Ability to Separate Situations with a Priori Coalition Structures by

Means of Symmetric Solutions

Jos

e Miguel Gim

enez

Department of Mathematics, Technical University of Catalonia, Avda. Bases de Manresa 61, E-08242 Manresa, Spain

Keywords:

Cooperative game, Coalition structure, Marginal contribution, Semivalue, Separability.

Abstract:

We say that two situations described by cooperative games are inseparable by a family of solutions, when they

obtain the same allocation by all solution concept of this family. The situation of separability by a family of

linear solutions reduces to separability from the null game. This is the case of the family of solutions based

on marginal contributions weighted by coefﬁcients only dependent of the coalition size: the semivalues. It is

known that for games with four or more players, the spaces of inseparable games from the null game contain

games different to zero-game. We will prove that for ﬁve or more players, when a priori coalition blocks are

introduced in the situation described by the game, the dimension of the vector spaces of inseparable games

from the null game decreases in an important manner.

1 INTRODUCTION

Probabilistic values as a solution concept for coop-

erative games were introduced in (Weber, 1988). The

payoff that a probabilistic value assigns to each player

is a weighted sum of its marginal contributions to the

coalitions, where the weighting coefﬁcients form a

probabilistic distribution over the coalitions to which

it belongs. A particular type of probabilistic values is

formed by the semivalues that were deﬁned in (Dubey

et al., 1981). In this case the weighting coefﬁcients

are independent of the players and they only depend

on the coalition size. Semivalues represent a natu-

ral generalization of both the Shapley value (Shapley,

1953) and the Banzhaf value (Banzhaf, 1965; Owen,

1975). According to this approach, many works deal

with the semivalues, with general properties as in

(Carreras and Gim

enez, 2011), or applied to simple

games as in (Carreras et al., 2003), and many others.

It is possible to ﬁnd two cooperative games that

obtain the same payoff vector for each semivalue. We

say that these games are inseparable by semivalues.

By the linearity property of semivalues, we can re-

duce the problem of separability between games to

separability from the null game. The vector subspace

of inseparable games from the null game by semival-

ues is called in (Amer et al., 2003) shared kernel and

its dimension is 2

− n

+ n − 2, where n denotes the

number of players. For spaces of cooperative games

with four or more players, the shared kernel contains

games different to zero-game

The semivalues form an important family of solu-

tions. We can evaluate their amplitude according to

their faculty to separate games. Two games are sepa-

rable if their difference does not belong to the shared

kernel. The dimension of this subspace would mark

the separation impossibility. In this paper we consider

coalition structures in the player set. It is not difﬁ-

cult to ﬁnd in the literature many papers devoted to

the modiﬁed semivalues by coalition structures, for

instance (Albizuri, 2009) or (Gim

enez and Puente,

2015), among others. Our purpose is to reduce the di-

mension of the vector subspace of inseparable games

from the null game. For cooperative games with ﬁve

or more players, modiﬁed semivalues for games with

coalition structure (Amer and Gim

enez, 2003) are

able to reduce in a signiﬁcant way the dimension of

the shared kernel.

In addition, once an a-priori ordering is chosen in

the player set, we can see in (Amer et al., 2003) that

the shared kernel is spanned by speciﬁc {−1, 0,1}-

valued games. These games are known as commuta-

tion games. Now, we will prove that the vector sub-

space of inseparable games from the null game by

modiﬁed semivalues is spanned by games introduced

here with the name of expanded commutation games.

The paper is organized as follows. In Section 2

we remember the solution concepts of semivalue and

semivalue modiﬁed for games with a coalition struc-

ture whose allocations can be computed by means of

242

GimÃl’nez J.

Ability to Separate Situations with a Priori Coalition Structures by Means of Symmetric Solutions.

DOI: 10.5220/0006116802420249

In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 242-249

ISBN: 978-989-758-218-9

the multilinear extension (Owen, 1972) of each game.

Also, nomenclature and main results for inseparable

games by semivalues are described. Section 3 shows

that commutation games that are the solution for the

problem of separability by semivalues does not have

in general the same properties with respect to separa-

bility by modiﬁed semivalues. In section 4 two sufﬁ-

cient conditions for separability by modiﬁed semival-

ues are proposed. Finally, in Section 5 we determine

the dimension and a basis of the vector subspace of

inseparable games from the null game by modiﬁed

semivalues.

