The Owen and the Owen–Banzhaf Values Applied to the Study of the

Madrid Assembly and the Andalusian Parliament in Legislature

2015–2019

Jos

e Miguel Gim

enez and Mar

ıa Albina Puente

Department of Mathematics and Engineering School of Manresa, Technical University of Catalonia, Spain

Keywords:

Cooperative Game, Shapley Value, Banzhaf Value, Coalition Structure, Multilinear Extension.

Abstract:

This work focuses on the Owen value and the Owen–Banzhaf value, two classical concepts of solution deﬁned

on games with structure of coalition blocks. We provide a computation procedure for these solutions based on

a method of double-level work obtained from the multilinear extension of the original game. Moreover, two

applications to several possible political situations in the Madrid Assembly and the Andalusian Parliament

(legislatures 2015–2019) are also given.

1 INTRODUCTION

Introduced in (Aumann and Dr

eze, 1974), the notion

of game with a coalition structure gave a new impul-

sion to the development of value theory. These au-

thors extended the Shapley value to this new frame-

work in such a manner that the game really splits into

subgames played by the unions isolatedly from each

other, and every player receives the payoff allocated

to him by the Shapley value in the subgame he is play-

ing within his union.

A second approach was used in (Owen, 1977),

when introducing the ﬁrst coalitional value, called

now the Owen value. The Owen value is the result

of a two–step procedure: ﬁrst, the unions play a quo-

tient game among themselves, and each one receives

a payoff which, in turn, is shared among its players in

an internal game. Both payoffs, in the quotient game

for unions and within each union for its players, are

given by the Shapley value.

The same procedure is applied in (Owen, 1982)

to the Banzhaf value and it is obtained the modi-

ﬁed Banzhaf value or Owen–Banzhaf value. In this

case the payoffs at both levels (unions in the quotient

game and players within each union) are given by the

Banzhaf value.

In (Alonso and Fiestras, 2002), the authors sug-

gested to modify the two–step allocation scheme and

use the Banzhaf value for sharing in the quotient

game and the Shapley value within unions. This gave

rise to the symmetric coalitional Banzhaf value or

Alonso–Fiestras value. On the other hand, in (Amer

et al., 2002) was considered a sort of “counterpart”

of the Alonso–Fiestras value where the Shapley value

is used in the quotient game and the Banzhaf value

within unions.

The multilinear extension of a cooperative game

was introduced in (Owen, 1972) and then it was

applied to the calculus of the Shapley value. The

computing technique based on the multilinear ex-

tension has been applied to many values: to the

Banzhaf value in (Owen, 1975); to the Owen value

in (Owen and Winter, 1992); to the Owen–Banzhaf

value in (Carreras and Magana, 1994); to the quo-

tient game in (Carreras and Magana, 1997); to the bi-

nomial semivalues and multinomial probabilistic in-

dices in (Puente, 2000); to the coalitional semivalues

in (Amer and Gim

enez, 2003); to the α–decisiveness

and Banzhaf α–indices in (Carreras, 2004); to the

Alonso–Fiestras value in (Alonso et al., 2005); to the

symmetric coalitional binomial semivalues in (Car-

reras and Puente, 2011); to all semivalues in (Carreras

and Gim

enez, 2011); and to coalitional multinomial

probabilistic values in (Carreras and Puente, 2013).

The present paper focuses on giving a new com-

putational procedure for the Owen and the Owen–

Banzhaf value by means of the multilinear extension

of the game.

The organization of the paper is as follows. In

Section 2, some preliminaries are provided. Section

3 is devoted to the computation of the Owen and the

Owen–Banzhaf values in terms of the multilinear ex-

Giménez, J. and Puente, M.

The Owen and the Owen-Banzhaf Values Applied to the Study of the Madrid Assembly and the Andalusian Parliament in Legislature 2015-2019.

DOI: 10.5220/0007297000450052

In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 45-52

ISBN: 978-989-758-352-0

tension. Section 4 contains two applications of these

values to the analysis of the Madrid Assembly and the

Andalusian Parliament (legislatures 2015–2019).

