Associated consistency, value and graphs

Abstract

This article presents an axiomatic characterization of a new value for cooperative games with incomplete communication. The result is obtained by slight modifications of associated games proposed by Hamiache (Games Econ Behav 26:59–78, 1999; Int J Game Theory 30:279–289, 2001). This new associated game can be expressed as a matrix formula. We generate a series of successive associated games and show that its limit is an inessential game. Three axioms (associated consistency, inessential game, continuity) characterize a unique sharing rule. Combinatorial arguments and matrix tools provide a procedure to compute the solution. The new sharing rule coincides with the Shapley value when the communication is complete.

Appendix

Proof of Lemma 8:

First of all it is easy to see that $<0, y_{_T}>$ are eigenpairs of matrix $M_g$ for non-connected coalitions T, when $y_{_T}[S]=1$ if ${S=T}$ and $y_{_T}[S]=0$ if ${S\not =T}$. In the following, we will assume that $\lambda \not =0$.

We already know that n vectors $x_{\{i\}}$ for $i\in N$, as defined by Eq. (4), are independent eigenvectors of matrix $M_c$ related to eigenvalue 1 and that they are also eigenvectors of $M_g$. We will denote the other eigenpairs of matrix $M_c$ by $<\lambda _{_S},\,x_{_S}>$ for $S\subseteq N$ and $\#S\ge 2$. Let $<\lambda , w>$ be an eigenpair of matrix $M_g=P_g \,M_{c}\,P_g$. Vector x can be expressed as a linear combination of eigenvectors of matrix $M_c$, $w= \sum _{\emptyset \not =S\subseteq N}c_{_S}\,x_{_S}.$ We will prove below that the eigenvalues of matrix $M_g$ have a norm smaller than or equal to one.

$$\begin{aligned}&P_g \,M_{c}\,P_g \, \Big [\sum _{\mathop {i}\limits _{i\in {}N}} c_{_{\{i\}}}\,x_{_{\{i\}}} +\mathop {\sum }\limits _{\mathop {S}\limits _{\mathop {S\subseteq N}\limits _{\#S\ge 2}}}c_{_S}\,x_{_S}\Big ] = P_g \,M_{c}\, \Big [\sum _{\mathop {i}\limits _{i\in {}N}} c_{_{\{i\}}}\,x_{_{\{i\}}} +\mathop {\sum }\limits _{\mathop {S}\limits _{\mathop {S\subseteq N}\limits _{\#S\ge 2}}}c_{_S}\,x_{_S}\Big ]\\&\quad = P_g \, \Big [\sum _{\mathop {i}\limits _{i\in {}N}} c_{_{\{i\}}}\,x_{_{\{i\}}} +\sum _{\mathop {S}\limits _{\#S\ge 2}}\lambda _{_S}\,c_{_S}\,x_{_S}\Big ] = \sum _{\mathop {i}\limits _{i\in {}N}} c_{_{\{i\}}}\,x_{_{\{i\}}} +P_g \,\sum _{\mathop {S}\limits _{\#S\ge 2}}\lambda _{_S}\,c_{_S}\,x_{_S}\\&\quad =\lambda \, \Big [\sum _{\mathop {i}\limits _{i\in {}N}} c_{_{\{i\}}}\,x_{_{\{i\}}} +\mathop {\sum }\limits _{\mathop {S}\limits _{\mathop {S\subseteq N}\limits _{\#S\ge 2}}}c_{_S}\,x_{_S}\Big ]. \end{aligned}$$

The first equality uses the fact that the $<1,\,w>$ is an eigenpair of matrix $P_g$. The second equality is true since $<\lambda _{_S},\,x_{_S}>$ is an eigenpair of matrix $M_c$. Let us consider the norm of the two last terms,

Matrix $P_g$ is diagonalizable (Property 3), its eigenvalues are thus semi-simple. It is well known that in that case there exists a matrix norm verifying where $\rho (P_g)$ is the spectral radius which is equal to one. Moreover, there exists a vector norm compatible with the considered matrix norm (theorem 5.7.13 p. 324 Horn and Johnson), which means that