2 PRELIMINARIES

2.1 Cooperative Games and Semivalues

A cooperative game with transferable utility is a pair

(N,v), where N is a ﬁnite set of players and v : 2

→

R is the so-called characteristic function, which as-

signs to every coalition S ⊆ N a real number v(S), the

worth of coalition S, and satisﬁes the natural condi-

tion v(

0) = 0. With G

we denote the set of all co-

operative games on N. For a given set of players N,

we identify each game (N,v) with its characteristic

function v.

The multilinear extension MLE (Owen, 1972) of

cooperative game v ∈ G

is a function f

: [0,1]

→ R

deﬁned as

,..., x

) =

∑

S⊆N

∏

i∈S

∏

j /∈S

(1 −x

)v(S), (1)

so that it provides all information of the game con-

tained in its characteristic function v.

A function ψ : G

→ R

is called a solution and

it represents a method to measure the negotiation

strength of the players in the game. The payoff vec-

tor space R

is also called the allocation space. The

semivalues (Dubey et al., 1981) as solution concept

were introduced and axiomatically characterized by

Dubey, Neyman and Weber in 1981. The payoff to

the players for a game v ∈ G

by a semivalue ψ is an

average of marginal contributions of each player:

[v] =

∑

S3i

[v(S) − v(S\{i})] ∀i ∈ N, (2)

where the weighting coefﬁcients p

only depend on

the coalition size and verify

∑

s=1



n−1

s−1



= 1 and

≥ 0 for 1 ≤ s ≤ n. With Sem(G

) we denote the

set of all semivalues on G

Given a number α ∈ R, 0 < α < 1, we call bi-

nomial semivalue ψ

to the semivalue whose coefﬁ-

cients are p

α,s

= α

s−1

(1−α)

n−s

. The extreme cases

correspond to values α = 0 and α = 1. For α = 0

we obtain the dictatorial index ψ

, with coefﬁcients

(1,0, ..., 0), whereas for α = 1 we obtain the marginal

index ψ

, with coefﬁcients (0,..., 0, 1):

(ψ

)

[v] = v({i}) ∀i ∈ N,

(ψ

)

[v] = v(N) − v(N \ {i}) ∀i ∈ N.

It is proven in (Amer and Gim

enez, 2003) that n

different binomial semivalues form a reference sys-

tem for the set of semivalues on G

. Given n dif-

ferent numbers α

in [0,1], for every semivalue ψ ∈

Sem(G

) they exist unique coefﬁcients λ

, 1 ≤ j ≤ n,

such that ψ =

∑

j=1

The Banzhaf value (Banzhaf, 1965; Owen, 1975)

is the binomial semivalue for α = 1/2. As it happens

for the Banzhaf value, we see in (Amer and Gim

enez,

2003) that the allocation by every binomial semivalue

can calculate replacing in the partial derivatives of

MLE the variables by value α:

(ψ

)

[v] =

∂ f

∂x

(α) ∀i ∈ N, where α = (α,...,α).

In addition, the allocation for every semivalue can

be computed by means of a product of two matrices,

ψ[v] = B Λ, (3)

where the matrix B depends on each reference system

of semivalues B = (b

i j

)

1≤i, j≤n

with b

i j

= (ψ

)

[v] =

∂ f

∂x

(α

) and Λ is the column matrix of the coefﬁcients

of ψ in this reference system, Λ

= (λ

·· · λ

) if

ψ =

∑

j=1

. Thus, a (n×n)-matrix summarizes

the payments by any semivalue to all players of a

given game v.

2.2 Cooperative Games and Coalition

Structures

The formation of coalition blocks in the player set N

gives rise to the construction of modiﬁed solutions in

attention to this circumstance. It is the case of the

Owen coalition value (Owen, 1977) from the Shap-

ley value (Shapley, 1953) or the modiﬁed Banzhaf

value for games with coalition structure (Owen, 1981)

from the Banzhaf value. If we denote by B =

,..., B

} the coalition structure in N, in both

cases, the construction of the modiﬁed solutions fol-

lows a parallel way. It is considered a modiﬁed quo-

tient game for each coalition S ⊆ B

and it is applied

the Shapley or Banzhaf value. This action deﬁnes a

game in B

and there it is now applied the same solu-

tion obtaining for each i ∈ B

the modiﬁed allocations.