2 PRELIMINARIES

2.1 Cooperative Games and Values

Let N be a ﬁnite set of players and 2

be the set of

its coalitions (subsets of N). A cooperative game on

N is a function v : 2

→ R, that assigns a real number

v(S) to each coalition S ⊆ N, with v(

0) = 0. A game v

is monotonic if v(S) ≤ v(T ) whenever S ⊆ T ⊆ N and

simple if, moreover, v(S) = 0 or 1 for every S ⊆ N.

A player i ∈ N is a dummy in v if v(S ∪ {i}) = v(S) +

v({i}) for all S ⊆ N\{i}, and null in v if, moreover,

v({i}) = 0. Two players i, j ∈ N are symmetric in v

if v(S ∪{i}) = v(S ∪ { j}) for all S ⊆ N\{i, j}. Given

a nonempty coalition T ⊆ N, the restriction to T of a

given game v on N is the game v

on T that we will

call a subgame of v and is deﬁned by v

(S) = v(S)

for all S ⊆ T .

Endowed with the natural operations for real–

valued functions, i.e. v + v

and λv for all λ ∈ R, the

set of all cooperative games on N is a vector space

. For every nonempty coalition T ⊆ N, the una-

nimity game u

is deﬁned by u

(S) = 1 if T ⊆ S and

(S) = 0 otherwise, and it is easily checked that the

set of all unanimity games is a basis for G

, so that

dim(G

) = 2

− 1 if n = |N|. Each game v ∈ G

can then be uniquely written as a linear combina-

tion of unanimity games, and its components are the

Harsanyi dividends (Harsanyi, 1959):

v =

∑

T ⊆N: T 6=

, where

= α

(v) =

∑

S⊆T

(−1)

t−s

v(S)

and, as usual, t = |T | and s = |S|.

By a value on G

we will mean a map f : G

→

, that assigns to every game v a vector f [v] with

components f

[v] for all i ∈ N.

Particularly, the Shapley value (Shapley, 1953) ϕ,

and the Banzhaf value (Owen, 1975) β, are deﬁned by

[v] =

∑

S⊆N\{i}

1/n



n − 1



[v(S ∪ {i}) − v(S)]

and

[v] =

∑

S⊆N\{i}

n−1

[v(S ∪ {i}) − v(S)]

for all i ∈ N and all v ∈ G

As it is well known, the Shapley value is the

unique value that satisﬁes:

(i) additivity: ϕ[v+v

] = ϕ[v]+ϕ[v

], for all v,v

∈ G

;

(ii) anonymity: ϕ

θi

[θv] = ϕ

[v] for all permutation θ

on N, i ∈ N, and v ∈ G

;

(iii) dummy player property: if i ∈ N is a dummy in

game v, then ψ

[v] = v({i}).

(iv) efﬁciency:

∑

i∈N

[v] = v(N)

The Banzhaf value follows a similar scheme, sat-

isfying the total power property (Owen, 1975)

∑

i∈N

[v] =

∑

i∈N

∑

S⊆N\{i}

n−1

[v(S ∪ {i}) − v(S)]

for all v ∈ G

, instead of additivity.

Notice that these two values are deﬁned for each

N. In fact, these values are deﬁned on cardinali-

ties rather than on speciﬁc player sets. When nec-

essary, we shall write ϕ

(n)

and β

for the Shapley and

Banzhaf values on cardinality n. In both cases, ϕ

(n)

and β

induce values ϕ

(t)

and β

for all cardinalities

t < n.

The multilinear extension (Owen, 1972) of a game

v ∈ G

is the real–valued function deﬁned on R

) =

∑

S⊆N

∏

i∈S

∏

j∈N\S

(1 − x

)v(S).

where X

denotes the set of variables x

for i ∈ N.

As it is well known, both the Shapley and Banzhaf

values of any game v can be easily obtained from its

multilinear extension. Indeed, ϕ[v] can be calculated

by integrating the partial derivatives of the multilin-

ear extension of the game along the main diagonal

= x

= · ·· = x

of the cube [0,1]

(Owen, 1972),

while the partial derivatives of that multilinear exten-

sion evaluated at point (1/2,1/2,. .. ,1/2) give β[v]

(Owen, 1975). This latter procedure extends well to

any p–binomial semivalue evaluating the derivatives

at point (p, p,. .., p), as we can see in (Puente, 2000),

(Freixas and Puente, 2002), or (Amer and Gim

enez,

2003).