We thus obtain,

for $\#S=s\ge 2$ the eigenvalues of $M_c$ are $\lambda _{_S}=1-s\tau$,

Parameter $\tau$ is positive and arbitrarily small. Thus if $(\mid {}\lambda \mid -1)$ is positive, the term of the left hand side of the inequality is positive and we can choose parameter $\tau$ to be sufficiently small to contradict the inequality. So $(\mid {}\lambda \mid -1)$ cannot be strictly positive, which leads to $\mid {}\lambda \mid \le 1.$$\square$

Proof of Lemma 9:

Let $<\lambda ,\,w>$ be an eigenpair of matrix $M_g$ when $w=\sum _{\emptyset \not =S\subseteq N}c_{_S}\,x_{_S}$,

$$\begin{aligned} P_gM_cP_g\sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}}c_{_S}\,x_{_S} =\lambda \sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}}c_{_S}\,x_{_S}. \end{aligned}$$

We know that $<1,\,w>$ is an eigenpair of matrix $P_g$,

$$\begin{aligned} P_gM_c\sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}}c_{_S}\,x_{_S} =\lambda \,P_g\sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}}c_{_S}\,x_{_S}. \end{aligned}$$

Separating the singletons,

$$\begin{aligned} P_gM_c\sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}}c_{_S}\,x_{_S} =P_gM_c\sum _{\mathop {i}\limits _{i\in N}}c_{_{\{i\}}}\,x_{_{\{i\}}} +P_gM_c\sum _{\mathop {S}\limits _{\#S\ge 2}}c_{_S}\,x_{_S} =P_g\lambda \sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}} c_{_S}\,x_{_S}. \end{aligned}$$

Since $<1,\,x_{_{\{i\}}}>$ for all $i\in N$ and $<1-s\,\tau ,\,x_{_S}>$ for all coalitions S verifying $\#S=s\ge 2$ are eigenpairs of matrix $M_c$,

$$\begin{aligned}&P_g \sum _{\mathop {i}\limits _{i\in N}} c_{_{\{i\}}}x_{_{\{i\}}} +P_g \mathop {\sum }\limits _{\mathop {S}\limits _{\mathop {S\subseteq N}\limits _{\#S\ge 2}}} (1-s\,\tau )c_{_S}x_{_S}\\&\quad = P_g \sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}} c_{_S}x_{_S} -P_g\tau \mathop {\sum }\limits _{\mathop {S}\limits _{\mathop {S\subseteq N}\limits _{\#S\ge 2}}} {}sc_{_S}x_{_S} = P_g\lambda \sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}} c_{_S}x_{_S}. \end{aligned}$$

For all connected coalitions T we have, after assembling a few terms,

$$\begin{aligned} \tau \,\mathop {\sum }\limits _{\mathop {S}\limits _{\mathop {S\subseteq N}\limits _{\#S\ge 2}}} \,s\,c_{_S}\,x_{_S}[T] =(1-\lambda )\,\sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}} c_{_S}\,x_{_S}[T]. \end{aligned}$$

Since $w=\sum _{\emptyset \not =S\subseteq N} c_{_S}\,x_{_S}$ is an eigenvector, there exists at least a connected coalition T such that $w[T]=\sum _{\emptyset \not =S\subseteq N} c_{_S}\,x_{_S}[T]\not =0$. Isolating $\lambda$,

$$\begin{aligned} \lambda =1 -\tau \,\frac{\displaystyle {\sum _{\mathop {S}\limits _{\#S\ge 2}}\,s\,c_{_S}\,x_{_S}[T]}}{\displaystyle {\sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}}c_{_S}\,x_{_S}[T]}}. \end{aligned}$$

Since matrix $P_gM_cP_g$ is real, ${{\overline{\lambda }}}$, the complex conjugate of $\lambda$, is also one of its eigenvalues, and the next equality is true,

$$\begin{aligned} {\overline{\lambda }}= & {} 1 -\tau \,\frac{\displaystyle {\sum _{\mathop {S}\limits _{\#S\ge 2}}\, s\,\overline{c_{_S}}\,x_{_S}[T]}}{\displaystyle {\sum _{\mathop {S}\limits _{\emptyset \not =S\subseteq N}} \overline{c_{_S}}\,x_{_S}[T]}}.\\ \quad \lambda \,{\overline{\lambda }}= & {} \mid {}\lambda {}\mid ^2 =\frac{1}{\mid {}w[T]{}\mid ^2} \Big [w[T] -\tau \sum _{\mathop {S}\limits _{\#S\ge 2}}\, s\,{c_{_S}}\,x_{_S}[T]\Big ] \Big [{\overline{w}}[T] -\tau \sum _{\mathop {S}\limits _{\#S\ge 2}}\, s\,\overline{c_{_S}}\,x_{_S}[T]\Big ]. \end{aligned}$$