Given a semivalue ψ ∈ Sem(G

) with weighting

coefﬁcients p

, the recursively obtained numbers

= p

m+1

+ p

m+1

s+1

1 ≤ s ≤ m < n,

Ability to Separate Situations with a Priori Coalition Structures by Means of Symmetric Solutions

243

deﬁne a induced semivalue ψ

(Dragan, 1999) on the

space of cooperative games with m players. Adding

the own semivalue, the family of induced semivalues

{ψ

∈ Sem(G

)/ 1 ≤ m ≤ n} allows us to deﬁne the

concept of semivalue modiﬁed for games with coali-

tion structure (Amer and Gim

enez, 2003) following

the same procedure as above. For a player i belongs

to coalition block B

the modiﬁed allocation has by

expression

[v;B] =

∑

S⊆B

\{i}

∑

T ⊆M\{ j}

s+1

t+1



[

t∈T

∪ S ∪ {i}



− v



[

t∈T

∪ S



(4)

For the extreme coalition structures, individual

blocks and grand coalition, the modiﬁed allocations

agree with the allocation by the initial semivalue.

Also, the allocations by modiﬁed semivalues can be

computed by means of a product of matrices, once

a reference system of binomial semivalues has been

chosen:

[v;B] = Λ

A(i) Λ. (5)

Matrix Λ is like in expression (3). The terms

(i), 1 ≤ p,q ≤ n, of matrix A(i) can be obtained

by means of the following rules:

(i) Obtain the MLE f

= f

,..., x

) of game v.

(ii) For each t ∈ M, t 6= j, and each m ∈ B

replace

the variable x

by y

. Thus, a new function of the

variables x

, y

for k ∈ B

and t ∈ M \{ j} is obtained.

(iii) In the above function, reduce all exponents

that appear in y

to 1, that is, replace y

(r > 1) by y

obtaining another multilinear function g

) k ∈

and t ∈ M \ { j}.

(iv) Calculate the derivative of the function g

with

respect to variable x

(v) Replace each x

with α

and each y

with α

Then,

(i) =

∂g

∂x

(α

,α

) for 1 ≤ p,q ≤ n. (6)

2.3 Separability in Cooperative Games

We say that two cooperative games v,v

∈ G

are sep-

arable by a solution ψ on G

if ψ[v] 6= ψ[v

] for v 6= v

When we study separability between games accord-

ing to semivalues, we can only consider separability

from the null game, since these solutions verify lin-

earity property.

For each G

, the linear subspace of all cooperative

games inseparable by semivalues from the null game

is called in (Amer et al., 2003) shared kernel C

. It

is proven that the dimension of C

is 2

− n

+ n − 2,

since games in C

have to satisfy conditions:

∑

S3i,|S|=s

v(S) = 0 for all i ∈ N and 1 ≤ s ≤ n. (7)

Grouping these conditions according to coalition

sizes, the freedom degrees for each s with 2 ≤ s ≤

n−2 are





−n, whereas v(S) = 0 for |S| = 1, n−1, n.

This way, the dimension of C

is 2

− n

+ n − 2 for

|N| = n ≥ 2 and C

= {0} if |N| = 2, 3.

In game spaces G

with cardinality |N| ≥ 4, for a

given coalition S ⊆ N and players i, j ∈ S and k, l ∈

N \ S, we deﬁne the commutation game v

S,i, j,k,l

= 1

+ 1

S∪{k,l}\{i, j}

− 1

S∪{k}\{i}

− 1

S∪{l}\{ j}

(8)

where 1

is the unity game in G

(S) = 1 and

(T ) = 1 otherwise). If v ∈ G

is a commutation

game, then v ∈ C

. In (Amer et al., 2003), it is

proven that the shared kernel is spanned by commuta-

tion games. Since each commutation game takes non

null values uniquely on coalitions of a single size, the

number of selected games in the proof of this property





−n for coalitions S with 2 ≤ s ≤ n − 2 (|S| = s).

