# Necessary players and values

J.C. Gonçalves-Dosantos1, I. García-Jurado1,
J. Costa2, J.M. Alonso-Meijide3

(September 3, 2024)

###### Abstract

In this paper we introduce the $\Gamma$ value, a new value for cooperative
games with transferable utility. We also provide an axiomatic characterization
of the $\Gamma$ value based on a property concerning the so-called necessary
players. A necessary players of a game is one without which the characteristic
function is zero. We illustrate the performance of the $\Gamma$ value in a
particular cost allocation problem that arises when the owners of the
apartments in a building plan to install an elevator and share its
installation cost; in the resulting example we compare the proposals of the
$\Gamma$ value, the equal division value and the Shapley value in two
different scenarios. In addition, we propose an extension of the $\Gamma$
value for cooperative games with transferable utility and with a coalition
structure. Finally, we provide axiomatic characterizations of the coalitional
$\Gamma$ value and of the Owen and Banzhaf-Owen values using alternative
properties concerning necessary players.

11footnotetext: Grupo MODES, CITIC and Departamento de Matemáticas,
Universidade da Coruña, Campus de Elviña, 15071 A Coruña,
Spain.22footnotetext: Grupo MODES, Departamento de Matemáticas, Universidade
da Coruña, Campus de Elviña, 15071 A Coruña, Spain.33footnotetext: Grupo
MODESTYA, Departamento de Estatística, Análise Matemática e Optimización,
Universidade de Santiago de Compostela, Campus de Lugo, 27002 Lugo, Spain.

Keywords: cooperative game, necessary player, coalition structure, value.

##  1 Introduction

The Shapley value, introduced in Shapley (1953), is a rule for distributing
the benefits that a set of agents $N$ can generate, taking into account the
contributions of each agent and of each possible subset of $N$. The Shapley
value is one of the most important solutions of cooperative game theory and it
has many applications in a wide variety of fields. For instance, in recent
years the Shapley value has been applied to cancer research (see Albino et
al., 2008), to machine learning (see Strumbelj and Kononenko, 2010), to data
envelopment analysis (see Yang and Zhang, 2015), to image classification (see
Gurran et al., 2016), to project management (see Bergantiños et al., 2018, and
Gonçalves-Dosantos et al., 2020), etc. Moretti and Patrone (2008) is a survey
explaining the transversality of the Shapley value.

The game theory literature provides many alternatives to the Shapley value,
such as the nucleolus (Schmeidler, 1969), the Banzhaf Value (Owen, 1975), the
$\tau$-value (Tijs, 1981), the equal-surplus division value (Driessen and
Funaki, 1991) or, more recently, the consensus value (Ju et al., 2007) and the
ie-Banzhaf value (Alonso-Meijide et al., 2019b). All those alternative values
have appealing properties and could be used instead of the Shapley value. In
order to decide what is the most appropriate value for a particular problem it
is helpful to know the properties that are essentially connected to each
value. This is why game theory is interested in the so-called
characterizations: to characterize a value in a class of games is to find a
set of properties so that it is the only value that fulfills them in that
class. For instance, in Luchetti et al. (2010), two relevance indexes for
genes are compared, one based on the Shapley value and the other based on the
Banzhaf value, and for that purpose they are characterized in the
corresponding class of games, the so-called microarray games.

In this article we introduce a value for cooperative games that results from
proposing a new property for so-called necessary players that, in a way,
corrects the properties for such players met by the Shapley and Banzhaf
values. Informally, necessary players are those without whom the
characteristic function of the game would be zero. These players have
attracted the attention of game theorists for axiomatic studies in the last
years. For instance, Alonso-Meijide et al. (2019a) and Béal and Navarro (2020)
are two recent papers dealing with necessary players and characterizations.
Apart from introducing a new value, in this paper we provide an axiomatic
characterization of it, which allows to compare the new value with other
solution concepts for cooperative games. Furthermore, we extend and
characterize the new value for cooperative games with a coalition structure. A
cooperative game with a coalition structure models those situations where the
agents in a set $N$ aim to distribute the benefits they generate taking into
account the contributions of each agent and of each possible subset of $N$, as
well as a coalition structure (a partition of $N$) that conditions the
distribution, in the sense that distribution among the classes of the
partition is made first and, then, a distribution within those classes is
performed. Cooperative games with a coalition structure have been applied in
several fields like political analysis (see, for instance, Carreras and
Puente, 2015), infrastructure management (see Costa, 2016), cost allocation
(see Fragnelli and Iandolino, 2004), etc. The Owen value (Owen, 1977) and the
Banzhaf-Owen value (Owen, 1982) are, respectively, the variations of the
Shapley value and the Banzhaf value for cooperative games with a coalition
structure. In this paper we also provide new characterizations of the Owen and
the Banzhaf-Owen values using properties involving necessary players.

The structure of this paper is as follows. In Section 2 we introduce the
$\Gamma$ value, a new value for cooperative games. We also provide an
axiomatic characterization of the $\Gamma$ value and illustrate its behaviour
in a practical example that arises in a problem of sharing the costs of
installing an elevator. In Section 3 we provide new characterizations of the
Owen and Banzhaf-Owen values and introduce and characterize an extension of
the $\Gamma$ value for cooperative games with a coalitional structure. We
finish the paper with a section of concluding remarks.

##  2 Values and necessary players

A cooperative game is a pair $(N,v)$ given by a finite set of players $N$ and
a characteristic function $v:2^{N}\rightarrow\mathbb{R}$, that assigns to each
coalition $S\subseteq N$ a real number $v(S)$ that indicates the benefits that
coalition $S$ is able to generate; by definition $v(\emptyset)=0$. We denote
by $\mathcal{G}_{N}$ the family of all cooperative games with player set $N$.

A value for cooperative games is a map $f$ that assigns to every game
$(N,v)\in\mathcal{G}_{N}$ a vector $f(N,v)\in\mathbb{R}^{N}$. Two of the most
important values for cooperative games are the Shapley value (Shapley, 1953)
and the Banzhaf value (Owen, 1975). A number of characterizations of these two
values can be found on the literature. For example, Alonso-Meijide et al.
(2019a) provides characterizations of those values using only three properties
for each of them: two common properties and one extra property concerning the
so-called necessary players that differs for the Shapley and the Banzhaf
values. In this paper we concentrate on characterizations of values involving
necessary players. Let us first remember the formal definition of a necessary
player.

###### Definition 1

A player $i\in N$ is said to be necessary in the cooperative game $(N,v)$ if
$v(S)=0$ for all $S\subseteq N\setminus\\{i\\}$.

In words, a necessary player is one without whom cooperation does not produce
any results. In fact, notice that if $i$ is necessary in $(N,v)$, then
$v(S)=\sum_{j\in S}v(\\{j\\})=0$ for all $S\subseteq N\setminus\\{i\\}$;
hence, the game resulting after the elimination of $i$ is additive and null.
Necessary players often arise in real situations. Take, for instance, the
following example.

###### Example 2

Consider a council formed by three entities with $24,15$ and $9$ votes,
respectively. Any proposal must receive at least $25$ votes to be approved. In
the resulting voting game, it is easy to see that the entity with $24$ votes
is a necessary player because, without it, the other two entities cannot get
any proposals approved.

In some specific problems, as in the example above, the necessary players
arise in a natural way and, therefore, a characterization based on such
players can be relevant in deciding what value to use in those problems. We
start by remembering the characterizations of the Shapley and Banzhaf values
in Alonso-Meijide et al. (2019a) and some other preliminary material. The
Shapley value $\varphi$ is defined as

$\varphi_{i}\left(N,v\right)=\frac{1}{n}\sum_{S\subseteq
N\setminus\\{i\\}}\frac{1}{{n-1\choose
s}}(v\left(S\cup\\{i\\}\right)-v\left(S\right))$

for all $(N,v)\in\mathcal{G}_{N}$ and all $i\in N$; $n$ and $s$ denote the
cardinalities of $N$ and $S$, respectively. The Banzhaf value $\beta$ is
defined as

$\beta_{i}\left(N,v\right)=\frac{1}{2^{n-1}}\sum_{S\subseteq
N\setminus\\{i\\}}(v\left(S\cup\\{i\\}\right)-v\left(S\right))$

for all $(N,v)\in\mathcal{G}_{N}$ and all $i\in N$. Both the Shapley and the
Banzhaf value are additive. This means that they satisfy the following
condition.

Additivity. A value for cooperative games $f$ satisfies the property of
additivity if for any pair of cooperative games
$\left(N,v\right),\left(N,w\right)$ it holds that

$f\left(N,v+w\right)=f\left(N,v\right)+f\left(N,w\right).$

Additivity is a good property because, at the same time that it is natural and
easily interpretable, it greatly facilitates the mathematical analysis of the
values that comply with it and the calculation of such values; for instance,
Benati et al. (2019) provides a method to approximate additive values in
cooperative games that is useful when the number of players is large.

Another reasonable property that is satisfied by the Shapley and Banzhaf value
concerns null players. Remember that a null player of $(N,v)$ is an $i\in N$
such that $v(S)=v(S\cup\\{i\\})$ for all $S\subseteq N\setminus\\{i\\}$.

