show independence of the axioms, we need to find elements in $\mathcal {Y}_G$ which satisfy one but not the other property. The Shapley-Position value satisfies BLC but (trivially) does not satisfy CLP w. r. t. the Banzhaf value. Now consider the allocation rule which shares the Banzhaf-link power of a component equally within this component, that is, the rule given by

$$\begin{aligned}W_i(N,v,g):=\frac{\sum \limits _{\lambda \in g|_{\mathcal {C}_{i}}}Ba_{\lambda }(g|_{\mathcal {C}_{i}},v^N|_{\mathcal {C}_{i}})}{|\mathcal {C}_{i}|}\in \mathcal {Y}_G.\end{aligned}$$

By definition and using Theorem 4.2, W satisfies CLP w. r. t. the Banzhaf value. To see that W does not satisfy BLC, consider the following TU-game with a network structure:

$$\begin{aligned}N = \{1, 2, 3, 4\},\quad v(K)={\left\{ \begin{array}{ll} 1,&{}\{1,2\}\subseteq K\\ 1,&{}\{1,3\}\subseteq K\\ 1,&{}\{2,3,4\}\subseteq K\\ 0,&{}\text { otherwise} \end{array}\right. },\quad g=\{12,23,34\}\end{aligned}$$

All players are connected in (N, g) resulting in $C_i=N$ for all $i\in N$. For the Banzhaf values of links we obtain $Ba_{\{12\}}=\frac{3}{2^{3-1}}=\frac{3}{4}$ (3 possibilities of creating 1 when materializing) and $Ba_{\{23\}}=Ba_{\{34\}}=\frac{1}{4}$ (1 possibility of creating 1 when materializing) resulting in a component power of . Equal sharing implies for $i=1,2,3,4$. For BLC consider players 1 and 2. We have

$$\begin{aligned}&\sum \limits _{\lambda \in g_1}\left[ W_2(N,v,g)-W_2(\mathcal {C}(2)(N,g-\lambda ),v|_{\mathcal {C}(2)(N,g-\lambda )},g|_{\mathcal {C}(2)(N,g-\lambda )})\right] \\&\quad =\frac{5}{16}-W_2(\{2,3,4\},v|_{\{2,3,4\}}, \{23,34\})\\ \text { and}&\\&\sum \limits _{\lambda \in g_2}\left[ W_1(N,v,g)-W_1(\mathcal {C}(1)(N,g-\lambda ),v|_{\mathcal {C}(1)(N,g-\lambda )},g|_{\mathcal {C}(1)(N,g-\lambda )})\right] \\&\quad =\frac{5}{16}-W_1(\{1\},v|_{\{1\}},\emptyset )+\frac{5}{16}-W_1(\{1,2\},v|_{\{1,2\}},\{12\}) \end{aligned}$$

In the restricted game $(\{2,3,4\},v|_{\{2,3,4\}}, \{23,34\})$, each link has a Banzhaf power of $\frac{1}{2^{2-1}}$ and equal sharing implies for $i=2,3,4$. In the restricted game $(\{1,2\},v|_{\{1,2\}},\{12\})$, the link has a Banzhaf power of $\frac{1}{2^{1-1}}$ and equal sharing implies for $i=1,2$. We hence obtain $-\frac{1}{48}$ in the first case but $\frac{1}{8}\ne -\frac{1}{48}$ in the second case. $\Rightarrow$ BLC is violated. $\square$

Appendix B: Supplementary material political example

See Tables 7, 8 and 9.

Table 7 Swings & Shapley/Banzhaf values (to 100%) for Parties (2nd line expresses less restrictive case)

Table 8 Swings & Shapley/Banzhaf values (to 100%) for connections

Table 9 Classic centrality approaches