2.2 Games with Coalition Structure and

Coalitional Values

Given N = {1,2,.. ., n}, we will denote by B(N) the

set of all partitions of N. Each B ∈ B(N) is called

a coalition structure in N, and a union each member

of B. The so–called trivial coalition structures are

= {{1},{2},. .. ,{n}} (individual coalitions) and

= {N} (grand coalition). A cooperative game with

a coalition structure is a pair [v;B], where v ∈ G

and B ∈ B(N) for a given N. Each partition B gives

a pattern of cooperation among players. We denote

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

by G

= G

× B(N) the set of all cooperative games

with a coalition structure and player set N.

If [v; B] ∈ G

and B = {B

,. .. ,B

}, the quo-

tient game v

is the cooperative game played by the

unions or, rather, by the quotient set M = {1, 2,..., m}

of their representatives, as follows:

(R) = v(

[

r∈R

) for all R ⊆ M.

By a coalitional value on G

we will mean a map

g : G

→ R

, which assigns to every pair [v;B] a vec-

tor g[v;B] with components g

[v;B] for each i ∈ N.

If f is a value on G

and g is a coalitional value

on G

, it is said that g is a coalitional value of f iff

g[v;B

] = f [v] for all v ∈ G

The Owen value (Owen, 1977) is the coalitional

value Φ deﬁned by

[v;B] =

∑

R⊆M\{k}

∑

T ⊆B

\{i}



m−1





−1



[v(Q ∪ T ∪ {i}) − v(Q ∪ T )]

for all i ∈ N and [v;B] ∈ G

, where B

∈ B is the union

such that i ∈ B

and Q =

r∈R

It was axiomatically characterized in (Owen,

1977) as the only coalitional value that satisﬁes the

following properties: the natural extensions to this

framework of

− efﬁciency

− additivity

− the dummy player property

and also

− symmetry within unions: if i, j ∈ B

are symmetric

in v then

[v;B] = Φ

[v;B]

− symmetry in the quotient game: if B

∈ B are

symmetric in [v;B] then

∑

i∈B

[v;B] =

∑

j∈B

[v;B].

The Owen value is a coalitional value of the Shap-

ley value ϕ in the sense that Φ[v; B

] = ϕ[v] for all

v ∈ G

. Besides, Φ[v; B

] = ϕ[v]. Finally, as Φ is de-

ﬁned for any N, the following property makes sense

and is also satisﬁed:

− quotient game property: for all [v; B] ∈ G

∑

i∈B

[v;B] = Φ

] for all B

∈ B.

The Owen value can be viewed as a two–step al-

location rule. First, each union B

receives its payoff

in the quotient game according to the Shapley value;

then, each B

splits this amount among its players by

applying the Shapley value to a game played in B

as follows: the worth of each subcoalition T of B

the Shapley value that T would get in a “pseudoquo-

tient game” played by T and the remaining unions on

the assumption that B

\T leaves the game, i.e. the

quotient game after replacing B

with T . This is the

way to bargain within the union: each subcoalition T

claims the payoff it would obtain when dealing with

the other unions in absence of its partners in B

The Owen–Banzhaf value Γ (Owen, 1982) fol-

lows a similar scheme. The resulting formula par-

allels that of the Owen value given above with

the sole change of coefﬁcient 1/mb



m−1



−1



1−m

1−b

. This value, which is a coalitional value

of the Banzhaf value β, does not satisfy efﬁciency, but

neither symmetry in the quotient game nor the quo-

tient game property. The bargaining interpretation is

the same as in the case of the Owen value by replac-

ing everywhere the Shapley value with the Banzhaf

value.

3 A NEW COMPUTATIONAL

PROCEDURE

In this section we give a new computational procedure

to calculate the Owen and the Owen–Banzhaf values

in terms of the MLE of the game. First of all we need

the following lemma.

Lemma 3.1. Let [v; B] ∈ G

, B = {B

,. .. ,B

} a

coalition structure in N. Then, the allocations given

by Φ and Γ to players belonging to a union B

can be

obtained as a linear combination of the allocations to

unanimity games u

, where T = V ∪ W , V ⊆ B

and

W ∈ 2

B\B

Proof. Each game v ∈ G

can be uniquely written as

linear combination of unanimity games

v =

∑

T ⊆N: T 6=

where α

= α

(v) =

∑

S⊆T

(−1)

t−s

v(S).