3 COMMUTATION GAMES AND

COALITION STRUCTURES

Let us remember that with C

we denote the linear

subspace of all cooperative games in G

inseparable

from the null game by semivalues.

Proposition 3.1. Let f

= f

,..., x

) be the

MLE of game v ∈ G

v ∈ C

⇔ ∇ f

(α) = 0 ∀α ∈ [0,1], α = (α, . ..,α).

Proof. If v ∈ C

, then ψ[v] = 0 ∀ψ ∈ Sem(G

In particular, for all binomial semivalue ψ

with α ∈

[0,1], ψ

[v] = ∇ f

(α) = 0 where α = (α,α,...,α).

Conversely, since n binomial semivalues form

a reference system in Sem(G

), every semivalue

ψ ∈ Sem(G

) can uniquely be written like ψ =

∑

j=1

with α

∈ [0, 1] for 1 ≤ j ≤ n. Then,

ψ[v] =

∑

j=1

[v] =

∑

j=1

∇ f

(α

) = 0

and game v belongs to the shared kernel C

. 

Example. Let N = {i, j, k,l} be the set of players.

For cooperative games with four players the coalition

S in the commutation games is only composed by two

players. For short, when S = {i, j} we write the com-

mutation game v

S,i, j,k,l

as v

i, j,k,l

, i. e.,

i, j,k,l

= 1

{i, j}

+ 1

{k,l}

− 1

{ j,k}

− 1

{i,l}

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

244

The MLE of this game is f

i, j,k,l

= x

−x

−

. It is easy to see that ∇ f

i, j,k,l

(α) = 0 ∀α ∈ [0,1],

α = (α, α, α, α).

Deﬁnition 3.2. We say that a cooperative game v ∈

is inseparable from the null game by semivalues

modiﬁed for games with coalition structure if and only

if ψ[v; B] = 0 for every semivalue ψ on G

and every

coalition structure B in N

The above deﬁnition introduces our central con-

cept of separability between games by modiﬁed semi-

values; linearity of these solutions allows us to reduce

the problem to separability from the null game. Now,

the commutation games that give the solution to the

problem of separability by semivalues, offer a differ-

ent answer according to the cardinality of the player

set.

Proposition 3.3. Let G

be the vector space of coop-

erative games with four players, |N| = 4. Condition

of inseparable by semivalues is equivalent to condi-

tion of inseparable by semivalues modiﬁed for games

with coalition structure.

Proof. For case |N| = 4, the shared kernel C

has

dimension 2. According to development in (Amer

et al., 2003), a basis for C

is formed by commutation

games v

1,4,3,2

and v

2,4,3,1

. For the commutation games

in a basis of C

, we will prove that condition of insep-

arability from the null game by semivalues extends

to condition of inseparability from the null game by

modiﬁed semivalues. For the remaining games in C

the property is veriﬁed by linearity.

We consider, for example, game v

2,4,3,1

and simul-

taneously all possible types of coalition structures in

N = {1, 2, 3, 4}. (a) Four individual blocks. (b) One

bipersonal block where game v

2,4,3,1

takes non-null

value and two individual blocks. (c) Like in (b) but

taking null value. (d) Two bipersonal blocks where

game v

2,4,3,1

takes non-null values. (e) Like in (d) but

taking null values. (f) One coalition block with three

players. (g) Only one coalition block with four play-

ers.

In cases (a) and (g), both allocations coincide:

ψ[v

2,4,3,1

;B] = ψ[v

2,4,3,1

] = 0 ∀ψ ∈ Sem(G

), B =

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

From now, we will use the MLE f

2,4,3,1

= x

− x

Case (b). We consider, for instance, coalition

structure B = {{1,2},{3},{4}}. According to rules

that lead to coefﬁcients in expression (6) for obtaining

value ψ

2,4,3,1

;B] by means of a product of matrices

as in (5), we ﬁrst determine modiﬁed MLE g

) = x

+ x

− y

− x

;

∂g

∂x

= y

− x

⇒ a

(1) =

∂g

∂x

(α

,α

) = α

− α

for 1 ≤ p, q ≤ 4.