Null Player. A value for cooperative games $f$ satisfies the property of null
player if for any cooperative game $\left(N,v\right)$ and for any $i\in N$
null player of $\left(N,v\right)$, it holds that $f_{i}\left(N,v\right)=0$.

Now let us see two alternative properties for necessary players introduced in
Alonso-Meijide et al. (2019a) and the main result concerning them.

Necessary Players Get the Weighted Mean. A value for cooperative games $f$
satisfies the property of necessary players get the weighted mean if, for all
cooperative game $\left(N,v\right)$ and for all $i\in N$ necessary player in
$\left(N,v\right)$, it holds that

$f_{i}\left(N,v\right)=\frac{1}{n}\sum_{S\subseteq N,i\in
S}\frac{1}{{n-1\choose s-1}}v(S).$

Necessary Players Get the Mean. A value for cooperative games $f$ satisfies
the property of necessary players get the mean if, for all cooperative game
$\left(N,v\right)$ and for all $i\in N$ necessary player in
$\left(N,v\right)$, it holds that

$f_{i}\left(N,v\right)=\frac{1}{2^{n-1}}\sum_{S\subseteq N,i\in S}v(S).$

Observe that the two properties above are similar. Both establish that a
necessary player must receive the average of the values of the coalitions to
which that player belongs, although the former takes into account the size of
such coalitions and the latter does not.

###### Theorem 3

(Alonso-Meijide et al., 2019a).
1\. The Shapley value is the unique value for cooperative games that satisfies
the properties of additivity, null player and necessary players get the
weighted mean.
2\. The Banzhaf value is the unique value for cooperative games that satisfies
the properties of additivity, null player and necessary players get the mean.

Now we remember two widely known properties for values that will be relevant
in the subsequent discussion.

Efficiency. A value for cooperative games $f$ satisfies the property of
efficiency if for all cooperative game $\left(N,v\right)$, it holds that

$\sum\limits_{i\in N}f_{i}\left(N,v\right)=v\left(N\right).$

We say that players $i,j\in N$ are symmetric in $\left(N,v\right)\in{\cal
G}_{N}$ if $v(S\cup\\{i\\})=v(S\cup\\{j\\})$ for every $S\subseteq
N\setminus\left\\{i,j\right\\}$.

Symmetry. A value for cooperative games $f$ satisfies the property of symmetry
if for all cooperative game $\left(N,v\right)$ and for all $i,j\in N$
symmetric players in $\left(N,v\right)$, it holds that

$f_{i}\left(N,v\right)=f_{j}\left(N,v\right).$

It is well-known that the Shapley and Banzhaf values satisfy the symmetry
property. However, only the Shapley value is efficient. In some problems,
efficiency is not an essential property for a value, see for example
microarray games in Lucchetti et al. (2010). In many cases, however,
efficiency will be required for a value to make sense; this happens, for
example, when we are faced with cost allocation problems. One question we can
ask is whether there is a value that fulfills the necessary players get the
mean property and the efficiency property. The answer is negative because
those properties are incompatible. Indeed, assume that a value for cooperative
games $f$ satisfies both properties and for every non-empty $S\subseteq N$
denote by $(N,e_{S})$ the cooperative game in ${\cal G}_{N}$ given, for every
$T\subseteq N$, by:

$e_{S}(T)=\left\\{\begin{array}[]{cc}1&\mbox{if $T=S$,}\\\
0&\mbox{otherwise.}\\\ \end{array}\right.$ (1)

Since $f$ satisfies efficiency, it holds that

$\sum\limits_{i\in N}f_{i}\left(N,e_{N}\right)=1.$ (2)

Notice now that every $i\in N$ is necessary in $(N,e_{N})$ and then, since $f$
satisfies the necessary players get the mean property, it holds that

$\sum\limits_{i\in N}f_{i}\left(N,e_{N}\right)=\sum\limits_{i\in
N}\frac{1}{2^{n-1}}=\frac{n}{2^{n-1}}.$ (3)

Observe that (2) and (3) are incompatible for $n>2$, which implies that
necessary players get the mean and efficiency are incompatible properties.
Such incompatibility vanishes when we consider the next weak version of the
former property.

(Weak) Necessary Players Get the Mean. A value for cooperative games $f$
satisfies the (weak) necessary players get the mean property if, for all
cooperative game $\left(N,v\right)$ with $v(N)=0$ and for all $i\in N$
necessary player in $\left(N,v\right)$, it holds that

$f_{i}\left(N,v\right)=\frac{1}{2^{n-1}}\sum_{S\subseteq N,i\in S}v(S).$

With this new property we can prove the following proposition.

###### Theorem 4

There exists a unique value for cooperative games that satisfies the
properties of additivity, (weak) necessary players get the mean, efficiency
and symmetry. This value that we denote by $G$ is given, for all
$(N,v)\in{\cal G}_{N}$ and all $i\in N$, by:

$G_{i}\left(N,v\right)=\frac{1}{2^{n-1}}\left(\sum_{S\subset N,i\in
S}v(S)-\sum_{S\subset N,i\not\in S}\frac{s}{n-s}v(S)\right)+\frac{v(N)}{n}.$
(4)

Proof. (Existence). It is clear that $G$ satisfies additivity. To check that
it satisfies the (weak) necessary players get the mean property, take a
cooperative game $\left(N,v\right)$ with $v(N)=0$ and such that $i\in N$ is a
necessary player in $\left(N,v\right)$. Then expression (4) reduces to

$G_{i}(N,v)=\frac{1}{2^{n-1}}\sum_{S\subset N,i\in
S}v(S)=\frac{1}{2^{n-1}}\sum_{S\subseteq N,i\in S}v(S).\\\ $

To check that $G$ satisfies efficiency notice that, for every cooperative game
$(N,v)$,

$\displaystyle\sum_{i\in N}G_{i}(N,v)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{n-1}}\sum_{i\in N}\left(\sum_{S\subset N,i\in
S}v(S)-\sum_{S\subset N,i\not\in S}\frac{s}{n-s}v(S)\right)+v(N)$
$\displaystyle=$ $\displaystyle\frac{1}{2^{n-1}}\left(\sum_{S\subset
N}sv(S)-\sum_{S\subset N}(n-s)\frac{s}{n-s}v(S)\right)+v(N)$ $\displaystyle=$
$\displaystyle v(N).$

To check that $G$ satisfies symmetry take a cooperative game $(N,v)$ and a
pair of symmetric players in $(N,v)$ $i,j\in N$. Notice that

$\displaystyle\sum_{S\subset N,i\in S}v(S)-\sum_{S\subset N,i\not\in
S}\frac{s}{n-s}v(S)$ $\displaystyle=$ $\displaystyle\sum_{S\subseteq
N\setminus\\{i,j\\}}\left(v(S\cup\\{i\\})\right)+\sum_{S\subset
N\setminus\\{i,j\\}}\left(v(S\cup\\{i,j\\})\right)$
$\displaystyle-\sum_{S\subseteq
N\setminus\\{i,j\\}}\left(\frac{s}{n-s}v(S)+\frac{s+1}{n-s-1}v(S\cup\\{j\\})\right).$

Now, since $i,j$ are symmetric in $(N,v)$, the last expression is equal to

$\sum_{S\subseteq
N\setminus\\{i,j\\}}\left(v(S\cup\\{j\\})\right)+\sum_{S\subset
N\setminus\\{i,j\\}}\left(v(S\cup\\{i,j\\})\right)-\sum_{S\subseteq
N\setminus\\{i,j\\}}\left(\frac{s}{n-s}v(S)+\frac{s+1}{n-s-1}v(S\cup\\{i\\})\right)$

and then it is clear that $G_{i}(N,v)=G_{j}(N,v)$.

(Uniqueness). Take $f$, a value for cooperative games that satisfies
efficiency, symmetry, (weak) necessary players get the mean and additivity and
take a cooperative game $(N,v)$. We prove now that $f(N,v)=G(N,v)$. Indeed,
consider the canonical basis of the vector space of characteristic functions
of cooperative games with set of players $N$: $\\{e_{S}\\}_{S\in
2^{N}\setminus\emptyset}$ (see expression (1)). Observe that $v$ can be
written in a unique way as a linear combination of the elements of the
canonical basis: $v=\sum_{S\in 2^{N}\setminus\emptyset}v(S)e_{S}$. Since $f$
satisfies additivity,

$f(N,v)=\sum_{S\in 2^{N}\setminus\emptyset}f(N,v(S)e_{S}).$

Note that efficiency, symmetry and (weak) necessary players get the mean
characterize a unique value in the class $\\{(N,v(S)e_{S})\ |\ S\subset
N,S\neq\emptyset\\}$. Besides, efficiency and symmetry characterize a unique
value for $(N,v(N)e_{N})$. Hence $f(N,v)=G(N,v)$. $\Box$

Surprisingly enough, the new value $G$ introduced in Proposition 4 looks a lot
like the e-Banzhaf value defined in Alonso-Meijide et al. (2019b) but it is
not the same, because $\frac{n-s}{s}$ is changed by $\frac{s}{n-s}$ and,
moreover, those two parameters do not multiply the same summands in the
expressions of $G$ and of the e-Banzhaf value. Indeed, such parameters do not
seem to have a clear interpretation from the point of view of fairness, which
leads us to think that perhaps the (weak) necessary players get the mean
property should be reformulated. In fact, it is more reasonable to ask that a
necessary player be entitled to the average of the per capita values of the
coalitions that contain it rather than the average of the values of those
coalitions; in fact, such players are necessary for coalitions to have a value
other than zero, but they require the other coalition members to generate such
a value. Thus we propose the new property formulated below.

