Evaluating groups with the generalized Shapley value

Abstract

Following the original interpretation of the Shapley value as a priori evaluation of the prospects of a player in a multi-person interaction situation, we intend to apply the Shapley generalized value (introduced formally in Marichal et al. in Discrete Appl Math 155:26–43, 2007) as a tool for the assessment of a group of players that act as a unit in a coalitional game. We propose an alternative axiomatic characterization which does not use a direct formulation of the classical efficiency property. Relying on this valuation, we also analyze the profitability of a group. We motivate this use of the Shapley generalized value by means of two relevant applications in which it is used as an objective function by a decision maker who is trying to identify an optimal group of agents in a framework in which agents interact and the attained benefit can be modeled by means of a transferable utility game.

Appendix

Theorem 1

The unique group value over the set of all games ${\mathcal {G}}$ verifying G-null player, G-linearity, G-CBC, and G-SPB is the Shapley group value $\phi ^g$.

Proof

Let us first prove that the Shapley group value $\phi ^g$ satisfies the previous four properties.

Let us arbitrarily fix the sets C and N, $C\subset N\in {\mathcal {N}}$, and let $v\in {\mathcal {G}}_N$.

To check properties P1 and P2 the reader is referred to Marichal et al. (2007).

With respect to property P3, G-CBC, let $C\subset N\in {\mathcal {N}}$ be any two finite sets, and let v any game in ${\mathcal {G}}_N$. First let us remark that condition (3) is equivalent to:

$$\begin{aligned}&\xi ^g({C\cup i};N,v)-\xi ^g({C\cup j};N,v) \nonumber \\&\quad =(\xi ^g({C\cup i};N\backslash j,v_{-j})-\xi ^g(C;N\backslash j,v_{-j}))\nonumber \\&\qquad -(\xi ^g({C\cup j};N\backslash i,v_{-i})-\xi ^g(C;N\backslash i,v_{-i})). \end{aligned}$$

(10)

Also note that, by definition of the merging game and the Shapley value, for any $i,j\in N{\setminus } C$ the following equalities hold:

$$\begin{aligned} \displaystyle \phi ^g({C\cup i};N,v)= & {} \displaystyle \sum _{\begin{array}{c} S\subset N{\setminus } C \\ i,j\notin S \end{array}} \Bigl ( \frac{s!(n-c-s-1)!}{(n-c)!}(v(S\cup C\cup i)-v(S))\\&+\frac{(s+1)!(n-c-s-2)!}{(n-c)!}(v(S\cup C\cup i\cup j)-v(S\cup j))\Bigr ), \\ \phi ^g(C;N\backslash j,v_{-j})= & {} \displaystyle \sum _{\begin{array}{c} S\subset N{\setminus } C\\ i,j\notin S \end{array}} \Bigl (\frac{s!(n-c-s-1)!}{(n-c)!}(v(S\cup C)-v(S))\\&+\frac{(s+1)!(n-c-s-2)!}{(n-c)!}(v(S\cup C\cup i)-v(S\cup i))\Bigr ), \\ \phi ^g ({C\cup j};N\backslash i,v_{-i})= & {} \sum _{\begin{array}{c} S\subset N{\setminus } C\\ i,j\notin S \end{array}} \frac{s!(n-c-s-2)!}{(n-c-1)!}(v(S\cup C\cup j)-v(S)). \end{aligned}$$

Analogous expressions hold for $\phi ^g({C\cup j};N,v)$, $\phi ^g (C;N{\setminus } i,v_{-i})$ and $\phi ^g ({C\cup i}; N{\setminus } j, v_{-j})$. Now it is enough to check that for every $S\in N$ with $i,j\notin S$ the coefficients of $v(S\cup C\cup i\cup j)$, $v(S\cup C\cup i)$, $v(S\cup C\cup j)$, $v(S\cup C)$, $v(S\cup i)$, $v(S\cup j)$ and v(S) are the same in both sides of the equation in (10), and this is easily deduced from the previous expressions. We leave the details to the reader.

It remains property P4, G-SPB. Consider the unanimity game with respect to the grand coalition $(N,u_N)$, and a non-empty group $C\subset N\in {\mathcal {N}}$. It is straightforward to see that the merging game $(N_C,(u_N)_C)$ is the unanimity game $(N_C,u_{N_C})$ with respect to the grand coalition $N_C$, so $\phi ^g (C;N,u_N):=\phi _{\mathbf {c}}(N_C,(u_N)_C)=\frac{1}{n-c+1}$, as desired. $\square$

Proof

We have proved that the properties hold for the Shapley group value, so we are left with the question of uniqueness.