Written any semivalue ψ as linear combination of

four different binomial semivalues, we can conclude

that

2,4,3,1

;B] = Λ

A(1) Λ = 0 ∀ψ ∈ Sem(G

since, in this case, matrix A(1) satisﬁes a

(1) =

−a

(1) for 1 ≤ p, q ≤ 4. In a similar way,

2,4,3,1

;B] = 0 ∀ψ ∈ Sem(G

Now, for obtaining value ψ

2,4,3,1

;B], we deter-

mine modiﬁed MLE g

) = y

+ y

− x

− y

;

∂g

∂x

= y

− y

⇒ a

(3) =

∂g

∂x

(α

,α

) = 0

for 1 ≤ p, q ≤ 4.

Then ψ

2,4,3,1

;B] = 0 and, also, ψ

2,4,3,1

;B] =

Case (c). Possible coalition structure B =

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

) = y

+ x

− y

− x

;

∂g

∂x

= y

− y

⇒ a

(1) =

∂g

∂x

(α

,α

) = 0

for 1 ≤ p, q ≤ 4.

Consequently, ψ

2,4,3,1

;B] = 0. In a similar way,

2,4,3,1

;B] = 0 and ψ

2,4,3,1

;B] = ψ

2,4,3,1

;B] =

Similar manipulations of MLE f

2,4,3,1

in cases

(d), (e) and (g) give rise to the same conclusion

ψ[v

2,4,3,1

;B] = 0.

Conversely, if a game is inseparable from the null

game by modiﬁed semivalues, in particular, it is in-

separable from the null game by semivalues. It suf-

ﬁces to consider the coalition structure formed by in-

dividual blocks. 

Proposition 3.4. For vector spaces of cooperative

games G

with ﬁve or more players, every commu-

tation game is separable from the null game by semi-

values modiﬁed for games with coalition structure.

Proof. In G

with |N| ≥ 5, the commuta-

tion gamev

S,i, j,k,l

= 1

+ 1

S∪{k,l}\{i, j}

− 1

S∪{k}\{i}

−

S∪{l}\{ j}

, with i, j ∈ S and k,l ∈ N \ S, has by MLE

S,i, j,k,l

= [x

+ x

− x

]

∏

p∈S\{i, j}

∏

q∈N\(S∪{k,l})

(1 −x

For coalitions S with 2 ≤ |S| < n−2, we con-

sider coalition structure B

= {S,N \S}. The modiﬁed

MLE g

for players in block S is

= x

(1 −y

)

∏

p∈S\{i, j}

Ability to Separate Situations with a Priori Coalition Structures by Means of Symmetric Solutions

245

and

∂g

∂x

= x

(1 −y

)

∏

p∈S\{i, j}

where N \ (S ∪ {k,l}) 6=

0 since |S| < n−2.

Then, modiﬁed Banzhaf value β separates game

S,i, j,k,l

, 2 ≤ |S| < n−2, from the null game:

S,i, j,k,l

] =

∂g

∂x

(1/2,1/2) =

6= 0.

For case |S| = n−2, S = N \{k, l} and the MLE is

N\{k,l},i, j,k,l

= [x

+ x

− x

]

∏

p∈N\{i, j,k,l}

Now, we consider coalition structure B

N\{k,l}

= {N \

{k, l},{k, l}} and we obtain the modiﬁed MLE g

for

players in block N \ {k,l}:

= [x

+ y

− x

]

∏

p∈N\{i, j,k,l}

where N \ {i, j,k,l} 6=

0 since |N| ≥ 5. Let h be

a player in N \ {i, j, k,l}. Again, modiﬁed Banzhaf

value β separates game v

N\{k,l},i, j,k,l

from the null

game:

∂g

∂x

= [x

+ y

− x

]

∏

p∈N\{h,i, j,k,l}

and

N\{k,l},i, j,k,l

N\{k,l}

] =

∂g

∂x

(1/2,1/2) =

n−3

6= 0.