Expanding the two last terms of the previous expression,

$$\begin{aligned} \mid {}\lambda {}\mid ^2= & {} \frac{1}{\mid {}w[T]{}\mid ^2} \Big [{\mid {}w[T]{}\mid ^2} -2\,\tau {}Re\Big (w[T]\,\sum _{\mathop {S}\limits _{\#S\ge 2}}\, s\,\overline{c_{_S}}\,x_{_S}[T]\Big )\\&+\tau ^2\mid \sum _{\mathop {S}\limits _{\#S\ge 2}}\, s\,{c_{_S}}\,x_{_S}[T]\mid ^2\Big ]. \end{aligned}$$

We can choose an eigenvector w such that ${\mid {}w[T]{}\mid }=1$,

$$\begin{aligned} \mid {}\lambda {}\mid ^2 \, ={1} -2\,\tau {}Re\Big (w[T]\,\sum _{\mathop {S}\limits _{\#S\ge 2}}\, s\,\overline{c_{_S}}\,x_{_S}[T]\Big ) +\tau ^2\mid \sum _{\mathop {S}\limits _{\#S\ge 2}}\, s\,{c_{_S}}\,x_{_S}[T]\mid ^2. \end{aligned}$$

Since $\mid {}\lambda {}\mid ^2\le 1$ we have,

$$\begin{aligned}&2\,\tau {}Re\Big (w[T]\,\sum _{\mathop {S}\limits _{\#S\ge 2}}\, s\,\overline{c_{_S}}\,x_{_S}[T]\Big ) -\tau ^2\mid \sum _{\mathop {S}\limits _{\#S\ge 2}}\, s\,{c_{_S}}\,x_{_S}[T]\mid ^2\ge 0.\\&\quad 0< \tau \le \frac{2\,Re\Big (w[T]\,\sum _{S:\#S\ge 2}\, s\,\overline{c_{_S}}\,x_{_S}[T]\Big )}{\mid \sum _{S:\#S\ge 2}\, s\,{c_{_S}}\,x_{_S}[T]\mid ^2}. \end{aligned}$$

Choosing parameter $\tau$ sufficiently small will ensure that $\mid {}\lambda {}\mid ^2<1$. $\square$

Auxiliary result:

$$\begin{aligned}&\sum _{\theta =1}^{m}\, {{j+\theta -1}\atopwithdelims (){\theta }}\, {{m}\atopwithdelims (){\theta }} \left( {\frac{\tau }{z_i}}\right) ^{\theta } =-1\\&\quad +\sum _{\theta =0}^{j-1} {{j-1}\atopwithdelims (){j-1-\theta }}\,{{m}\atopwithdelims (){j-1-\theta }} {\left( {\frac{\tau }{z_i}}\right) ^{j-1-\theta } \left( {1+\frac{\tau }{z_i}}\right) ^{m-j+1+\theta }}. \end{aligned}$$

Proof

$$\begin{aligned}&\sum _{\theta =0}^{m}\, {{\theta +j-1}\atopwithdelims (){\theta }}\, {{m}\atopwithdelims (){\theta }} \left( {\frac{\tau }{z_i}}\right) ^{\theta } =\sum _{\theta =0}^{m}\, \frac{(\theta +j-1)\dots (\theta +1)}{(j-1)!}\, {{m}\atopwithdelims (){\theta }} \left( {\frac{\tau }{z_i}}\right) ^{\theta }\nonumber \\&\quad =\frac{1}{(j-1)!}\,\sum _{\theta =0}^{m}\, \frac{d^{j-1}}{d\left( {\frac{\tau }{z_i}}\right) ^{j-1}}\left[ {{m}\atopwithdelims (){\theta }} \left( {\frac{\tau }{z_i}}\right) ^{\theta +j-1}\right] \nonumber \\&\quad =\frac{1}{(j-1)!}\, \frac{d^{j-1}}{d\left( {\frac{\tau }{z_i}}\right) ^{j-1}} \left[ \left( {\frac{\tau }{z_i}}\right) ^{j-1} \sum _{\theta =0}^{m}\, {{m}\atopwithdelims (){\theta }} \left( {\frac{\tau }{z_i}}\right) ^{\theta } \right] \nonumber \\&\quad =\frac{1}{(j-1)!}\, \frac{d^{j-1}}{d\left( {\frac{\tau }{z_i}}\right) ^{j-1}} \left[ \left( {\frac{\tau }{z_i}}\right) ^{j-1} \left( 1+{\frac{\tau }{z_i}}\right) ^{m} \right] . \end{aligned}$$