For all i ∈ B

, by linearity, Φ

[v;B] =

∑

T ⊆N: T 6=

] and it sufﬁces consider unanim-

ity games u

with

T = V ∪ A

∪ A

∪ . .. ∪ A

, V ⊆ B

,...,i

} ⊆ M \ { j},

0 6= A

⊆ B

, q = 1, ..., p.

According to the deﬁnition of the Owen value it is

easy to check that the allocations to players in B

only

depend on the allocations in the unanimity games de-

ﬁned on inside coalitions in B

and entire unions out-

side B

. That is,

;B] = Φ

V ∪A

∪A

∪...∪A

;B]

= Φ

V ∪B

∪B

∪...∪B

;B].

The Owen and the Owen-Banzhaf Values Applied to the Study of the Madrid Assembly and the Andalusian Parliament in Legislature

2015-2019

Analogously for the Owen–Banzhaf value. 

Notice that the number of unanimity games of this

form is (2

− 1)2

with b

= |B

| and m = |M|.

Proposition 3.2. Let B = {B

,. .. ,B

} be a coali-

tion structure in N. Fixed a union B

(i) The allocation to a player i belonging to B

a unanimity game u

, T = V ∪ B

∪ · ·· ∪ B

, V ⊆ B

and {i

,...,i

} ⊆ M \ { j}, given by the Owen value Φ

;B] =







h+1

i ∈ T

0 i /∈ T

where

h+1

and

are the induced coefﬁcients of ϕ

(h+1)

and ϕ

(v)

, respectively.

(ii) The allocation to a player i belonging to B

a unanimity game u

, T = V ∪ B

∪ · ·· ∪ B

, V ⊆ B

and {i

,...,i

} ⊆ M \{ j}, given by the Owen–Banzhaf

value Γ is

;B] =







−h

−(v−1)

= 2

−(h+v−1)

i ∈ T

0 i /∈ T

where

and

v−1

are the induced coefﬁcients of

(h+1)

and β

(v)

, respectively.

Proof. (i) For i ∈ T , we have

;B] =

∑

R⊆M\{ j}

1/m



m − 1



∑

S⊆B

\{i}

1/b



− 1



(Q ∪ S ∪ {i}) − u

(Q ∪ S)]

where Q =

[

r∈R

, b

= |B

|, and s = |S|.

Only u

(Q ∪ S ∪ {i}) − u

(Q ∪ S) does not vanish

for coalitions R such that {i

,...,i

} ⊆ R ⊆ M \ { j}

and for coalitions S such that V \ {i} ⊆ S ⊆ B

\ {i}.

Then,

;B] =

m−1

∑

r=h



m − 1 − h

r − h



1/m



m − 1



−1

∑

s=v−1



− v

s − v + 1



1/b



− 1



h + 1

In case of i 6∈ T , all marginal contributions u

(Q∪ S ∪

{i}) − u

(Q ∪ S) vanish.

(ii) For i ∈ T , we have

;B] =

∑

R⊆M\{ j}

−(m−1)

∑

S⊆B

\{i}

−(b

−1)

(Q ∪ S ∪ {i}) − u

(Q ∪ S)]

where Q =

[

r∈R

, b

= |B

|, and s = |S|.

Analogously to the previous case, only u

(Q∪S ∪

{i})− u

(Q∪ S) does not vanish for coalitions R such

that {i

,...,i

} ⊆ R ⊆ M\{ j} and for coalitions S such

that V \ {i} ⊆ S ⊆ B

\ {i}. Then,

;B] =

m−1

∑

r=h



m − 1 − h

r − h



−(m−1)

−1

∑

s=v−1



− v

s − v + 1



−(b

−1)

= 2

−(h+v−1)

In case of i 6∈ T , all marginal contributions u

(Q∪ S ∪

{i}) − u

(Q ∪ S) vanish. 

Example 3.3. On the set N = {1,2,3,4,5,6, 7}, let

B = {{1,2,3},{4, 5},{6}, {7}} be a coalition struc-

ture.