Necessary Players Get the Per Capita Mean. A value for cooperative games $f$
satisfies the necessary players get the per capita mean property if, for all
cooperative game $\left(N,v\right)$ with $v(N)=0$ and for all $i\in N$
necessary player in $(N,v)$, it holds that

$f_{i}\left(N,v\right)=\frac{1}{2^{n-1}}\sum_{S\subseteq N,i\in
S}\frac{v(S)}{s}.$

The next result introduces and characterizes a new value for cooperative
games.

###### Theorem 5

There exists a unique value for cooperative games that satisfies the
properties of additivity, necessary players get the per capita mean,
efficiency and symmetry. This value that we denote by $\gamma$ is given, for
all $(N,v)\in{\cal G}_{N}$ and all $i\in N$, by:

$\gamma_{i}\left(N,v\right)=\frac{1}{2^{n-1}}\left(\sum_{S\subset N,i\in
S}\frac{v(S)}{s}-\sum_{S\subset N,i\not\in
S}\frac{v(S)}{n-s}\right)+\frac{v(N)}{n}.$ (5)

Proof. (Existence). It is clear that $\gamma$ satisfies additivity. To check
that it satisfies the necessary players get the per capita mean property take
a cooperative game $\left(N,v\right)$ with $v(N)=0$ and such that $i\in N$ is
a necessary player in $\left(N,v\right)$. Then expression (5) reduces to

$\gamma_{i}(N,v)=\frac{1}{2^{n-1}}\sum_{S\subset N,i\in
S}\frac{v(S)}{s}=\frac{1}{2^{n-1}}\sum_{S\subseteq N,i\in S}\frac{v(S)}{s}.\\\
$

To check that $\gamma$ satisfies efficiency notice that, for every cooperative
game $(N,v)$,

$\displaystyle\sum_{i\in N}\gamma_{i}(N,v)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{n-1}}\sum_{i\in N}\left(\sum_{S\subset N,i\in
S}\frac{v(S)}{s}-\sum_{S\subset N,i\not\in S}\frac{v(S)}{n-s}\right)+v(N)$
$\displaystyle=$ $\displaystyle\frac{1}{2^{n-1}}\left(\sum_{S\subset
N}s\frac{v(S)}{s}-\sum_{S\subset N}(n-s)\frac{v(S)}{n-s}\right)+v(N)$
$\displaystyle=$ $\displaystyle v(N).$

To check that $\gamma$ satisfies symmetry take a cooperative game $(N,v)$ and
a pair of symmetric players in $(N,v)$ $i,j\in N$. Notice that

$\sum_{S\subset N,i\in S}\frac{v(S)}{s}-\sum_{S\subset N,i\not\in
S}\frac{v(S)}{n-s}=$ $\sum_{S\subseteq
N\setminus\\{i,j\\}}\frac{v(S\cup\\{i\\})}{s+1}+\sum_{S\subset
N\setminus\\{i,j\\}}\frac{v(S\cup\\{i,j\\})}{s+2}-\sum_{S\subseteq
N\setminus\\{i,j\\}}\left(\frac{v(S)}{n-s}+\frac{v(S\cup\\{j\\})}{n-s-1}\right).$

Now, since $i,j$ are symmetric in $(N,v)$, the last expression is equal to

$\sum_{S\subseteq
N\setminus\\{i,j\\}}\frac{v(S\cup\\{j\\})}{s+1}+\sum_{S\subset
N\setminus\\{i,j\\}}\frac{v(S\cup\\{i,j\\})}{s+2}-\sum_{S\subseteq
N\setminus\\{i,j\\}}\left(\frac{v(S)}{n-s}+\frac{v(S\cup\\{i\\})}{n-s-1}\right)$

and then it is clear that $\gamma_{i}(N,v)=\gamma_{j}(N,v)$.

(Uniqueness). Take $f$, a value for cooperative games that satisfies
efficiency, symmetry, necessary players get the per capita mean and additivity
and take a cooperative game $(N,v)$. We prove now that $f(N,v)=\gamma(N,v)$.
Indeed, consider the basis of the vector space of characteristic functions of
cooperative games with set of players $N$: $\\{e_{S}\\}_{S\in
2^{N}\setminus\emptyset}$ (see expression (1)). Observe that $v$ can be
written in a unique way as a linear combination of the elements of the basis:
$v=\sum_{S\in 2^{N}\setminus\emptyset}v(S)e_{S}$. Since $f$ satisfies
additivity,

$f(N,v)=\sum_{S\in 2^{N}\setminus\emptyset}f(N,v(S)e_{S}).$

Notice that efficiency, symmetry and necessary players get the per capita mean
characterize a unique value in the class of games $\\{(N,v(S)e_{S})\ |\
S\subset N,S\neq\emptyset\\}$. Besides, efficiency and symmetry characterize a
unique value for $(N,v(N)e_{N})$. Hence $f(N,v)=\gamma(N,v)$. $\Box$

A very desirable property for values for cooperative games is the invariance
to S-equivalence, which we remember below. Two cooperative games with the same
sets of players $(N,v)$ and $(N,w)$ are said to be $S$-equivalent if there
exist $a\in\mathbb{R}$ with $a>0$ and $b\in\mathbb{R}^{N}$ such that, for
every $T\subseteq N$, it holds that

$w(T)=av(T)+\sum_{j\in T}b_{j}.$

When $(N,v)$ and $(N,w)$ are $S$-equivalent we can transform $v$ into $w$
simply by changing the scale and translating the players’ utilities. In these
conditions it seems reasonable to ask a value for cooperative games $f$ that
$f(N,v)$ is transformed into $f(N,w)$ by doing the corresponding change of
scale and translations.

Invariance to $S$-equivalence (INV). A value for cooperative games $f$
satisfies invariance to S-equivalence if for any pair of S-equivalent
cooperative games $(N,v)$ and $(N,w)$ such that $w(T)=av(T)+\sum_{j\in
T}b_{j}$ for all $T\subseteq N$ (with $a\in\mathbb{R}$, $a>0$ and
$b\in\mathbb{R}^{N}$) it holds that, for every $i\in N$,

$f_{i}(N,w)=af_{i}(N,v)+b_{i}.$

Unfortunately, the value $\gamma$ defined by (5) is not invariant to
$S$-equivalence. Then, we make an adjustment of $\gamma$ that leads us to the
$\Gamma$ value for cooperative games that we define below.

###### Definition 6

The $\Gamma$ value for cooperative games is given for every $(N,v)\in{\cal
G}_{N}$ and every $i\in N$ by:

$\Gamma_{i}(N,v)=v(\\{i\\})+\gamma_{i}(N,v^{0}),$ (6)

where $v^{0}(S)=v(S)-\sum_{j\in S}v(\\{j\\})$ for all $S\subseteq N$.

It is easy to check that $\Gamma$ satisfies the invariance to $S$-equivalence.
In order to characterize it, we introduce below a new property concerning the
necessary players.

Necessary Players Get the $0$-Normalized Per Capita Mean. A value for
cooperative games $f$ satisfies the necessary players get the $0$-normalized
per capita mean property if, for all cooperative game $\left(N,v\right)$ with
$v(N)=\sum_{j\in N}v(\\{j\\})$ and for all $i\in N$ necessary player in
$(N,v)$, it holds that

$f_{i}\left(N,v\right)=v(\\{i\\})+\frac{1}{2^{n-1}}\sum_{S\subseteq N,i\in
S}\frac{v^{0}(S)}{s}.$

###### Theorem 7

$\Gamma$ is the unique value for cooperative games that satisfies the
properties of additivity, necessary players get the $0$-normalized per capita
mean, efficiency and symmetry.