(14)

Applying Leibnitz’s Theorem for differentiation of a product,

$$\begin{aligned} \frac{d^t}{dx^t}(u\cdot {}v) =\sum _{\theta =0}^{t} {{t}\atopwithdelims (){\theta }} \frac{d^\theta }{dx^\theta }(u) \frac{d^{t-\theta }}{dx^{t-\theta }}(v), \end{aligned}$$

to Eq. (14), with $u=(\tau /z_i)^{j-1}$, $v=(1+\tau /z_i)^m$ and $t=j-1$ we obtain,

$$\begin{aligned}= & {} \frac{1}{(j-1)!}\, \sum _{\theta =0}^{j-1} {{j-1}\atopwithdelims (){\theta }} \frac{d^{\theta }}{d\left( {\frac{\tau }{z_i}}\right) ^{\theta }} \left( \frac{\tau }{z_i}\right) ^{j-1} \frac{d^{j-1-\theta }}{d\left( {\frac{\tau }{z_i}}\right) ^{j-1-\theta }} \left( 1+{\frac{\tau }{z_i}}\right) ^{m} \\= & {} \frac{1}{(j-1)!}\, \sum _{\theta =0}^{j-1} {{j-1}\atopwithdelims (){\theta }} \frac{(j-1)!}{(j-\theta -1)!} \left( \frac{\tau }{z_i}\right) ^{(j-1-\theta )}\\&\frac{(m)!}{(m-j+\theta +1)!} \left( 1+{\frac{\tau }{z_i}}\right) ^{(m-j+1+\theta )} \\= & {} \sum _{\theta =0}^{j-1} {{j-1}\atopwithdelims (){j-\theta -1}} {{m}\atopwithdelims (){j-\theta -1}} \left( \frac{\tau }{z_i}\right) ^{(j-1-\theta )} \left( 1+{\frac{\tau }{z_i}}\right) ^{(m-j+1+\theta )}, \end{aligned}$$

which proves the Auxiliary result. $\square$

Proof of Lemma 10:

If T is non-connected, all the terms of the series are equal to 0 and Lemma 10 is true. So let us consider instead that T is connected.

$$\begin{aligned} (P_gM_cP_g)^{m}[S,T]= \sum _{\theta =1}^{m}A^{\theta }[S,T]\, {{m}\atopwithdelims (){\theta }} \tau ^{\theta }+P_g[{S,T}], \end{aligned}$$

where the parameters $A^{\theta }[S,T]$ are the successive coefficients of the powers of x in the Maclaurin development of the generating function F(x). Those coefficients are the value of the relevant derivatives of F(x) at point $x=0$.

$$\begin{aligned} A^{\theta }[S,T]=\frac{1}{\theta !}\frac{d^\theta \,F(0)}{d\,x^\theta }. \end{aligned}$$

(15)

Performing the euclidean division of Eq. (11), we can rewrite the generating function of Lemma 4 as,

$$\begin{aligned} F(x)=\frac{R(x)}{Q(x)} =\frac{\alpha _0+\alpha _{1}\,x+ \cdots +\alpha _{q-\mu -1}\,x^{q-\mu -1}}{1+b_1\,x+b_2\,x^2+\cdots +b_{q-\mu +1}\,x^{q-\mu +1}}+\frac{a_{q-\mu }}{b_{q-\mu }}, \end{aligned}$$

where $\alpha _i=a_{i}-\frac{a_{q-\mu }}{b_{q-\mu }}b_{i}$ for $1\le i \le q-\mu -1$ and $\alpha _0=-\frac{a_{q-\mu }}{b_{q-\mu }}$.