4 SUFFICIENT CONDITIONS OF

SEPARABILITY

For games with ﬁve or more players, the commuta-

tion games are not a solution for the problem of in-

separability by semivalues modiﬁed for games with

coalition structure. In this section we provide two

sufﬁcient conditions of separability, that is, two nec-

essary conditions of inseparability from the null game

by modiﬁed semivalues.

Proposition 4.1. Let us consider vector spaces of co-

operative games G

with |N| ≥ 4. If there exists a

coalition S with v(S) 6= v(N \ S), then game v is sep-

arable from the null game by semivalues modiﬁed for

games with coalition structure.

Proof. Let us suppose S

a coalition with smallest

size that veriﬁes v(S

) 6= v(N \ S

). If |S

| = 1, game

v is separable from the null game by semivalues and

also by modiﬁed semivalues. We can consider that

| = s

≥ 2 and s

≤ n/2. Then, the MLE of game v

can be written as

∑

S:2≤|S|≤s

∏

i∈S

∏

j∈N\S

(1 −x

)v(S)+

∏

i∈N\S

∏

j∈S

(1 −x

)v(N \ S)

∑

S:s

<|S|<n−s

∏

i∈S

∏

j∈N\S

(1 −x

)v(S).

Now, we choose the coalition structure B

,N \ S

}. In such a case, the modiﬁed MLE g

for

players in coalition block S

has by expression

∑

S⊂S

,s≥2

(1 −y

)

∏

i∈S

∏

j∈S

(1 −x

∏

i∈S

∏

j∈S

(1 −x

)

v(S)+

(1 −y

)

∏

i∈S

v(S

) +y

∏

j∈S

(1 −x

)v(N\S

because terms for coalitions S containing elements as

much in S

as in N \S

vanish in MLE g

. If k is a

player in S

∂g

∂x

∑

S⊂S

,s≥2, S3k

(1 − y

)

∏

i∈S\{k}

∏

j∈S

(1 − x

)−

∏

i∈S

∏

j∈S\{k}

(1 − x

)

v(S)+

∑

S⊂S

,s≥2, S63k

− (1 − y

)

∏

i∈S

∏

j∈S

\(S∪{k})

(1 − x

∏

i∈S

\(S∪{k})

∏

j∈S

(1 − x

)

v(S)+

+ (1 − y

)

∏

i∈S

\{k}

v(S

) − y

∏

j∈S

\{k}

(1 − x

)v(N\S

Then

∂g

∂x

(1/2,1/2) =



v(S

) −v(N\S

)



and the modiﬁed Banzhaf value β separates game v

from the null game:

[v;B

] =

∂g

∂x

(1/2,1/2) 6= 0 for k ∈ S

. 

Proposition 4.2. For spaces of cooperative games

with |N| ≥ 6, let us consider a game v that sat-

isﬁes v(S) = v(N \ S) ∀S ⊆ N and v({i}) = 0 ∀i ∈ N.

If there exists a coalition S with

v(S) 6=

∑

T ⊂S, |T |=2

v(T ) and 3 ≤ |S| ≤ n/2, (9)

then game v is separable from the null game by semi-

values modiﬁed for games with coalition structure.

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

246

Proof. The MLE of game v that satisﬁes the two

ﬁrst conditions of the statement can be written as

∑

S: 2≤|S|<n/2

∏

i∈S

∏

j∈N\S

(1 −x

) +

∏

i∈N\S

∏

j∈S

(1 −x

)

v(S)+

∑

S: |S|=n/2

∏

i∈S

∏

j∈N\S

(1 −x

)v(S),

(10)

where the second sum only appears in case n even

number. Let us suppose S

a coalition with smallest

size that veriﬁes (9) for |S

| < n/2. In such a case, we

choose coalition structure B

= {S

,N \ S

} and write

modiﬁed MLE g

for players in coalition block S

∑

S⊂S

,2≤s<s

(1 −y

)

∏

i∈S

∏

j∈S

(1 −x

∏

i∈S

∏

j∈S

(1 −x

)

v(S)+

(1 −y

)

∏

i∈S

+ y

∏

j∈S

(1 −x

)

v(S

Next, we consider a player j

in block S

, compute

the partial derivative of MLE g

with respect to vari-

able x

and replace all variables by generic value α

grouping the sums as follows:

∂g

∂x

(α,α) =

∑

S⊂S

,S3 j

,|S|=2



α(1 −α)

−1

− α

−1

(1 −α)



v(S)+

∑

S⊂S

,S3 j

,2<s<s



s−1

(1 −α)

−s+1

− α

−s+1

(1 −α)

s−1



v(S)+

∑

S⊂S

,S63 j

,2≤s<s

−1



−s

(1 −α)

− α

(1 −α)

−s

]v(S)+



α(1 −α)

−1

− α

−1

(1 −α)



v(S

\{ j

}) −v(S

)



All terms for coalitions S with S 63 j

and 2 ≤ s <

− 1 can be written by means of coalitions T with

T 3 j

and 3 ≤ t < s

. Then,

∂g

∂x

(α,α) = α (1 − α)



(1 −α)

−2

− α

−2

]



∑

S⊂S

,S3 j

,|S|=2

v(S) + v(S

\{ j

}) −v(S

)

∑

S⊂S

,S3 j

,2<s<s



s−1

(1 −α)

−s+1

− α

−s+1

(1 −α)

s−1



v(S)+

∑

T ⊂S

,T 3 j

,2<t<s



−t+1

(1 −α)

t−1

− α

t−1

(1 −α)

−t+1



v(T \{ j

}).

We shorten polynomial (1 − α)

−2

− α

−2

means of p

(α) and write v(S

\{ j

}) as a sum of all

values on contained bipersonal coalitions:

∂g

∂x

(α,α) = α (1 − α)p

(α)

∑

S⊂S

,S3 j

,|S|=2

v(S)+

∑

T ⊆S

\{ j

},|T |=2

v(T ) − v(S

)

∑

S⊂S

,S3 j

,2<s<s



s−1

(1 −α)

−s+1

− α

−s+1

(1 −α)

s−1





v(S) − v(S\{ j

})



(11)

It is possible to ﬁnd coalitions S with S ⊂ S

, S 3 j

and 2 < s < s

only in case s

≥ 4. Then, the last sum

in the above expression can be written as

∑

S⊂S

,S3 j

,3≤s<1+s



s−1

(1 −α)

−s+1

− α

−s+1

(1 −α)

s−1





v(S) − v(S\{ j

})



∑

T ⊂S

,T 3 j

,1+s

/2<t≤s

−1



t−1

(1 −α)

−t+1

− α

−t+1

(1 −α)

t−1





v(T ) − v(T \{ j

})



where case s = 1 + s

/2 is not considered, since only

for s

even number, cardinality of S can take value

s = 1 + s

/2 but, in this case, coefﬁcient α

s−1

(1 −

α)

−s+1

− α

−s+1

(1 − α)

s−1

vanish. In the above

sums, we can identify coalitions S for 3 ≤ s < 1+s

with coalitions T for 1 + s

/2 < t ≤ s

− 1 by means

relation t = s

− s + 2. Then, both sums reduce to

∑

3≤s<1+s



s−1

(1 −α)

−s+1

− α

−s+1

(1 −α)

s−1



∑

S⊂S

,S3 j

,|S|=s



v(S) − v(S\{ j

})



−

∑

T ⊂S

,T 3 j

,|T |=s

−s+2



v(T ) − v(T \{ j

})



Let us suppose that S

= { j

, j

,. . . , j

}. For a

given cardinality s with 3 ≤ s < 1 + s

/2, the last dif-

ference of sums vanish, because it can be written as

∑

S⊂S

,S3 j

,|S|=s

∑

P⊂S,|P|=2

v(P) −

∑

Q⊆S\{ j

},|Q|=2

v(Q)

−

∑

T ⊂S

,T 3 j

,|T |=s

−s+2

∑

P⊂T,|P|=2

v(P)−

∑

Q⊂T \{ j

},|Q|=2

v(Q)

∑

S⊂S

,S3 j

,|S|=s

∑

P⊂S,P3 j

,|P|=2

v(P)

−

∑

T ⊂S

,T 3 j

,|T |=s

−s+2

∑

P⊂T,P3 j

,|P|=2

v(P)

Ability to Separate Situations with a Priori Coalition Structures by Means of Symmetric Solutions

247

∑

i=2



− 2

s − 2



−



− 2

− s



v({ j

, j

}) = 0.