Proof. (Existence). Since $\gamma$ satisfies additivity, efficiency and
symmetry, it is clear that $\Gamma$ also satisfies those properties. To check
that it fulfils the necessary players get the $0$-normalized per capita mean
property take a cooperative game $\left(N,v\right)$ with $v(N)=\sum_{j\in
N}v(\\{j\\})$ and such that $i\in N$ is a necessary player in
$\left(N,v\right)$. Then expression (6) reduces to

$\Gamma_{i}(N,v)=v(\\{i\\})+\frac{1}{2^{n-1}}\sum_{S\subset N,i\in
S}\frac{v^{0}(S)}{s}=v(\\{i\\})+\frac{1}{2^{n-1}}\sum_{S\subseteq N,i\in
S}\frac{v^{0}(S)}{s}.\\\ $

(Uniqueness). Take $f$ a value for cooperative games that satisfies
efficiency, symmetry, necessary players get the $0$-normalized per capita mean
and additivity and take a cooperative game $(N,v)$. We prove now that
$f(N,v)=\Gamma(N,v)$. Indeed, consider the basis of the vector space of
characteristic functions of cooperative games with set of players $N$ given
by:

$\\{e_{\\{i\\}}+e_{N}\ |\ i\in N\\}\cup\\{e_{S}\ |\ S\in 2^{N},|S|\geq 2\\}.$

Observe that $v$ can be written in a unique way as a linear combination of the
elements of this basis. Since $f$ satisfies additivity and, moreover, the
properties of efficiency, symmetry and necessary players get the
$0$-normalized per capita mean characterize a unique value in the games of the
basis, the proof is concluded. $\Box$

Now we analyse an example in order to make some comments on the $\Gamma$
value. It is based on a similar example in Alonso-Meijide et al. (2020).

###### Example 8

Consider a three-storey building with one apartment on each floor, the three
apartments having the same surface. The three corresponding owners have agreed
to install an elevator and share the corresponding cost. Such a cost is 120
(in thousands of euros), 50 of which correspond to the machine, 40 to the
works to make the hollow of the elevator (a fixed cost of 10 plus a cost of 10
for the owner of the apartment in the first floor that is incremented by 10
for the owner of the apartment in the second floor and by an additional 10 for
the owner of the apartment in the third floor), and 30 to the works to be done
on each floor to allow access to the elevator (10 in each of them). According
to this, the cost $c(i)$ in which each player is involved is:

  * •

50 (machine) + 10 (floor) + 20 (hollow) = 80 for $i=1$, the player of the
first floor,

  * •

50 (machine) + 10 (floor) + 30 (hollow) = 90 for $i=2$, the player of the
second floor,

  * •

50 (machine) + 10 (floor) + 40 (hollow) = 100 for $i=3$, the player of the
third floor.

The rest of the corresponding cost game is given by: $c(\\{1,2\\})=100$,
$c(\\{1,3\\})=c(\\{2,3\\})=110$, $c(N)=120$. Table 1 below shows the
distribution of costs for each of the apartments according to the Egalitarian
value, the Shapley value and $\Gamma$. In European city centres it is common
to find buildings coping with situations like the one described in this
example. It is not uncommon for the owners of the lower floors to be less
favourable to installing an elevator because of the costs involved. According
to Spanish legislation, when owners decide to make an investment in the common
elements of a building, the corresponding costs will be distributed in
proportion to the owners’ shares (which, in turn, sometimes depend only on the
surface areas of the apartments). Therefore, the distribution due to the
Egalitarian value will be the one proposed by the legislation in some
occasions. Note that the proposed Shapley value and $\Gamma$ distributions
tend to favour the owners of the lower floors. In short, $\Gamma$ seems to be
the least controversial distribution in view of the usual dynamics of
homeowners’ communities, because it tends to favour the owners of the lowest
floor, who are usually the most reluctant to bear the costs of installing an
elevator.

| Egalitarian | Shapley | $\Gamma$
---|---|---|---
$1$ | 40 | 33.3333 | 32.5
$2$ | 40 | 38.3333 | 38.75
$3$ | 40 | 48.3333 | 48.75
Table 1: The Egalitarian value, the Shapley value and $\Gamma$ for $(N,c)$

It is not uncommon that in real situations such as those described in this
example not all the owners are in favour of the elevator. When this occurs,
sometimes the elevator will not be installed immediately even if the owners in
favour of it have a majority. The reason for this is that the unfavourable
owners (generally those on the lower floors) may refuse to pay the financial
amounts due to them and the owners’ community can only force them to do so by
initiating legal proceedings which may be long, economically costly and which,
moreover, may profoundly damage coexistence in the building. The practical
consequence of this is that negotiations often take place within the owners’
community to try to ensure that the installation of the elevator is possible
without damaging coexistence in the building. One possible solution is that
the owners not in favour of the elevator give up its service; this means that
the elevator will not have stops on the corresponding floors, so that the
works to give access to the elevator on those floors will not be necessary and
the total cost of the installation will be lower. Assume, for instance that in
the three-storey building in this example the owner of the apartment in the
first floor is not in favour to install the elevator and, moreover, declares
that he will not pay any costs unless a court decision obliges him to do so.
Negotiation in the community may propose that the elevator does not serve the
first floor. In that case, the cost $d(i)$ in which each player is involved
is:

  * •

0 for $i=1$, the player of the first floor,

  * •

50 (machine) + 10 (floor) + 30 (hollow) = 90 for $i=2$, the player of the
second floor,

  * •

50 (machine) + 10 (floor) + 40 (hollow) = 100 for $i=3$, the player of the
third floor.

The rest of the corresponding cost game is given by: $d(\\{1,2\\})=90$,
$d(\\{1,3\\})=100$, $d(\\{2,3\\})=110$, $d(N)=110$. Table 2 below shows the
distribution of costs for each of the apartments according to the Egalitarian
value, the Shapley value and $\Gamma$. Note that the distribution given by the
Egalitarian rule does not seem to facilitate the agreement on the installation
of the elevator because the owner of the first floor will continue to pay a
considerable amount and, in addition, will give up the service of the
elevator. The distributions given by the Shapley value and by $\Gamma$,
however, do seem to facilitate a final settlement. According to the Shapley
value, the owner of the first floor will waive elevator service but pay
nothing in return. According to $\Gamma$, the owner of the first floor will
even receive a small compensation for the inconvenience caused to him by the
works and the installation.

| Egalitarian | Shapley | $\Gamma$
---|---|---|---
$1$ | 36.6666 | 0 | -6.6666
$2$ | 36.6666 | 50 | 53.3333
$3$ | 36.6666 | 60 | 63.3333
Table 2: The Egalitarian value, the Shapley value and $\Gamma$ for $(N,d)$

##  3 Coalitional values and necessary players

In this section we extend the $\Gamma$ value to cooperative games with a
coalition structure. We start by remembering the mean features concerning that
model.

We denote by $P(N)$ the set of all partitions of a finite set $N$. Each $P\in
P(N)$, of the form $P=\\{P_{1},\dots,P_{m}\\}$, is called a coalition
structure on $N$. We call unions of $P$ to its elements $P_{1},\dots,P_{m}$.
We denote by $M$ the set $\\{1,...,m\\}$.

A cooperative game with a coalition structure is a triple $(N,v,P)$ where
$(N,v)\in\mathcal{G}_{N}$ and $P\in P(N)$. $\mathcal{G}_{N}^{cs}$ denotes the
family of all cooperative games with a coalition structure and with player set
$N$. Note that the first two elements of a cooperative game with a coalition
structure, $(N,v)$, characterize a cooperative game.

By a coalitional value we mean a map $g$ that assigns to every game with a
coalition structure $(N,v,P)$ a vector $g(N,v,P)\in\mathbb{R}^{N}$ with
components $g_{i}(N,v,P)$, $i\in N$. Two of the most important coalitional
values are the Owen value (Owen, 1977) and the Banzhaf-Owen value (Owen,
1982). In a similar way to the Shapley and Banzhaf values, the value of a
particular player is a weighted sum of his contributions. In the case of the
Shapley and Banzhaf values all possible contributions are taken into account,
but for the coalitional values only the contributions to some coalitions are
used to compute the values.

The Owen value $\Phi$ is the coalitional value defined by:

$\Phi_{i}(N,v,P)=\frac{1}{m}\frac{1}{p_{k}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq
P_{k}\backslash\\{i\\}}\frac{1}{{\binom{m-1}{r}}}\frac{1}{{\binom{p_{k}-1}{t}}}\Big{[}v(\bigcup\limits_{r\in
R}P_{r}\cup T\cup\\{i\\})-v(\bigcup\limits_{r\in R}P_{r}\cup T)\Big{]}$

for all $(N,v,P)\in\mathcal{G}_{N}^{cs}$ and all $i\in N$, where $P_{k}\in P$
is the union such that $i\in P_{k}$; $m$, $p_{k}$, $r$ and $t$ are the
cardinalities of $M$, $P_{k}$, $R$ and $T$, respectively.

The Banzhaf–Owen value $\Psi$ is the coalitional value defined as

$\Psi_{i}(N,v,P)=\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq
P_{k}\backslash\\{i\\}}\Big{[}v(\bigcup\limits_{r\in R}P_{r}\cup
T\cup\\{i\\})-v(\bigcup\limits_{r\in R}P_{r}\cup T)\Big{]}$

for all $(N,v,P)\in\mathcal{G}_{N}^{cs}$ and all $i\in N$, where $P_{k}\in P$
is the union such that $i\in P_{k}$; $m$, $p_{k}$, $r$ and $t$ are the
cardinalities of $M$, $P_{k}$, $R$ and $T$, respectively.