From the partial fraction decomposition theorem, we can write the rational function, the first term of the right hand side of the previous equation, as a finite linear combination of terms of the form, $E(x)=(x-z)^{-w}$, where z is a root of the denominator and w is an integer at most equal to the algebraic multiplicity of z. Note that z could be a complex number.

$$\begin{aligned} F(x)=\sum _{i=1}^p\sum _{j=1}^{w_i}\beta _{i,j}\frac{1}{(x-z_i)^j} +\frac{a_{q-\mu }}{b_{q-\mu }}, \end{aligned}$$

where $z_1$, $\dots$, $z_p$ are the roots of Q(x), $w_1$, $\dots$, $w_p$ are their respective algebraic multiplicities and $\beta _{i,j}$ are the coefficients of the linear combination. The derivative of order $\theta$ is given by,

$$\begin{aligned} \frac{d^\theta \,F(x)}{d\,x^\theta } =\sum _{i=1}^p\sum _{j=1}^{w_i}\beta _{i,j} \frac{(-j)(-j-1)\dots (-j-\theta +1)}{(x-z_i)^{j+\theta }}, \end{aligned}$$

and the coefficients $A^{\theta }[S,T]$ are given by,

$$\begin{aligned} A^{\theta }[S,T]= & {} \frac{1}{\theta !} \frac{d^\theta \,F(0)}{d\,x^\theta } =\sum _{i=1}^p\sum _{j=1}^{w_i}\beta _{i,j} \frac{(-1)^\theta }{(-z_i)^{j+\theta }}\, {{j+\theta -1}\atopwithdelims (){\theta }}.\\ (P_gM_cP_g)^{m}[S,T]= & {} \sum _{\theta =1}^{m}\sum _{i=1}^p\sum _{j=1}^{w_i} \beta _{i,j}\frac{(-1)^\theta }{(-z_i)^{j+\theta }}\, {{j+\theta -1}\atopwithdelims (){\theta }}\, {{m}\atopwithdelims (){\theta }} \tau ^{\theta }+P_g[{S,T}]. \end{aligned}$$

Inverting the order of summations,

$$\begin{aligned} =\sum _{i=1}^p\sum _{j=1}^{w_i}\beta _{i,j} \frac{1}{(-z_i)^{j}}\, \sum _{\theta =1}^{m}\, {{j+\theta -1}\atopwithdelims (){\theta }}\, {{m}\atopwithdelims (){\theta }} \left( {\frac{\tau }{z_i}}\right) ^{\theta } +P_g[{S,T}]. \end{aligned}$$

(16)

Using the Auxiliary result we get,

$$\begin{aligned} (P_gM_cP_g)^{m}[S,T]= & {} -\sum _{i=1}^p\sum _{j=1}^{w_i}\beta _{i,j} \frac{1}{(-z_i)^{j}}+P_g[{S,T}]\nonumber \\&+\sum _{i=1}^p\sum _{j=1}^{w_i}\beta _{i,j} \frac{1}{(-z_i)^{j}}\, \sum _{\theta =0}^{j-1} {{j-1}\atopwithdelims (){j-1-\theta }}\,{{m}\atopwithdelims (){j-1-\theta }}\nonumber \\&{\times \left( {\frac{\tau }{z_i}}\right) ^{j-1-\theta } \left( {1+\frac{\tau }{z_i}}\right) ^{m-j+1+\theta }}. \end{aligned}$$

(17)

Since,

$$\begin{aligned} F(0)=\frac{R(0)}{Q(0)}=\sum _{i=1}^p\sum _{j=1}^{w_i}\beta _{i,j}\, \left( \frac{1}{-z_i}\right) ^j +\frac{a_{q-\mu }}{b_{q-\mu }}=0, \end{aligned}$$

we have,

$$\begin{aligned}&(P_gM_cP_g)^{m}[S,T] =\frac{a_{q-\mu }}{b_{q-\mu }}+P_g[{S,T}] \\&+\sum _{i=1}^p\sum _{j=1}^{w_i}\beta _{i,j} \frac{1}{(-z_i)^{j}}\\&\times \left[ \sum _{\theta =0}^{j-1}\! {{j-1}\atopwithdelims (){j-1-\theta }}\!\!{{m}\atopwithdelims (){j-1-\theta }} \!\!{\left( {\frac{\tau }{z_i}}\right) ^{\!j-1-\theta } \!\!\!\left( {1+\frac{\tau }{z_i}}\right) ^{\!m-j+1+\theta }}\right] \!\!. \end{aligned}$$