Since $\{(N,u_S)\}_{\begin{array}{c} S\subset N \\ S\ne \varnothing \end{array}}$ forms a basis of ${\mathcal {G}}_N$ for all $N\in {\mathcal {N}}$, by G-linearity it is sufficient to consider the games $(N,u_S)$, $\varnothing \ne S\subset N\in {\mathcal {N}}$. So let us see that $\xi ^g (C;N,u_S)=\phi ^g (C;N,u_S):=\phi _{\mathbf {c}} (N_C,(u_{S})_C)$ for all non-empty subsets $C,S\subset N\in {\mathcal {N}}$. When $C=\varnothing$, this equality trivially holds by definition of a group value.

The proof will consist in a double induction over the cardinality of the player set N (first induction) and the cardinality of the unanimous coalition $S\subset N$ (second induction).

First, we will prove that $\xi ^g (C;N,u_S)=\phi ^g (C;N,u_S)$ for all non-empty subsets $C,S\subset N$ whenever the cardinality of $N\in {\mathcal {N}}$ is $n\le 2$. For the unanimity game $(\{i\},u_i)$ with just one player $i\equiv N$, G-SPB property P4 implies that $\xi ^g (i;N,u_i)=1=\phi ^g(i;N,u_i)$. Now, let $(N,u_S)$ be a two-person unanimity game with $N=\{i,j\}$. For the unanimity game $u_N$, P4 implies that $\xi ^g ( \{i,j\};N,u_N)=\phi ^g (\{i,j\};N,u_N)$. For the unanimity game $(N,u_i)$, G-null player P1 implies:

$$\begin{aligned} \xi ^g (j;N,u_i)=\xi ^g (\varnothing ;N,u_i):=0 \;\text { and } \; \xi ^g (i;N,u_i)=\xi ^g (\{i,j\};N,u_i) , \end{aligned}$$

(11)

Now, G-CBC property P3 implies

$$\begin{aligned} \xi ^g (i;N,u_i)-\xi ^g (j;N,u_i)&=(\xi ^g (i;\{i \},u_i)-\xi ^g({\varnothing };\{i \},u_i)-(\xi ^g(j;\{j \}, u^0) \\&\quad -\xi ^g({\varnothing };\{j \}, u^0)),= 1-0-0+0, \end{aligned}$$

where $\xi ^g(i;N,u_i)=0$ for the trivial game $(\{i\},u^0)$ with $u^0(i)=0$, follows from P1. Therefore, taking into account (11), $\xi ^g (\cdot ;\{i,j\}, u_i)\equiv \phi ^g (\cdot ;\{i,j\}, u_i)$ holds. The same reasoning applies to $(N,u_j)$.

Let us fix a player set $N\in {\mathcal {N}}$ with $\vert N\vert =r$, and consider the unanimity game $(N,u_S)$ for a fix set $\varnothing \ne S\subset N$. We will prove that $\xi ^g (C;N,u_S)=\phi ^g (C;N,u_S)$ for all $C\subset N$. Two cases are possible:

(i):

If $S=N$, then G-SPB implies $\xi ^g (C;N,u_N)=\phi ^g (C;N,u_N)$ for any non-empty group C in N.

(ii):

Otherwise, if $\varnothing \ne S\subsetneq N$, we proceed by induction on the cardinality of C. Let us first prove the individual case $C=\{i\}$.

There is at least a player $j\in N\backslash S$ which by definition of unanimity game must be null. Let i be a player in S. Again by G-CBC, taking $C=\varnothing$, we obtain $\xi ^g (i;N,u_S)= \xi ^g (i;N\backslash j,{u_S} \vert _{N{\setminus } j})$, since G-null player implies $\xi ^g (j;N,u_S)=\xi ^g ({\varnothing };N,u_S)=0$, and taking into account that $u_S\vert _{N{\setminus } i}\equiv u^0\in {\mathcal {G}}_{N{\setminus } i}$.