In the literature, we can find several characterizations of the Owen and the
Banzhaf-Owen coalitional values; see for example Vázquez et al. (1997), Amer
et al. (2002), Khmelnitskaya and Yanovskaya (2007), Alonso-Meijide et al.
(2007), Casajus (2010) and Lorenzo-Freire (2016). We contribute to this
research line providing a new characterization of these two coalitional values
using necessary players. Only three properties are used in our results and the
difference between them is the assigned payoff to necessary players.

Necessary Players Get the Weighted Coalitional Mean. A coalitional value $g$
satisfies the property of necessary players get the weighted coalitional mean
if for any coalitional game $\left(N,v,P\right)$ and for any necessary player
$i\in P_{k}$ in $(N,v)$, it holds that

$g_{i}\left(N,v,P\right)=\frac{1}{m}\frac{1}{p_{k}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq
P_{k}}\frac{1}{{\binom{m-1}{r}}}\frac{1}{{\binom{p_{k}-1}{t-1}}}v(\bigcup\limits_{r\in
R}P_{r}\cup T).$

Necessary Players Get the Coalitional Mean. A coalitional value $g$ satisfies
the property of necessary players get the coalitional mean if for any
coalitional game $\left(N,v,P\right)$ and for any necessary player $i\in
P_{k}$ in $(N,v)$, it holds that

$g_{i}\left(N,v,P\right)=\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq P_{k}}v(\bigcup\limits_{r\in R}P_{r}\cup
T).$

Both properties propose that a necessary player must receive the average worth
over all coalitions that are compatible with the partitions (i.e., those that
are formed by some complete unions and a subset of another union), but the
first one takes into account the size of the coalitions while the second one
assigns the same weight to all compatible coalitions. With these new
properties we can prove the following results.

###### Theorem 9

The Banzhaf-Owen value is the unique coalitional value that satisfies the
properties of additivity, null player and necessary players get the
coalitional mean.

Proof. (Existence). It is known that the Banzhaf-Owen value satisfies
additivity and null player. Now let us see that it satisfies the property of
necessary players get the coalitional mean. Take a cooperative game with a
coalition structure $(N,v,P)$ and take $i\in P_{k}$ a necessary player in
$(N,v)$. Then the Banzhaf-Owen value is reduced to

$\displaystyle\Psi_{i}(N,v,P)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq
P_{k}\backslash\\{i\\}}\Big{[}v(\bigcup\limits_{r\in R}P_{r}\cup
T\cup\\{i\\})-v(\bigcup\limits_{r\in R}P_{r}\cup T)\Big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq
P_{k}\backslash\\{i\\}}v(\bigcup\limits_{r\in R}P_{r}\cup T\cup\\{i\\})$
$\displaystyle=$
$\displaystyle\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq P_{k}}v(\bigcup\limits_{r\in R}P_{r}\cup
T).$

(Uniqueness). For every $S\subseteq N$, $S\neq\emptyset$, the unanimity game
$(N,u_{S})$ is given, for every $T\subseteq N$, by:

$u_{S}(T)=\left\\{\begin{array}[]{cc}1&\mbox{if $S\subseteq T$,}\\\
0&\mbox{otherwise.}\\\ \end{array}\right.$ (7)

Take a coalitional value $g$ that satisfies additivity, null player and
necessary players get the coalitional mean and take a cooperative game with a
coalition structure $(N,v,P)$. We prove now that $g(N,v,P)=\Psi(N,v,P)$. Given
$S\subseteq N$, in the unanimity game $(N,u_{S})$ every $i\in S$ is a
necessary player and every $i\in N\backslash S$ is a null player. Let us fix
$P$, a finite set $S\subseteq N$ and $c\in\mathbb{R}$. By additivity it is
sufficient to prove that for all $i\in N$,
$g_{i}(N,cu_{S},P)=\Psi_{i}(N,cu_{S},P)$. If $i\in N\backslash S$, applying
the null player property $g_{i}(N,cu_{S},P)=\Psi_{i}(N,cu_{S},P)=0$. If $i\in
S$, applying neccesary players get the coalitional mean, we have that

$g_{i}(N,cu_{S},P)=\Psi_{i}(N,cu_{S},P)=c\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq
M\backslash k}\sum_{T\subseteq P_{k}}u_{S}(\bigcup\limits_{r\in R}P_{r}\cup
T),$

where $P_{k}$ is the union such that $i\in P_{k}$. $\Box$

###### Theorem 10

The Owen value is the unique coalitional value that satisfies the properties
of additivity, null player and necessary players get the weighted coalitional
mean.

Proof. (Existence). It is known that the Owen value satisfies additivity and
null player. Let us see that it satisfies the property of necessary players
get the weighted coalitional mean. Suppose that $i$ is a necessary player with
$i\in P_{k}$; then the Owen value is

$\displaystyle\Phi_{i}(N,v,P)$ $\displaystyle=$
$\displaystyle\frac{1}{m}\frac{1}{p_{k}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq
P_{k}\backslash\\{i\\}}\frac{1}{{\binom{m-1}{r}}}\frac{1}{{\binom{p_{k}-1}{t}}}\Big{[}v(\bigcup\limits_{r\in
R}P_{r}\cup T\cup\\{i\\})-v(\bigcup\limits_{r\in R}P_{r}\cup T)\Big{]}$
$\displaystyle=$ $\displaystyle\frac{1}{m}\frac{1}{p_{k}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq
P_{k}\backslash\\{i\\}}\frac{1}{{\binom{m-1}{r}}}\frac{1}{{\binom{p_{k}-1}{t}}}v(\bigcup\limits_{r\in
R}P_{r}\cup T\cup\\{i\\})$ $\displaystyle=$
$\displaystyle\frac{1}{m}\frac{1}{p_{k}}\sum_{R\subseteq
M\backslash\\{k\\}}\sum_{T\subseteq
P_{k}}\frac{1}{{\binom{m-1}{r}}}\frac{1}{{\binom{p_{k}-1}{t-1}}}v(\bigcup\limits_{r\in
R}P_{r}\cup T).$

(Uniqueness) Take a coalitional value $g$ that satisfies additivity, null
player and necessary players get the weighted coalitional mean and take a
cooperative game with a coalition structure $(N,v,P)$. We prove now that
$g(N,v,P)=\Phi(N,v,P)$. Given $S\subseteq N$, in the unanimity game
$(N,u_{S})$ every $i\in S$ is a necessary player and every $i\in N\backslash
S$ is a null player. Let us fix $P$, a finite set $S\subseteq N$ and
$c\in\mathbb{R}$. By additivity it is sufficient to prove that for all $i\in
N$, $g_{i}(N,cu_{S},P)=\Phi_{i}(N,cu_{S},P)$. If $i\in N\backslash S$,
applying the null player property

$g_{i}(N,cu_{S},P)=\Phi_{i}(N,cu_{S},P)=0.$

If $i\in S$, applying necessary players get the weighted coalitional mean, we
have that

$g_{i}(N,cu_{S},P)=\Phi_{i}(N,cu_{S},P)=c\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq
M\backslash k}\sum_{T\subseteq P_{k}}u_{S}(\bigcup\limits_{r\in R}P_{r}\cup
T),$

where $P_{k}$ is the union such that $i\in P_{k}$. $\Box$

We are now willing to extend the $\Gamma$ value, defined in Section 2, to
cooperative games with a coalition structure. We next remind some properties
that are relevant for our aim.

Symmetry Inside Unions. A coalitional value $g$ satisfies the property of
symmetry inside unions if for all cooperative game with a coalition structure
$\left(N,v,P\right)$, it holds that

$g_{i}\left(N,v,P\right)=g_{j}\left(N,v,P\right).$

for all $i,j$ symmetric players in $(N,v)$ with $i,j\in P_{k}$, $P_{k}\in P$.

We say that unions $P_{k},P_{l}\in P$ are symmetric in
$\left(N,v,P\right)\in\mathcal{G}_{N}^{cs}$ if $v(S\cup P_{k})=v(S\cup
P_{l})$, for every $S=\cup_{j\in R}P_{j}$ with $R\subseteq
M\backslash\\{k,l\\}$.

Symmetry Among Unions. A coalitional value $g$ satisfies the property of
symmetry among unions if for all cooperative game with a coalition structure
$\left(N,v,P\right)$, it holds that

$\sum_{i\in P_{k}}g_{i}\left(N,v,P\right)=\sum_{j\in
P_{r}}g_{j}\left(N,v,P\right)$

for all $P_{k},P_{r}\in P$, symmetric unions in $(N,v,P)$.

Given the properties of efficiency, additivity, symmetry inside unions and
symmetry among unions one can expect to extend the $\Gamma$ value to
cooperative games with a coalition structure and to characterize the new value
using a property for necessary players that somewhat adapts the necessary
players get the $0$-normalized per capita mean property. First at all, let us
see how to extend the $\gamma$ value, since the $\Gamma$ value depends on it.