Let us consider the characteristic polynomial in Eq. (8) and Q(x) as defined by Eq. (11). If $z_i$ is a root of Q(x), it is true that $\frac{1}{z_i}$ is a root of Eq. (8) and thus an eigenvalue of matrix A (note that $z_i\not =0$). We are now ready to show that $1+\frac{\tau }{z_i}$ is an eigenvalue of matrix $M_g$.

Since the columns of matrix $(P_gM_cP_g)$ corresponding to non-connected coalitions are zero, the non-zero eigenvalues are preserved when we delete from matrix $(P_gM_cP_g)$ the lines and columns corresponding to non-connected coalitions. Let us denote by $M^{s}_g$ that “simplified”  matrix. The corresponding sub-matrix of A is thus equal to $(A^{s})=(M^{s}_g-Id)/\tau$, which leads to $Id+\tau \,A^{s}= M^{s}_g$. As a consequence, the eigenvalues of $M^{s}_g$ are given by $1+\frac{\tau }{z_i}$, which are also non-zero eigenvalues of matrix $M_g$. Since the spectral radius of $M_g$ is one, and since $z_i\not =0$, the moduli of $1+\frac{\tau }{z_i}$ are strictly smaller than 1.

Let us focus now on the terms ${{m}\atopwithdelims (){j-1-\theta }}({1+\frac{\tau }{z_i}})^{m}$ as m tends to infinity. If $\theta =j-1$, the corresponding term reduces to $({1+\frac{\tau }{z_i}})^{m}$ and thus converges to 0. Let us assume now that $\theta =0,\,1,\dots ,j-2$.

$$\begin{aligned} \displaystyle { {{m}\atopwithdelims (){j-1-\theta }}\left| \left( {1+\frac{\tau }{z_i}}\right) \right| ^{m}}&=\displaystyle {\frac{m(m-1)(m-2)\ldots (m-j+2+\theta )}{(j-1-\theta )!}\left| \left( {1+\frac{\tau }{z_i}}\right) \right| ^m}\nonumber \\&\displaystyle {\le \frac{m^{j-1-\theta }}{(j-1-\theta )!}\left| \left( {1+\frac{\tau }{z_i}}\right) \right| ^m}. \end{aligned}$$

(18)

The logarithm of the last expression converges if and only if $(j-1-\theta )\,\log (m)+m\,\log \left| \left( 1+\frac{\tau }{z_i}\right) \right|$ converges as $m\rightarrow \infty$. Using the fact that $(\log m)/m \rightarrow 0$ as $m\rightarrow \infty$, the term in Expression (18) converges to 0 as m tends to infinity, which completes the proof of Lemma 10. $\square$

Proof of Lemma 11:

Let us write $(P_g\, M_c\, P_g)^n= (P_g\, M_c\, P_g)\, (P_g\, M_c\, P_g)^{n-1}$, and let W be the limit of the series $\{(P_g\, M_c\, P_g)^k\}_{k=1}^{\infty }$. It is thus true that $(P_g\, M_c\, P_g)\, W=W$. In words, the columns of matrix W are eigenvectors of matrix $(P_g\, M_c\, P_g)$ related to eigenvalue 1. We shall show that these eigenvectors are “inessential” vectors. Let $w=(w_{_S})_{_{\emptyset \not =S\subseteq N}}$ be an eigenvector associated to $\lambda =1$. We will solve the following system of linear equations, $(P_g\, M_c\, P_g)\,w=w$. Since we have, for non-connected coalitions S, $w_{_{S}}=\sum _{K\in S/g}w_{_{K}}$, we will concentrate only on connected coalitions. Considering Eq. (1), we obtain after few cancellations,

$$\begin{aligned} \sum _{\mathop {j}\limits _{j\in S^*{\setminus } S}} \big (w_{_{S\cup \{j\}}}-w_{_{S}}-w_{_{\{j\}}}\big )=0. \end{aligned}$$

(19)