We will obtain the allocations to players i ∈ B

ac-

cording to Φ for the unanimity games u

{1,2,4,6,7}

and

{1,2,4,5,6,7}

. They are

{1,2,4,6,7}

;B] =

, for i = 1, 2 and

{1,2,4,6}

;B] = 0,

In a similar way and according to Lemma 3.1, for

{1,2,4,5,6,7}

we obtain

{1,2,4,5,6,7}

;B] =

, for i = 1, 2 and

{1,2,4,5,6,7}

;B] = 0,

Notice that the allocations in both games are the same

because coalitions {1,2,4, 6,7} and {1,2, 4,5,6, 7}

intersect the same unions B

, B

and B

The computing technique based on the multilinear

extension has been applied to many coalitional val-

ues: to the Owen value in (Owen and Winter, 1992);

to the Owen–Banzhaf value in (Carreras and Magana,

1994); to the quotient game in (Carreras and Mag-

ana, 1997); to the coalitional semivalues in (Amer

and Gim

enez, 2003); to the Alonso–Fiestras value

in (Alonso et al., 2005); to the symmetric coalitional

binomial semivalues in (Carreras and Puente, 2011);

and to the coalitional multinomial probabilistic val-

ues in (Carreras and Puente, 2013). In next theorems

we present a new method to compute the Owen and

the Owen–Banhaf values by means of the multilinear

extension of the game.

Theorem 3.4. Given in N a game with coalition

structure, [v;B] ∈ G

, B = {B

,. .. ,B

} coalition

structure in N, the following steps lead to the Owen

value of any player i ∈ B

in [v;B].

1. Obtain the multilinear extension f (x

,. .. ,x

)

of game v.

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

2. For every r 6= j and all h ∈ B

, replace the variable

with y

. This yields a new function of x

for

k ∈ B

and y

for r ∈ M\{ j}.

3. In this new function, reduce to 1 all higher expo-

nents, i.e. replace with y

each y

such that q > 1.

This gives a new multilinear function denoted as

((x

)

k∈B

, (y

)

r∈M\{ j}

) –the modiﬁed multilin-

ear extension of union B

–

4. After some calculus, the obtained modiﬁed multi-

linear extension reduces to

((x

)

k∈B

,(y

)

r∈M\{ j}

) =

∑

V ⊆B

∑

W ⊆M\{ j}

V ∪W

∏

k∈V

∏

r∈W

5. Multiply each product

∏

k∈V

and each

product

∏

r∈W

w+1

obtaining a new multi-

linear function called g

6. Obtain the partial derivative of g

with respect to

evaluated at point (1,.. ., 1) and

[v;B] =

∂g

∂x

M\{ j}

Proof. Steps 1–3 have been already used in (Owen

and Winter, 1992), (Carreras and Magana, 1994),

(Puente, 2000), (Freixas and Puente, 2002), (Alonso

et al., 2005), (Carreras and Puente, 2011) and (Car-

reras and Puente, 2013) to obtain the modiﬁed multi-

linear extension of union B

. Step 4 shows the mod-

iﬁed multilinear extension as a linear combination of

multilinear extensions of unanimity games. Step 5

weights each unanimity game according to Proposi-

tion 3.2 so that step 6 gives as usual the marginal

contribution of player i and his allocation Φ

[v;B] is

obtained. 

Theorem 3.5. Given in N a game with coalition

structure, [v;B] ∈ G

, B = {B

,. .. ,B

} coalition

structure in N, the following steps lead to the Owen–

Banzhaf value of any player i ∈ B

in [v;B].

1. Obtain the multilinear extension f (x

,. .. ,x

)

of game v.

2. For every r 6= j and all h ∈ B

, replace the variable

with y

. This yields a new function of x

for

k ∈ B

and y

for r ∈ M\{ j}.

3. In this new function, reduce to 1 all higher expo-

nents, i.e. replace with y

each y

such that q > 1.

This gives a new multilinear function denoted as

((x

)

k∈B

, (y

)

r∈M\{ j}

) –the modiﬁed multilin-

ear extension of union B

–

4. After some calculus, the obtained modiﬁed multi-

linear extension reduces to

((x

)

k∈B

,(y

)

r∈M\{ j}

) =

∑

V ⊆B

∑

W ⊆M\{ j}

V ∪W

∏

k∈V

∏

r∈W

5. Multiply each product

∏

k∈V

by 2

−(v−1)

and

each product

∏

r∈W

by 2

−w

obtaining a new

multilinear function called g

6. Obtain the partial derivative of g

with respect to

evaluated at point (1,.. ., 1) and

[v;B] =

∂g

∂x

M\{ j}

Proof. Analogously to the previous theorem. 