###### Definition 11

The $\gamma^{C}$ value for cooperative games with a coalition structure is
given for every $(N,v,P)\in\mathcal{G}_{N}^{cs}$ and every $i\in P_{k}$ by:

$\begin{split}\gamma^{C}_{i}\left(N,v,P\right)&=\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\left(\sum_{R\subseteq
M\backslash k}\sum_{T\subset P_{k},i\in T}\frac{v(\bigcup\limits_{r\in
R}P_{r}\cup T)}{t}\right.\\\ &\left.-\sum_{R\subseteq M\backslash
k}\sum_{T\subset P_{k},i\notin T,T\neq\emptyset}\frac{v(\bigcup\limits_{r\in
R}P_{r}\cup T)}{p_{k}-t}\right)\\\
&+\frac{1}{2^{m-1}}\frac{1}{p_{k}}\left(\sum_{R\subset M,k\in
R}\frac{v(\bigcup\limits_{r\in R}P_{r})}{r}-\sum_{R\subseteq M\backslash
k}\frac{v(\bigcup\limits_{r\in
R}P_{r})}{m-r}\right)+\frac{v(N)}{mp_{k}}.\end{split}$ (8)

Let us see that $\gamma^{C}$ is an reasonable extension of $\gamma$. To check
it, we can see that $\gamma^{C}$ is a coalitional value of $\gamma$, that is
$\gamma^{C}(N,v,P^{n})=\gamma(N,v)$ for all
$(N,v,P^{n})\in\mathcal{G}_{N}^{cs}$ where $P^{n}=\\{\\{1\\},...,\\{n\\}\\}$.
In fact

$\begin{split}\gamma^{C}_{i}\left(N,v,P^{n}\right)&=\frac{1}{2^{m-1}}\left(\sum_{R\subset
M,k\in R}\frac{v(\bigcup\limits_{r\in R}P_{r})}{r}-\sum_{R\subseteq
M\backslash k}\frac{v(\bigcup\limits_{r\in
R}P_{r})}{m-r}\right)+\frac{v(N)}{mp_{k}}\\\
&=\frac{1}{2^{n-1}}\left(\sum_{S\subset N,i\in
S}\frac{v(S)}{s}-\sum_{S\subseteq
N\backslash\\{i\\}}\frac{v(S)}{n-s}\right)+\frac{v(N)}{n}=\gamma_{i}\left(N,v\right).\end{split}$

The next lemma proves that $\gamma^{C}$ satisfies an interesting property for
cooperative games with a coalition structure.

###### Lemma 12

The $\gamma^{C}$ value satisfies the quotient game property, i.e., that

$\sum_{i\in
P_{k}}\gamma^{C}_{i}\left(N,v,P\right)=\gamma^{C}_{k}\left(M,v^{P},P^{m}\right)$

for all $P_{k}\in P$, where $v^{P}(R)=v(\cup_{r\in R}P_{r})$ for all
$R\subseteq M$, and
$P^{m}=\left\\{\left\\{1\right\\},\ldots,\left\\{m\right\\}\right\\}$.

Proof. Take a cooperative game with a coalition structure $(N,v,P)$ and
$P_{k}\in P$. Then

$\begin{split}&\sum_{i\in P_{k}}\gamma^{C}_{i}\left(N,v,P\right)\\\
&=\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{i\in
P_{k}}\left(\sum_{R\subseteq M\backslash k}\sum_{T\subset P_{k},i\in
T}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup T)}{t}-\sum_{R\subseteq M\backslash
k}\sum_{T\subset P_{k},i\notin T,T\neq\emptyset}\frac{v(\bigcup\limits_{r\in
R}P_{r}\cup T)}{p_{k}-t}\right)\\\
&+\frac{1}{2^{m-1}}\frac{1}{p_{k}}\sum_{i\in P_{k}}\left(\sum_{R\subset M,k\in
R}\frac{v(\bigcup\limits_{r\in R}P_{r})}{r}-\sum_{R\subseteq M\backslash
k}\frac{v(\bigcup\limits_{r\in R}P_{r})}{m-r}\right)+\sum_{i\in
P_{k}}\frac{v(N)}{mp_{k}}\\\
&=\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\left(\sum_{R\subseteq M\backslash
k}\sum_{T\subset P_{k}}\frac{tv(\bigcup\limits_{r\in R}P_{r}\cup
T)}{t}-\sum_{R\subseteq M\backslash k}\sum_{T\subset
P_{k}}\frac{(p_{k}-t)v(\bigcup\limits_{r\in R}P_{r}\cup T)}{p_{k}-t}\right)\\\
&+\frac{1}{2^{m-1}}\left(\sum_{R\subset M,k\in R}\frac{v(\bigcup\limits_{r\in
R}P_{r})}{r}-\sum_{R\subseteq M\backslash k}\frac{v(\bigcup\limits_{r\in
R}P_{r})}{m-r}\right)+\frac{v(N)}{m}\\\
&=\frac{1}{2^{m-1}}\left(\sum_{R\subset M,k\in R}\frac{v(\bigcup\limits_{r\in
R}P_{r})}{r}-\sum_{R\subseteq M\backslash k}\frac{v(\bigcup\limits_{r\in
R}P_{r})}{m-r}\right)+\frac{v(N)}{m}=\gamma^{C}_{k}\left(M,v^{P},P^{m}\right).\end{split}$

$\Box$

The quotient game is an interesting property because it guarantees that the
total worth obtained by the players of a union coincides with the worth
obtained by the union in the game played by the unions with the trivial
coalition structure. Note that the Banzhaf-Owen value does not satisfy this
property; however, Alonso-Meijide and Fiestras-Janeiro (2002) introduces the
so-called symmetric coalitional Banzhaf value, which is an extension of the
Banzhaf value to cooperative games with a coalition structure that satisfies
the quotient game property.

In order to characterize $\gamma^{C}$, we introduce a new property for
necessary players.

Necessary Players Get the Per Capita Coalitional Mean. A coalitional value $g$
satisfies the property of necessary players get the per capita coalitional
mean if for any coalitional game $\left(N,v,P\right)$ with $v(N)=0$ and for
any necessary player $i\in P_{k}$ in $(N,v)$, it holds that

$g_{i}\left(N,v,P\right)=\frac{1}{2^{m-1}}\left[\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq
M\backslash k}\sum_{T\subset P_{k},i\in T}\frac{v(\bigcup\limits_{r\in
R}P_{r}\cup T)}{t}+\frac{1}{p_{k}}\sum_{R\subseteq M,k\in
R}\frac{v(\bigcup\limits_{r\in R}P_{r})}{r}\right]$

where $t=|T|$ and $r=|R|$ for all $T\subset P_{k}$ and $R\subseteq M$.

###### Theorem 13

The $\gamma^{C}$ value is the unique value for cooperative games with a
coalition structure that satisfies the properties of additivity, necessary
players get the per capita coalitional mean, efficiency, symmetry inside
unions and symmetry among unions.

Proof. (Existence). It is clear that $\gamma^{C}$ satisfies additivity. To
check that it satisfies the necessary players get the per capita coalitional
mean property take a cooperative game with a coalition structure
$\left(N,v,P\right)$ with $v(N)=0$ and such that $i\in P_{k}\subseteq N$ is a
necessary player in $\left(N,v\right)$. Then expression (8) reduces to

$\displaystyle\begin{split}\displaystyle\gamma^{C}_{i}(N,v,P)&=\displaystyle\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\left(\sum_{R\subseteq
M\backslash k}\sum_{T\subset P_{k},i\in T}\frac{v(\bigcup\limits_{r\in
R}P_{r}\cup T)}{t}\right)+\frac{1}{2^{m-1}}\frac{1}{p_{k}}\left(\sum_{R\subset
M,k\in R}\frac{v(\bigcup\limits_{r\in R}P_{r})}{r}\right)\\\
&\displaystyle=\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq
M\backslash k}\sum_{T\subset P_{k}}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup
T)}{t}+\frac{1}{2^{m-1}}\frac{1}{p_{k}}\sum_{R\subseteq
M}\frac{v(\bigcup\limits_{r\in R}P_{r})}{r}.\end{split}$

To check that $\gamma^{C}$ satisfies symmetry inside coalitions take a
cooperative game with a coalition structure $(N,v,P)$ and a pair of symmetric
players in $(N,v)$ $i,j\in P_{k}$ with $P_{k}\in P$. Notice that, for a fixed
$R\subseteq M\backslash k$,

$\displaystyle\begin{split}&\sum_{T\subset P_{k},i\in
T}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup T)}{t}-\sum_{T\subset P_{k},i\notin
T,T\neq\emptyset}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup T)}{p_{k}-t}\\\
&=\sum_{T\subseteq P_{k}\backslash\\{i,j\\}}\frac{v(\bigcup\limits_{r\in
R}P_{r}\cup T\cup\\{i\\})}{t+1}+\sum_{T\subset
P_{k}\backslash\\{i,j\\}}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup
T\cup\\{i,j\\})}{t+2}\\\ &-\sum_{T\subseteq
P_{k}\backslash\\{i,j\\},T\neq\emptyset}\frac{v(\bigcup\limits_{r\in
R}P_{r}\cup T)}{p_{k}-t}-\sum_{T\subseteq
P_{k}\backslash\\{i,j\\}}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup
T\cup\\{j\\})}{p_{k}-t-1}.\end{split}$

Now, since $i,j$ are symmetric in $(N,v)$, the last expression is equal to

$\displaystyle\begin{split}&\sum_{T\subseteq
P_{k}\backslash\\{i,j\\}}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup
T\cup\\{j\\})}{t+1}+\sum_{T\subset
P_{k}\backslash\\{i,j\\}}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup
T\cup\\{i,j\\})}{t+2}\\\ &-\sum_{T\subseteq
P_{k}\backslash\\{i,j\\},T\neq\emptyset}\frac{v(\bigcup\limits_{r\in
R}P_{r}\cup T)}{p_{k}-t}-\sum_{T\subseteq
P_{k}\backslash\\{i,j\\}}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup
T\cup\\{i\\})}{p_{k}-t-1}\end{split}$

and then it is clear that $\gamma^{C}_{i}(N,v,P)=\gamma^{C}_{j}(N,v,P)$.