So we learn that for all connected coalitions S, the related coefficient $w_S$ is defined uniquely as a linear combination of $w_{N\cup \{j\}}$ and $w_{\{j\}}$ for $j\in {}S^*{\setminus } S$. Noting that for $S=N{\setminus } \{i\}$, we have $w_{_{N{\setminus } \{i\}}}=w_{_{N}}-w_{_{\{i\}}},$ we can conclude that $w_S$ is a linear combination of $w_{N}$ and a selection of $w_{\{j\}}$. We show below that for all connected coalitions S such that $1\le \#S\le {}n-1$, we have,

$$\begin{aligned} w_{_{S}}=w_{_{N}}-\sum _{\mathop {i}\limits _{i\not \in S}}w_{_{\{i\}}}. \end{aligned}$$

(20)

Equation (20) is of course true for $S=N$ and for all the connected coalitions with $n-1$ elements. Let us assume that Eq. (20) is true for all connected coalitions of size s and above. We will show that Eq. (20) is also true for connected coalitions T verifying $\#T=s-1$.

$$\begin{aligned} -\sum _{\mathop {j}\limits _{j\in {}T^*{\setminus } T}}w_{_{\{j\}}} -(\#T^*-\#T)\,w_{_{T}} +\sum _{\mathop {j}\limits _{j\in T^*{\setminus } T}}w_{_{T\cup \{j\}}} =0. \end{aligned}$$

Using the induction hypothesis,

$$\begin{aligned} -\sum _{\mathop {j}\limits _{j\in {}T^*{\setminus } T}}w_{_{\{j\}}} -(\#T^*-\#T)\,w_{_{T}} +\sum _{\mathop {j}\limits _{j\in T^*{\setminus } T}} \Big [w_{_{N}} -\sum _{\mathop {m}\limits _{m\in N{\setminus }(T\cup \{j\})}}w_{_{\{m\}}}\Big ] =0. \end{aligned}$$

Taking into account that $N{\setminus }(T\cup \{j\})=(N{\setminus }{}T)\setminus \{j\}$,

$$\begin{aligned}&-\sum _{\mathop {j}\limits _{j\in {}T^*{\setminus } T}}w_{_{\{j\}}} -(\#T^*-\#T)\,w_{_{T}}+(\#T^*-\#T)\,w_{_{N}} \\&+\sum _{\mathop {m}\limits _{m\in T^*{\setminus } T}}w_{_{\{m\}}} -(\#T^*-\#T)\sum _{\mathop {m}\limits _{m\in N{\setminus }(T)}}w_{_{\{m\}}} =0. \end{aligned}$$

After relevant cancellations,

$$\begin{aligned} w_{_{T}}=w_{_{N}} -\sum _{\mathop {m}\limits _{m\in N{\setminus }{}T}}w_{_{\{m\}}}, \end{aligned}$$

which proves that Eq. (20) is true for all non-empty connected coalitions. As a consequence,

$$\begin{aligned} w_{_{\{j\}}}= & {} w_{_{N}}-\sum _{\mathop {i}\limits _{i\in N{\setminus }\{j\}}}w_{_{\{i\}}}, \end{aligned}$$

(21)

$$\begin{aligned} w_{_{N}}= & {} \sum _{\mathop {i}\limits _{i\in N}}w_{_{\{i\}}}. \end{aligned}$$

(22)

Combining Eqs. (20) and (22),

$$\begin{aligned} w_{_{S}}=w_{_{N}} -\sum _{\mathop {i}\limits _{i\in N{\setminus } S}} w_{_{\{i\}}} =\sum _{\mathop {i}\limits _{i\in {}N}} w_{_{\{i\}}} -\sum _{\mathop {i}\limits _{i\in N{\setminus } S}} w_{_{\{i\}}} =\sum _{\mathop {i}\limits _{i\in {}S}} w_{_{\{i\}}}. \end{aligned}$$

(23)

The eigenvectors of matrix $P_g\, M_c\, P_g$ related to eigenvalue 1 are “inessential”  vectors. We have thus proved that $(P_g\, M_c\, P_g)^\infty [S,\,T] =\sum _{i\in S}(P_g\, M_c\, P_g)^\infty [\{i\},\,T]$. Which ensures that the limit game ${\tilde{v}}$ is inessential. $\square$