Example 3.6. Let us consider the 4–person weighted

majority game v ≡ [65;48, 37,27, 17] and the coali-

tion structure B = {{1,4},{2}, {3}}. We will com-

pute Φ[v;B].

The set of minimal winning coalitions is

(v) = {{1,2},{1, 3},{1, 4},{2,3,4}},

and the multilinear extension of v is given by

f (x

) = x

+ x

− x

Notice that players 2 and 3 became null in the

quotient game and, by the quotient game property,

[v;B] = Φ

[v;B] = 0 and it is not necessary to

compute the corresponding modiﬁed multilinear ex-

tensions g

and g

Steps 1–4 in Theorem 3.4 give the modiﬁed multi-

linear extension g

of union B

) = x

+ x

+ y

− x

Step 5 leads to g

) =

−

Step 6 yields

[v;B] =

4 TWO APPLICATIONS TO THE

POLITICAL ANALYSIS

Example 4.1. We consider here the Madrid Assembly

in legislature 2015–2019.

Four parties elected members to the Madrid As-

sembly (129 seats) in the elections held on 24 May

2015. The seat distribution of the parties are as fol-

lows.

The Owen and the Owen-Banzhaf Values Applied to the Study of the Madrid Assembly and the Andalusian Parliament in Legislature

2015-2019

Table 1: Classical measures of power in the Madrid Assembly, legislature 2015–2019.

(a) (b) (c)

(–) (R) (L) (–) (R) (L) (–) (R) (L)

1. PP 0.5000 0.6666 0.0000 0.7500 0.7500 0.0000 0.7500 0.7500 0.0000

2. PSOE 0.1666 0.0000 0.3333 0.2500 0.0000 0.2500 0.2500 0.0000 0.3333

3. Podemos 0.1666 0.0000 0.3333 0.2500 0.0000 0.2500 0.2500 0.0000 0.3333

4. C’s 0.1666 0.3333 0.3333 0.2500 0.2500 0.2500 0.2500 0.2500 0.3333

1: PP (Partido Popular), conservative party: 48

seats.

2: PSOE (Partido Socialista Obrero Espa

nol), mod-

erate left–wing party: 37 seats.

3: Podemos, radical left–wing party: 27 seats

4: C’s (Ciudadanos), Spanish nationalist liberal

party: 17 seats.

Under the standard absolute majority rule, and as-

suming voting discipline within parties, the structure

of this parliamentary body can be represented by the

weighted majority game

v ≡ [65; 48,37, 27,17].

Therefore, the strategic situation given by means of

the set of minimal wining coalitions

(v) = {{1,2},{1, 3},{1, 4},{2,3,4}},

shows that players 2, 3 and 4 are symmetric in v, and

the multilinear extension of v is

f (x

) =x

+ x

− x

A main feature of the Madrid Assembly issued from

the elections was the absence of a party enjoying ab-

solute majority, so a coalition government was ex-

pected to form. We will not try to give here a full

description of the complexity of the Madrid politics.

We wish only to state that the politically most likely

coalitions to form, and the corresponding coalition

structures to the analysis of which we will limit our-

selves, were clearly the following:

• PP + C’s, the “right”–wing majority alliance:

= {{1, 4},{2},{3}}.

• PSOE + Podemos + C’s, the “left”–wing majority

alliance: B

= {{1}, {2,3,4}}.

We would like to analyze these two situations. Of

course, our main interest will center on the strategic

possibilities of party 4 (C’s), whose position is crucial

in the two–alternative scenario we are considering.

A classical approach to study the problem would

consist in using either (a) the Shapley value and the

Owen value, (b) the Banzhaf value and the Owen–

Banzhaf value, or (c) the Banzhaf value and the sym-

metric coalitional Banzhaf value, in order to evaluate

the strategic possibilities of each party under both hy-

potheses. The results are given in Table 1, where (–)

means no coalition formation, (R) means that PP +

C’s forms, and (L) means that PSOE + Podemos +

C’s forms.

According to (a), C’s gets the same proﬁt in both

alliances. The same conclusion is obtained according

to (b). Instead, according to (c), C’s would strictly

prefer joining PSOE and Podemos instead of PP.