Now, we may assume by the first induction that for every unanimity game $(N',u_S\vert _{N'})$ with $N'\subsetneq N$ we have $\xi ^g (C;N',u_S\vert _{N'})=\phi ^g (C;N',u_S\vert _{N'})$ for any group C in $N'$. Thus, $\xi ^g (i;N\backslash j,{u_S} \vert _{N{\setminus } j})= \phi ^g (i;N\backslash j,{u_S} \vert _{N{\setminus } j})$, which in turn is equal to $\frac{1}{s}$ by definition, and then $\xi ^g (i;N,u_S) =\frac{1}{s}=\phi _i(N,u_S)=\phi ^g(i;N,u_S)$. Note that every $i\notin S$ is a null player in $u_S$ and therefore G-null player implies $\xi ^g (i;N,u_S)=0=\phi ^g (i;N,u_S)$. So we are done with the individual case $C=\{i\}$.

Now, in order to prove $\xi ^g (C;N,u_S)=\phi ^g (C;N,u_S)$ for all groups C with $c>1$, we proceed by induction on the cardinality of C (second induction). So we take now $1<r'\le r$, and we may assume that $\xi ^g (C;N,u_S)=\phi ^g (C;N,u_S)$ holds for any $C\subset N$ with $|C|<r'$. We will check that $\xi ^g (D;N,u_S)=\phi ^g (D;N,u_S)$ for all $D\subset N$ with $|D|=r'$.

Let D be a fixed subset of N of cardinality $r'$. Since $S\varsubsetneq N$, again there is a null player j in $u_S$. So, if $j\in D$, then $D{\setminus } j$ is a coalition of cardinal $r'-1$ and, thus, G-null player and the second induction hypothesis imply

$$\begin{aligned} \xi ^g (D;N,u_S)=\xi ^g ({D\backslash j};N,u_S)=\phi ^g ({D\backslash j};N,u_S)=\phi ^g(D;N,u_S). \end{aligned}$$

Otherwise, if D does not contain any null player, then let i be a player in $D\subset S$. By the second induction hypothesis and G-null player property it holds

$$\begin{aligned} \xi ^g ({(D\backslash i)\cup j};N,u_S)= & {} \xi ^g ({(D\backslash i)};N,u_S)=\phi ^g ({(D\backslash i)};N,u_S)\\= & {} \phi ^g ({(D\backslash i)\cup j};N,u_S). \end{aligned}$$

Note also that $\xi ^g (C;N\backslash i,{u_S}\vert _{N{\setminus } i}))=0=\phi ^g (C;N\backslash i,{u_S}\vert _{N{\setminus } i}))$ for all $C\subset N$, since $u_S\vert _{N{\setminus } i}\equiv u^0\in {\mathcal {G}}_{N{\setminus } i}$. Hence, by G-CBC, taking $C=D\backslash i$, and the first induction,

$$\begin{aligned}&\xi ^g (D;N,u_S)- \phi ^g ({(D\backslash i)\cup j};N,u_S) \\&\quad = (\phi ^g (D;N\backslash j,{u_S}\vert _{N{\setminus } j})-\phi ^g ({D\backslash i};N\backslash j,{u_S}\vert _{N{\setminus } j})) \\&\qquad -(\phi ^g ({(D\backslash i)\cup j};N\backslash i,{u_S}\vert _{N{\setminus } i})-\phi ^g ({D\backslash i};N\backslash i,{u_S}\vert _{N{\setminus } i})) \\&\quad =\phi ^g (D;N,u_S)- \phi ^g ({(D\backslash i)\cup j};N,u_S). \end{aligned}$$

So we have proved the uniqueness for the unanimity games $(N,u_S)$ for all $S\subset N\in {\mathcal {N}}$ and we are done. $\square$

Aside from the previous considerations regarding the alternative axioms of Marichal et al. (2007), it must be remarked that it is not a trivial extension of a characterization of the Shapley value, since we do not impose any condition about the value of the individual agents out of the group C we are evaluating. In particular, we have been forced to use in the same characterization group linearity and group coalitional balanced contributions properties. In the following we will check that all the axioms above are necessary to guarantee the uniqueness of the Shapley group value $\phi ^g$.