Since $\gamma^{C}$ satisfies the quotient game property and it is a
coalitional value of $\gamma$, then

$\sum_{i\in
P_{k}}\gamma^{C}_{i}\left(N,v,P\right)=\gamma^{C}_{k}\left(M,v^{P},P^{m}\right)=\gamma_{k}(M,v^{P}).$

Now, the efficiency and the symmetry properties of $\gamma$ imply that
$\gamma^{C}$ satisfies symmetry among unions and efficiency.

(Uniqueness). Take $g$, a value for cooperative games with a coalition
structure that satisfies efficiency, symmetry inside unions, symmetry among
unions, necessary players get the per capita coalitional mean and additivity,
and take a cooperative game with a coalition structure $(N,v,P)$. We prove now
that $g(N,v,P)=\gamma^{C}(N,v,P)$. Indeed, consider the basis of the vector
space of characteristic functions of cooperative games with set of players $N$
given by: $\\{e_{S}\\}_{S\in 2^{N}\setminus\emptyset}$ (see expression (1)).
Observe that $v$ can be written in a unique way as a linear combination of the
elements of the basis: $v=\sum_{S\in 2^{N}\setminus\emptyset}v(S)e_{S}$. Since
$g$ satisfies additivity,

$g(N,v,P)=\sum_{S\in 2^{N}\setminus\emptyset}g(N,v(S)e_{S},P).$

Notice that efficiency, symmetry inside unions, symmetry among unions, and
necessary players get the per capita coalitional mean characterize a unique
value in the class of games $\\{(N,v(S)e_{S},P)\ |\ S\subset
N,S\neq\emptyset\\}$. Besides, efficiency, symmetry inside unions and symmetry
among unions, characterize a unique value for $(N,v(N)e_{N},P)$. Hence
$g(N,v,P)=\gamma^{C}(N,v,P)$. $\Box$

Now, in an analogous way as we obtain $\Gamma$ from $\gamma$, we introduce the
following value.

###### Definition 14

The $\Gamma^{C}$ value for cooperative games with a coalition structure is
given for every $(N,v,P)\in{\cal G}_{N}^{cs}$ and every $i\in P_{k}$ by:

$\Gamma^{C}_{i}(N,v,P)=v(\\{i\\})+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(\\{j\\})}{p_{k}}+\gamma^{C}_{i}(N,v^{0^{\prime}},P),$ (9)

where $v^{0^{\prime}}(S)=v(S)-\sum_{r\in R}v(P_{r})-\sum_{j\in
S\backslash(\cup_{r\in R}P_{r})}v(\\{j\\})$ and $R=\\{r\in M\ |\
P_{r}\subseteq S\\}$ for all $S\subseteq N$.

As for $\gamma^{C}$, we check that $\Gamma^{C}$ is a coalitional value of
$\Gamma$. Take the cooperative game with a coalition structure $(N,v,P^{n})$.
Then

$\begin{split}\Gamma^{C}_{i}\left(N,v,P^{n}\right)&=v(\\{i\\})+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(\\{j\\})}{p_{k}}\\\ &+\frac{1}{2^{m-1}}\left(\sum_{R\subset M,k\in
R}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r})}{r}-\sum_{R\subseteq
M\backslash k}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in
R}P_{r})}{m-r}\right)+\frac{v^{0^{\prime}}(N)}{mp_{k}}\\\
&=v(\\{i\\})+\frac{1}{2^{n-1}}\left(\sum_{S\subset N,i\in
S}\frac{v^{0}(S)}{s}-\sum_{S\subseteq
N\backslash\\{i\\}}\frac{v^{0}(S)}{n-s}\right)+\frac{v^{0}(N)}{n}\\\
&=\Gamma_{i}\left(N,v\right).\end{split}$

Now we provide an axiomatic characterization of $\Gamma^{C}$. We start with a
lemma concerning the quotient game property.

###### Lemma 15

The $\Gamma^{C}$ value satisfies the quotient game property, i.e., that

$\sum_{i\in
P_{k}}\Gamma^{C}_{i}\left(N,v,P\right)=\Gamma^{C}_{k}\left(M,v^{P},P^{m}\right)$

for all $P_{k}\in P$, where $v^{P}(R)=v(\cup_{r\in R}P_{r})$ for all
$R\subseteq M$, and
$P^{m}=\left\\{\left\\{1\right\\},\ldots,\left\\{m\right\\}\right\\}$.

Proof. Take a cooperative game with a coalition structure $(N,v,P)$ and $i\in
N$ such that $P_{k}\in P$. Then

$\begin{split}&\sum_{i\in P_{k}}\Gamma^{C}_{i}\left(N,v,P\right)\\\
&=v(P_{k})+\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{i\in
P_{k}}\left(\sum_{R\subseteq M\backslash k}\sum_{T\subset P_{k},i\in
T}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r}\cup T)}{t}\right.\\\
&\left.-\sum_{R\subseteq M\backslash k}\sum_{T\subset P_{k},i\notin
T,T\neq\emptyset}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r}\cup
T)}{p_{k}-t}\right)\\\ &+\frac{1}{2^{m-1}}\frac{1}{p_{k}}\sum_{i\in
P_{k}}\left(\sum_{R\subset M,k\in R}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in
R}P_{r})}{r}-\sum_{R\subseteq M\backslash
k}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r})}{m-r}\right)+\sum_{i\in
P_{k}}\frac{v^{0^{\prime}}(N)}{mp_{k}}\\\ &=v(P_{k})\\\
&+\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\left(\sum_{R\subseteq M\backslash
k}\sum_{T\subset P_{k}}\frac{tv^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r}\cup
T)}{t}-\sum_{R\subseteq M\backslash k}\sum_{T\subset
P_{k}}\frac{(p_{k}-t)v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r}\cup
T)}{p_{k}-t}\right)\\\ &+\frac{1}{2^{m-1}}\left(\sum_{R\subset M,k\in
R}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r})}{r}-\sum_{R\subseteq
M\backslash k}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in
R}P_{r})}{m-r}\right)+\frac{v^{0^{\prime}}(N)}{m}\\\
&=v(P_{k})+\frac{1}{2^{m-1}}\left(\sum_{R\subset M,k\in
R}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r})}{r}-\sum_{R\subseteq
M\backslash k}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in
R}P_{r})}{m-r}\right)+\frac{v^{0^{\prime}}(N)}{m}\\\
&=\Gamma^{C}_{k}\left(M,v^{P},P^{m}\right).\end{split}$

$\Box$

In order to characterize $\Gamma^{C}$, we introduce a new property for
necessary players.

Necessary Players Get the $0$-Normalized Per Capita Coalitional Mean. A
coalitional value $g$ satisfies the property of necessary players get the
$0$-normalized per capita coalitional mean if for any coalitional game
$\left(N,v,P\right)$ with $v(N)=\sum_{r\in M}v(P_{r})$ and for any necessary
player $i\in P_{k}$ in $(N,v)$, it holds that

$\begin{split}g_{i}\left(N,v,P\right)&=v(\\{i\\})+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(\\{j\\})}{p_{k}}\\\
&+\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\sum_{R\subseteq M\backslash
k}\sum_{T\subset P_{k}}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r}\cup
T)}{t}+\frac{1}{2^{m-1}}\frac{1}{p_{k}}\sum_{R\subseteq
M}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r})}{r},\end{split}$

where $t=|T|$ and $r=|R|$ for all $T\subset P_{k}$ and $R\subseteq M$.

###### Theorem 16

$\Gamma^{C}$ is the unique coalitional value for cooperative games with a
coalition structure that satisfies the properties of additivity, necessary
players get the $0$-normalized per capita coalitional mean, efficiency,
symmetry inside unions and symmetry among unions.