Moreover, by symmetry, the power of C’s when there

is not a coalition formation coincides with the power

of PSOE. According to (a), when the “right”-wing al-

liance is formed, the outside parties are reduced to a

null position and the power of C’s increases regard-

ing to the initial power in v. The same happens when

the “left”–wing alliance is formed

As we have seen, in the present Legislature, stud-

ied here, in order to form a government coalition

the role of C’s was crucial. Thus, C’s was faced to

the dilemma of choosing among either a a “left”–

wing majority coalition with PSOE and Podemos or a

“right”–wing majority coalition with PP, which was

ﬁnally formed in 2015.

Example 4.2. We consider here the Andalusian Par-

liament (legislature 2015–2019).

Five parties elected members to the Andalucia

Parliament (109 seats) in the elections held on 22

March 2015. The seat distribution of the parties are

as follows.

1: PSOE (Partido Socialista Obrero Espa

nol), mod-

erate left–wing party: 47 seats.

2: PP (Partido Popular), conservative party: 33

seats.

3: Podemos, radical left–wing party: 15 seats

4: C’s (Ciudadanos), Spanish nationalist liberal

party: 9 seats.

5: IULV–CA, Coalition of eurocommunist parties,

federated to Izquierda Unida, and ecologist

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

Table 2: Classical measures of power in the Andalusian Parliament, legislature 2015–2019.

(a) (b)

(–) (R) (L) (N) (–) (R) (L) (N)

1. PSOE 0.5000 0.3333 0.6666 0.6666 0.7500 0.5000 0.7500 0.7500

2. PP 0.1666 0.1666 0.0000 0.0000 0.2500 0.2500 0.0000 0.0000

3. Podemos 0.1666 0.3333 0.3333 0.0000 0.2500 0.5000 0.2500 0.0000

4. C’s 0.1666 0.1666 0.0000 0.3333 0.2500 0.2500 0.0000 0.2500

5. IULV–CA 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

groups : 5 seats

Under the standard absolute majority rule, and

assuming voting discipline within parties, the struc-

ture of this parliamentary body can be represented by

the weighted majority game

v ≡ [55; 47,33, 15,9, 5].

Therefore, the strategic situation is given by

(v) = {{1,2},{1, 3},{1, 4},{2,3,4}},

so that players 2, 3 and 4 are symmetric in v, player 5

is null and the multilinear extension of v is

f (x

) =x

+ x

− x

As in the same case as the Madrid Assembly, a coali-

tion government was expected to form. We will ana-

lyze the following alliances:

• PP + C’s, the “right” alliance: B

{{2,4}, {1},{3}, {5}}.

• PSOE + Podemos, the “left”–wing majority al-

liance: B

= {{1, 3},{2},{4}, {5}}.

• PSOE + C’s, the “neutral”–wing majority al-

liance: B

= {{1, 4},{2},{3}, {5}}

To study the problem we will use either (a) the Shapley

value and the Owen value and (b) the Banzhaf value

and the Owen–Banzhaf value, in order to evaluate the

strategic possibilities of each party under the three

hypotheses. The results are given in Table 2, where

(–) means no coalition formation, (R) means that PP

+ C’s forms, and (L) means that PSOE + Podemos

forms and (N) that PSOE + C’s forms.

In general, we can conclude that the formation of

a two-person coalition block is favorable for its mem-

bers and, especially, for the one that was obtained a

fewer number of seats.

5 CONCLUSIONS

We have obtained the allocations to the players ac-

cording to solution concepts modiﬁed by coalition

structures following a double-level procedure based

on the multilinear extension of the game. The two

levels are (i) the modiﬁcation of the multilinear ex-

tension according to the quotient game and (ii) the

weighting of each product in the modiﬁed multilinear

extension according to the solution concept that we

want to compute.

This procedure has been effective for the compu-

tation of allocations according to the Owen value and

the Owen–Banzhaf value by means of simple steps.

In this way, the calculus in Section 4 of several situ-

ations in two territorial Spanish Parliaments has been

easy to compute.

ACKNOWLEDGEMENTS

Research supported by Grant SGR 2017–758 of the

Catalonia Government (Generalitat de Catalunya)

and Grant MTM 2015–66818P of the Economy and

Competitiveness Spanish Ministry.

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