G-null player. Let $\alpha \in (0,1)$. Define a value $\xi ^g$ in the following manner. If (N, v) is a null game, $\xi ^g_C(N,v)$=0 for every $C\in N$. Given a non-null unanimity game $u_S$, with $S\subsetneq N$, define $\xi ^g_C(u_S)$ as

$$\begin{aligned} \xi ^g_C(N,u_S)={\left\{ \begin{array}{ll} \displaystyle \sum _{k=0}^{c-1} \alpha ^{n-k}, &{} \text { if }C\cap S=\varnothing , \\ \displaystyle \frac{1}{s-\vert S\cap C\vert +1} + \sum _{k=0}^{n-s-1} \alpha ^{n-k}, &{} \text { if }C\cap S\ne \varnothing , \end{array}\right. } \end{aligned}$$

(12)

$\xi ^g_C(N,u_N)=\frac{1}{n-c+1}$, for all $C\subseteq N$, and then extend the value by additivity. It is clear that $\xi ^g_i (N,u_S)=\alpha ^n >0$ for all $i\notin S$. So, G-null player does not hold. Let us check that $\xi ^g$ verifies G-CBC over the class of unanimity games. If $(N,u_N)$, then G-CBC condition (10) trivially holds. Let $(N,u_S)$ be a unanimity game with $S\subsetneq N$. If $\vert S\vert =n-1$, then two cases are possible:

• (a) If $i,j\in S$, then $\xi ^g_{C\cup i} (N,u_S)= \frac{1}{s-\vert S\cap C\vert } + \sum _{k=0}^{n-s-1} \alpha ^{n-k}=\xi ^g_{C\cup j} (N,u_S)$ and (10) holds, since $u_S\vert _{N{\setminus } i}$ and $u_S\vert _{N{\setminus } j}$ are null games.

• (b) If $i\in S$ and $j\notin S$, then $S=N{\setminus } j$, $u_S\vert _{N{\setminus } i}\equiv 0$ and $u_S\vert _{N{\setminus } j}$ is a unanimity game w.r.t. the grand coalition $N{\setminus } j$. Thus (10) holds:

  $$\begin{aligned}&\xi ^g_{C\cup i} (N{\setminus } j, u_{N{\setminus } j}\vert _{N{\setminus } j})-\xi ^g_{C} (N{\setminus } j, u_{N{\setminus } j}\vert _{N{\setminus } j}) \\&\quad =\frac{1}{(n-1)-(c+1)+1} -\frac{1}{(n-1)-c+1}=\xi ^g_{C\cup i} (N, u_{N{\setminus } j})-\xi ^g_{C\cup j} (N u_{N{\setminus } j}) . \end{aligned}$$

If $\vert S\vert <n-1$, then three cases are possible:

• (a) If $i,j\in S$, then $\xi ^g_{C\cup i} (N,u_S)= \frac{1}{s-\vert S\cap C\vert } + \sum _{k=0}^{n-s-1} \alpha ^{n-k}=\xi ^g_{C\cup j} (N,u_S)$ and (10) holds.

• (b) If $i\in S$ and $j\notin S$, then

  $$\begin{aligned} \xi ^g_{C\cup i} (N{\setminus } j, u_{S}\vert _{N{\setminus } j}) = \frac{1}{s-\vert S\cap C\vert } + \sum _{k=1}^{n-s-1} \alpha ^{n-k} \end{aligned}$$

  and

  $$\begin{aligned} \xi ^g_{C} (N{\setminus } j, u_{S}\vert _{N{\setminus } j}) = {\left\{ \begin{array}{ll} \displaystyle \sum _{k=1}^{c} \alpha ^{n-k}, &{} \text { if }C\subseteq N{\setminus } S, \\ \displaystyle \frac{1}{s-\vert S\cap C\vert +1} + \sum _{k=1}^{n-s-1} \alpha ^{n-k}, &{} \text { otherwise.} \end{array}\right. } \end{aligned}$$

  Thus, if $C\subseteq N{\setminus } S$:

  $$\begin{aligned}&\xi ^g_{C\cup i} (N{\setminus } j, u_{S}\vert _{N{\setminus } j})-\xi ^g_{C} (N{\setminus } j, u_{S}\vert _{N{\setminus } j}) \\&\quad =\frac{1}{s}+ \sum _{k=1}^{n-s-1} \alpha ^{n-k} - \sum _{k=1}^{c} \alpha ^{n-k} =\xi ^g_{C\cup i} (N, u_{S})-\xi ^g_{C\cup j} (N, u_{S}) \end{aligned}$$