Proof. (Existence). Since $\gamma^{C}$ satisfies additivity and symmetry
inside unions, it is clear that $\Gamma^{C}$ also satisfies those properties.
To check that it fulfils the necessary players get the $0$-normalized per
capita coalitional mean take a cooperative game with a coalition structure
$\left(N,v,P\right)$ with $v(N)=\sum_{r\in M}v(P_{r})$ and such that $i\in N$,
with $i\in P_{k}\in P$, is a necessary player in $\left(N,v\right)$. Then
expression (9) reduces to

$\Gamma^{C}_{i}(N,v,P)=v(\\{i\\})+\frac{v(P_{k})-\sum_{j\in
P_{k}}v(\\{j\\})}{p_{k}}\\\ $
$+\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\left(\sum_{R\subseteq M\backslash
k}\sum_{T\subset P_{k}}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r}\cup
T)}{t}\right)+\frac{1}{2^{m-1}}\frac{1}{p_{k}}\left(\sum_{R\subset
M}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r})}{r}\right).$

Since this solution satisfies the quotient game property and it is a
coalitional value of $\Gamma$, for a cooperative game with a coalition
structure $(N,v,P)$ and $P_{k}\in P$ it holds that

$\sum_{i\in
P_{k}}\Gamma^{C}_{i}\left(N,v,P\right)=\Gamma^{C}_{k}\left(M,v^{P},P^{m}\right)=\Gamma_{k}(M,v^{P}).$

Then it is easy to check that $\Gamma^{C}$ satisfies symmetry among unions and
efficiency taking into account that $\Gamma$ satisfies efficiency and
symmetry.

(Uniqueness). Take $g$, a value for cooperative games with a coalition
structure that satisfies efficiency, symmetry, necessary players get the
$0$-normalized per capita coalitional mean and additivity and take a
cooperative game with a coalition structure $(N,v,P)$. We prove now that
$g(N,v,P)=\Gamma^{C}(N,v,P)$. Indeed, consider the basis of the vector space
of characteristic functions of cooperative games with set of players $N$ given
by:

$\\{e_{P_{r}}+e_{N}\ |\ r\in M\\}\cup\\{e_{S}\ |\ S\in 2^{N},S\neq P_{r},r\in
M\\}.$

Observe that $v$ can be written in a unique way as a linear combination of the
elements of this basis. Since $g$ satisfies additivity and, moreover, the
properties of efficiency, symmetry and necessary players get the
$0$-normalized per capita coalitional mean characterize a unique value in the
games of the basis, the proof is concluded. $\Box$

##  4 Concluding Remarks

Notice that $G$, $\gamma$ and $\Gamma$, the three values introduced in Section
2, satisfy the properties of additivity, efficiency and symmetry and then they
can be written using the formula provided in Ruiz et al. (1998). Moreover, it
is clear that $G$ and $\gamma$ satisfy the property of coalitional
monotonicity dealt with in Wang et al. (2019) and thus, in view of Theorem 3.2
in Wang et al. (2019), they belong to the family of ideal values. Moreover, it
is not difficult to prove that $\Gamma$ also satisfies the property of
coalitional monotonicity and then it is also an ideal value. We provide next
such a proof; to start with, we remember the property of coalitional
monotonicity.

Coalitional Monotonicity. A value for cooperative games $f$ satisfies the
property of coalitional monotonicity if for any pair of cooperative games
$\left(N,v\right)$ and $\left(N,w\right)$ fulfilling that there exists
$T\subseteq N$ with $v(T)>w(T)$ and $v(S)=w(S)$ for all $S\subseteq N$, $S\neq
T$, it holds that

$f_{i}(N,v)\geq f_{i}(N,w)$

for all $i\in T$.

In view of expressions (4) and (5), it is clear that $G$ and $\gamma$ satisfy
the property of coalitional monotonicity. With respect to $\Gamma$ notice
that, in view of expressions (5) and (6), for every TU-game $(N,u)$ and for
every $i\in N$, $\Gamma_{i}(N,u)$ can be written as:

$\displaystyle\Gamma_{i}(N,u)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{n-1}}\sum_{S\subset N,i\in
S}\frac{1}{s}\left(u(S)-\sum_{j\in S}u(j)\right)$
$\displaystyle-\frac{1}{2^{n-1}}\sum_{S\subset N,i\not\in
S}\frac{1}{n-s}\left(u(S)-\sum_{j\in S}u(j)\right)$
$\displaystyle+\frac{1}{n}\left(u(N)-\sum_{j\in N}u(j)\right)+u(i).$

Take now $\left(N,v\right)$, $\left(N,w\right)$ and $T\subseteq N$ as in the
statement of coalitional monotonicity. Using (4) it is clear that if $T$ has
two or more elements, then $\Gamma_{i}(N,v)\geq\Gamma_{i}(N,w)$ for all $i\in
T$. Assume now that $T=\\{i\\}$ ($i\in N$). According to (6), the coefficients
of $v(i)$ and $w(i)$ in $\Gamma_{i}(N,v)$ and $\Gamma_{i}(N,w)$ are identical
and given by:

$\displaystyle-\frac{1}{2^{n-1}}\sum_{S\subset N,i\in S,S\neq
i}\frac{1}{s}-\frac{1}{n}+1$ $\displaystyle=$
$\displaystyle-\frac{1}{2^{n-1}}\sum_{s=2}^{n-1}\frac{1}{s}\left(\begin{array}[]{c}n-1\\\
s-1\end{array}\right)-\frac{1}{n}+1$ $\displaystyle=$
$\displaystyle-\frac{1}{2^{n-1}}\sum_{s=2}^{n-1}\frac{1}{n}\left(\begin{array}[]{c}n\\\
s\end{array}\right)-\frac{1}{n}+1$ $\displaystyle=$
$\displaystyle-\frac{1}{2^{n-1}}\frac{1}{n}(2^{n}-1-n-1)-\frac{1}{n}+1=\frac{n-3}{n}+\frac{2+n}{2^{n-1}n}.$

It is easy to check that $\frac{n-3}{n}+\frac{2+n}{2^{n-1}n}>0$ for all $n\geq
1$, which implies that $\Gamma_{i}(N,v)>\Gamma_{i}(N,w)$ and completes the
proof.

We finish this paper with a remark on the relation between our new values and
the equal division and the equal surplus division values, that we denote by
$ED$ and $ESD$ (see, for instance Alonso-Meijide et al., 2020). It is clear
that, for every $(N,v)\in{\cal G}_{N}$ and every $i\in N$,

$\gamma_{i}\left(N,v\right)=ED_{i}(N,v)+\frac{1}{2^{n-1}}\left(\sum_{S\subset
N,i\in S}\frac{v(S)}{s}-\sum_{S\subset N,i\not\in S}\frac{v(S)}{n-s}\right),$

$\Gamma_{i}\left(N,v\right)=ESD_{i}(N,v)+\frac{1}{2^{n-1}}\left(\sum_{S\subset
N,i\in S}\frac{v^{0}(S)}{s}-\sum_{S\subset N,i\not\in
S}\frac{v^{0}(S)}{n-s}\right).$

Now, if $ED^{U}$ and $ESD2^{U}$ are the extensions of $ED$ and $ESD$ for
cooperative games with a coalition structure introduced in Alonso-Meijide et
al. (2020), then for every $(N,v,P)\in{\cal G}_{N}^{cs}$ and every $i\in N$ it
holds that

$\begin{split}&\gamma^{C}_{i}\left(N,v,P\right)=ED^{U}_{i}(N,v,P)\\\
&+\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\left(\sum_{R\subseteq M\backslash
k}\sum_{T\subset P_{k},i\in T}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup
T)}{t}-\sum_{R\subseteq M\backslash k}\sum_{T\subset P_{k},i\notin
T,T\neq\emptyset}\frac{v(\bigcup\limits_{r\in R}P_{r}\cup
T)}{p_{k}-t}\right)\\\ &+\frac{1}{2^{m-1}}\frac{1}{p_{k}}\left(\sum_{R\subset
M,k\in R}\frac{v(\bigcup\limits_{r\in R}P_{r})}{r}-\sum_{R\subseteq
M\backslash k}\frac{v(\bigcup\limits_{r\in R}P_{r})}{m-r}\right),\end{split}$

$\begin{split}&\Gamma^{C}_{i}\left(N,v,P\right)=ESD2^{U}_{i}(N,v,P)\\\
&+\frac{1}{2^{m-1}}\frac{1}{2^{p_{k}-1}}\left(\sum_{R\subseteq M\backslash
k}\sum_{T\subset P_{k},i\in T}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in
R}P_{r}\cup T)}{t}-\sum_{R\subseteq M\backslash k}\sum_{T\subset P_{k},i\notin
T,T\neq\emptyset}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r}\cup
T)}{p_{k}-t}\right)\\\ &+\frac{1}{2^{m-1}}\frac{1}{p_{k}}\left(\sum_{R\subset
M,k\in R}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in
R}P_{r})}{r}-\sum_{R\subseteq M\backslash
k}\frac{v^{0^{\prime}}(\bigcup\limits_{r\in R}P_{r})}{m-r}\right).\end{split}$

## Acknowledgements

This work has been supported by the ERDF, the MINECO/AEI grants
MTM2017-87197-C3-1-P, MTM2017-87197-C3-3-P, and by the Xunta de Galicia
(Grupos de Referencia Competitiva ED431C-2016-015 and ED431C-2017-38 and
Centro Singular de Investigación de Galicia ED431G/01).

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