  If $C\cap S\ne \varnothing$, then

  $$\begin{aligned}&\xi ^g_{C\cup i} (N{\setminus } j, u_{S}\vert _{N{\setminus } j})-\xi ^g_{C} (N{\setminus } j, u_{S}\vert _{N{\setminus } j}) \\&\quad =\frac{1}{s-\vert S\cap C\vert }+ \sum _{k=1}^{n-s-1} \alpha ^{n-k} - \left( \frac{1}{s-\vert S\cap C\vert +1}+ \sum _{k=1}^{n-s-1} \alpha ^{n-k} \right. \\&\quad \left. =\xi ^g_{C\cup i} (N, u_{S})-\xi ^g_{C\cup j} (N, u_{S}) \right) \end{aligned}$$

• c) If $i,j\in N{\setminus } S$, then $C\cup i\subseteq N{\setminus } S$ if, and only if, $C\cup j\subseteq N{\setminus } S$ and, therefore condition (10) can be easily checked.

Now, since G-CBC condition is additive $\xi ^g$ satisfies it over $\bigcup _{n\ge 1} {\mathcal {G}}_n$

G-linearity. Let $\xi ^g$ be another group value over (N, v) which is defined in the following way:

$(\mathrm{A}_1)$ :

If there is at least a null player in N, or (N, v) is the unanimity game with respect to the grand coalition N, then $\xi ^g_C(N,v)=\phi ^g_C(N,v)$ for every group C in N.

$(\mathrm{A}_2)$ :

Otherwise, $\xi ^g_C(N,v)=\phi ^g_C(N,v)+k$, being $k\ne 0$ a fixed constant.

It is easily checked that G-null-player and G-SPB hold for $\xi ^g$. We will check that the property of coalitional balanced contributions G-BMC also holds for this value. Note that all differences in (10) match $\xi ^g_{D} (L,w)$ with $\xi ^g_{D'} (L,w)$, for some coalition $L\in \{N,N{\setminus } i, N{\setminus } j\}$ and some game $w\in \{v,v_{-i},v_{-j}\}$.

Taking into account that $\phi ^g=\xi ^g$ in case (N, v) is some of the games in the first case $(A_1)$, and in the second one the $k's$ cancel, it holds:

$$\begin{aligned} \xi ^g_{C\cup i} (N,v)- \xi ^g_{C\cup j} (N,v)&= \phi ^g_{C\cup i} (N,v)- \phi ^g_{C\cup j} (N,v),\\ \xi ^g_{C\cup i} (N{\setminus } j,v_{-j})- \xi ^g_{C} (N{\setminus } j,v_{-j})&= \phi ^g_{C\cup i} (N{\setminus } j,v_{-j})- \phi ^g_{C} (N{\setminus } j,v_{-j}),\\ \xi ^g_{C\cup j} (N{\setminus } i,v_{-i})- \xi ^g_{C} (N{\setminus } i,v_{-i})&= \phi ^g_{C\cup j} (N{\setminus } i,v_{-i})- \phi ^g_{C} (N{\setminus } i,v_{-i}), \end{aligned}$$

G-CBC property holds for $\phi ^g$, and so the property does so for $\xi ^g$, and we are done.

Note that we can modify $\xi ^g$ defining $\xi ^g_i(N,v)=\Phi _i (N,v)$, for all $i\in N$, and $\xi ^g_N(N,v)=v(N)$, for all $N\subseteq \mathbb {N}$, and the same result holds.

G-CBC. Define a value $\xi ^g$ in the following manner. If (N, v) is a null game, $\xi ^g_C(N,v)$=0 for every $C\in N$. Given a non-null unanimity game $u_S$ with $S\subsetneq N$, define $\xi ^g_C(u_S)$ as

$$\begin{aligned} \xi ^g_C(u_S)={\left\{ \begin{array}{ll} 0 &{} \text { if }C\cap S=\varnothing , \\ \xi ^g_{C}(u_S)= 1, &{} \text { if }S\subseteq C,\\ |S\cap C|\times |S\backslash C| &{} \text { if }C\cap S\ne \varnothing \text { and }C\cap S\ne S, \end{array}\right. } \end{aligned}$$

(13)

$\xi ^g_C(N,u_N)=\frac{1}{n-c+1}$, for all $C\subseteq N$, and then extend the value by additivity.

Observe that all axioms but G-CBC hold. The unique one that needs a bit of discussion is the G-null player axiom, which holds when considering the base of unanimity games because in $u_S$ the null players are precisely the players outside S, and therefore:

$$\begin{aligned} \xi ^g_{C\cup i} (N,u_S)&= 0= \xi ^g_{C} (N,u_S), \text { if }S\cap C=\varnothing ,\\ \xi ^g_{C\cup i} (N,u_S)&= 1= \xi ^g_{C} (N,u_S), \text { if }S\subseteq C,\text { and}\\ \xi ^g_{C\cup i} (N,u_S)&= |S\cap (C\cup i)|\times |S\backslash (C\cup i)|=|S\cap C|\times |S\backslash C|= \xi ^g_{C} (N,u_S), \text { otherwise,} \end{aligned}$$

for all player $i\notin S$. Then, taking into account that the Harsanyi dividend $c_S (N,v)$ of any coalition S containing null players in the game (N, v) equals zero, G-null player property holds for any n-person game $(N,v)\in {\mathcal {G}}_n$, for all $N\subseteq \mathbb {N}$.

Let us check by means of a concrete example that the G-CBC axiom fails in this case. Consider $(N,u_S)$ with $|N|=3$, and $S=\{1,2\}$. In the notation of the axiom, take $C=\{1\}$, $i=2$, and $j=3$. Then:

$$\begin{aligned} \xi ^g_{\{1,2\}}(N,u_S)= & {} 1, \; \xi ^g_{\{1,3\}}(N,u_S)=1\times 1=1, \;\xi ^g_{\{1,2\}}(N\backslash 3, u_S\vert _{N \backslash 3}) \\= & {} 1, \;\xi ^g_{1}(N\backslash 3, u_S\vert _{N \backslash 3})=1/2, \end{aligned}$$

and $\xi ^g_{\{1,3\}}(N\backslash 2, u_S\vert _{N \backslash 2})=0=\xi ^g_{1}(N\backslash 2, u_S\vert _{N \backslash 2})$, because the game $(N\backslash 2, u_S\vert _{N \backslash 2})$ is null. It is clear now that the two sides of the equalities that define the axiom do not coincide in this case.

Note that we can modify $\xi ^g$ defining $\xi ^g_i(N,u_S)=\Phi _i (N,v)$, for all $i\in N$, and $\varnothing \ne S\subsetneq N$, for all $N\subseteq \mathbb {N}$, and the same result holds. Probably it is easy to find more examples by defining the value over $C\cap S$ (when $C\cap S\ne \varnothing$) as another appropriate function of $(|C\cap S|,|S\backslash C|)$.

G-symmetry over pure bargaining games. Define a value $\xi ^g$ in the following manner. If (N, v) is a null game, $\xi ^g_C(N,v)$=0 for every $C\in N$. Given a non-null unanimity game $u_S$, define $\xi ^g(N,u_S)$ as $\xi ^g_C(N,u_S)=\frac{\vert C\cap S\vert }{\vert S\vert }$, for all group $C\subseteq N$, and then extend the value by additivity.

Observe that all axioms but G-SPB hold. The G-null player axiom trivially holds when considering the base of unanimity games. Then, since $c_S (N,v)=0$ for all S containing null players in the game (N, v), G-null player property holds in general. Moreover, G-additivity follows from the definition.

Let us check that the G-CBC axiom holds. Let (N, v) be any n-person game with $n\ge 2$. Let C be any group in N of cardinality $c\le n-2$, and let $i,j\in N{\setminus } C$. Then:

• If $i,j\in N{\setminus } S$, then (10) holds since all the involved differences are zero because i and j are null players in the three games.

• If $i,j\in S$, then $\vert (C\cup i)\cap S\vert =\vert C\cap S\vert + 1=\vert (C\cup j)\cap S\vert$. Thus, $\xi ^g_{C\cup i} (N,u_S)- \xi ^g_{C\cup j} (N,u_S)=0$, and (10) holds because the games $(N{\setminus } i, u_S\vert _{N{\setminus } i})$ and $(N{\setminus } j, u_S\vert _{N{\setminus } j})$ are null.

• If $i\in S$ and $j\notin S$, then $\xi ^g_{C\cup i} (N,u_S)- \xi ^g_{C\cup j} (N,u_S)=\frac{1}{s}=\xi ^g_{C\cup i} (N{\setminus } j, u_S\vert _{N{\setminus } j})- \xi ^g_{C} (N{\setminus } j, u_S\vert _{N{\setminus } j})$, and (10) holds because $u_S\vert _{N{\setminus } i}\equiv 0$.

Clearly, G-SPB fails, so we are done.

Proposition 1

Let $N\in {\mathcal {N}}$ be any finite set of players, and v be any game in ${\mathcal {G}}_N$. Then, the Shapley group value $\phi ^g$ verifies the following properties:

(i):

Group Rationality: $\phi ^g (C;N,v) \ge v(C)$ for every $C\subset N \in {\mathcal {N}}$ if the game $v\in {\mathcal {G}}_N$ is superadditive, and

(ii):

Monotonicity: $\phi ^g (C;N,v) \le \phi ^g (D;N,v)$ for every pair of coalitions $C\subset D\subset N\in {\mathcal {N}}$ if the game $v\in {\mathcal {G}}_N$ is monotonic.

Proof

Group rationality follows from the individual rationality of the Shapley value. Note that every merging game $(N_C,v_C)$, $C\subset N$, is superadditive if it is (N, v), for all $C\subset N\in {\mathcal {N}}$ and $v\in {\mathcal {G}}_N$.

Monotonicity follows from being

$$\begin{aligned} \phi ^g ({C\cup i};N,v) - \phi ^g (C;N,v)&=\sum _{\begin{array}{c} S\subset N{\setminus } C\\ i\notin S \end{array}} \frac{s! (n-c-s)!}{(n-c+1)!} \Bigl ( v(S\cup i\cup C)-v(S\cup C)\Bigr ) \nonumber \\&\quad +\frac{(s+1)! (n-c-s-1)!}{(n-c+1)!} \Bigl ( v(S\cup i)-v(S) \Bigr ) \ge 0, \end{aligned}$$

(14)

for all coalitions $C\subset N\in {\mathcal {N}}$, and all players $i\notin C$, whenever the game v is monotonic. $\square$

Theorem 2

Let $N\in {\mathcal {N}}$ be any finite set of players, and v be any game in ${\mathcal {G}}_N$. Let $C\subset N$ be any group in N, and let $i\notin C$. Then, the marginal contribution of player $i\in N{\setminus } C$ to the Shapley group value of C equals:

$$\begin{aligned} MC_i^{g} (C;N,v):=\phi ^g ({C\cup i};N,v)-\phi ^g (C;N,v)=\phi _i (N{\setminus } C, v\vert _{N{\setminus } C}) + \psi _{\mathbf {c}i} (N_C,v_C). \end{aligned}$$

(15)

Proof

Let us arbitrarily fix the sets C and N and the player i, $C\subsetneq N\in {\mathcal {N}}$, $i\in N{\setminus } C$, and let $v\in {\mathcal {G}}_N$. Then, adding and subtracting the amount

$$\begin{aligned} \sum _{\begin{array}{c} S\subset N{\setminus } C\\ i\notin S \end{array}} \frac{s! (n-c-s-1)!}{(n-c)!} \Bigl ( v(S\cup i)-v(S)\Bigr ) \end{aligned}$$

to the above expression (14) of the marginal contribution $MC_i^{g} (C;N,v)$ follows that:

$$\begin{aligned} MC_i^{g} (C;N,v)=&\sum _{\begin{array}{c} S\subset N{\setminus } C\\ i\notin S \end{array}} \frac{s! (n-c-s-1)!}{(n-c)!} \Bigl ( v(S\cup i)-v(S)\Bigr ) \\&+ \sum _{\begin{array}{c} S\subset N{\setminus } C\\ i\notin S \end{array}} \frac{s! (n-c-s)!}{(n-c+1)!} \Bigl ( v(S\cup i\cup C)-v(S\cup C)\\&-v(S\cup i) + v(S)\Bigr ). \end{aligned}$$

The first term is precisely the Shapley value of player i in the restricted game $(N{\setminus } C, v\vert _{N{\setminus } C})$, where players in C do not play a role. The second term can be expressed by means of the second-order difference operators for the pair of players $i,\mathbf {c}\in N_C$, as follows:

$$\begin{aligned}&\sum _{\begin{array}{c} S\subset N{\setminus } C\\ i\notin S \end{array}} \frac{s! (n-c-s)!}{(n-c+1)!} \Bigl ( v(S\cup i\cup C)-v(S\cup C)-v(S\cup i) + v(S)\Bigr )\\&\quad =\sum _{\begin{array}{c} S\subset N{\setminus } C\\ i\notin S \end{array}} \frac{s! (n-c-s)!}{(n-c+1)!} \Delta ^2_{ic} (S;N_C,v_C)=:\psi _{\mathbf {c}i} (N_C,v_C) . \end{aligned}$$

$\square$