Thus, from expression (11), we can write the mod-

iﬁed binomial semivalue ψ

for player j

∈ S

(ψ

)

[v;B

] =

∂g

∂x

(α,α) =

α(1 −α)p

(α)

∑

T ⊂S

,|T |=2

v(T ) − v(S

)

Since α = 1/2 is the unique real zero of poly-

nomial p

for values s

≥ 3 and game v satisﬁes

inequality (9) for coalition S

, we conclude that

(ψ

)

[v;B

] 6= 0 for values α ∈ (0, 1/2) ∪ (1/2,1)

and these modiﬁed semivalues separate game v from

the null game.

It only lack to see case in which |S| = n/2 is the

smallest size of coalitions that verify (9). Here, n is

a even number and all coalitions in the second sum

of expression (10) can be grouped by pairs: S and

N \ S. The selected coalition S

will belong to one or

another half of coalitions with size n/2; we choose

half that contains coalition S

and describe the second

sum with S and N \ S, as the same way that the ﬁrst

sum in (10). Then, by repeating the same procedure as

in case |S| < n/2, we arrived at the same conclusion.



5 EXPANDED COMMUTATION

GAMES

We denote with D

the vector subspace of all cooper-

ative games in G

inseparable from the null game by

semivalues modiﬁed for games with coalition struc-

ture.

Deﬁnition 5.1. In G

with |N| ≥ 5, we consider a

commutation game with coalition size 2, v

i, j,k,l

, k, l ∈

N \ {i, j}. The expanded game of commutation game

i, j,k,l

is the sum of all commutation games in G

P,i, j,k,l

, with the same commuted players, i.e.,

i, j,k,l

∑

P3i, j,P⊆N\{k,l}

P,i, j,k,l

Lemma 5.2. In G

with |N| ≥ 5 an expanded com-

mutation game v

i, j,k,l

, k,l ∈ N \{i, j}, satisﬁes the fol-

lowing properties:

(a) v

i, j,k,l

(S) = v

i, j,k,l

(N \ S) ∀S ⊆ N;

(b) v

i, j,k,l

(S) =

∑

T ⊂S, |T |=2

i, j,k,l

(T ) ∀S ⊆ N and 3 ≤

|S| ≤ |N|;

i, j,k,l

= x

+ x

− x

Proof. It is easy to prove sections (a) and (b); it

sufﬁces to check if players i, j, k,l belong or not to

coalitions S, since the only bipersonal coalitions that

take non-null values in game v

i, j,k,l

are {i, j}, {k,l},

{ j, k} and {i,l}. In order to verify section (c) we can

write MLE of game v

i, j,k,l



+ x

− x



∏

q∈N\{i, j,k,l}

(1 −x

) + f

∑

Q⊆N\{i, j,k,l}

where games 1

are considered in G

N\{i, j,k,l}

. Since

∑

Q⊆N\{i, j,k,l}

(T ) = 1 ∀T ⊆ N \ {i, j,k,l}, T 6=

(Q 6=

0), its MLE equals the unity in N \ {i, j,k, l}

and section (c) follows. 

Proposition 5.3. In spaces of cooperative games G

with |N| ≥ 5, every expanded commutation game

i, j,k,l

, k,l ∈ N \{i, j} belongs to vector subspace D

Proof. Section (c) in above Lemma proves that

MLE of expanded commutation game v

i, j,k,l

, k,l ∈

N \{i, j} in G

with |N| ≥ 5 agrees with MLE of com-

mutation game v

i, j,k,l

in a space of cooperative games

with only four players, {i, j,k,l}.

In order to demonstrate that game v

i, j,k,l

, k,l ∈

N \ {i, j}, is inseparable by modiﬁed semivalues, we

can consider that players i, j,k,l are distributed in

different coalition blocks in the same way that in

the proof of Proposition 3.3. The remaining players

N \{i, j, k,l} will be distributed in the different blocks

next to players i, j,k,l or they will form new coalition

blocks.

Since variables that correspond to players in N \

{i, j,k,l} does not appear in the MLE of game v

i, j,k,l

when we compute allocations for players i, j,k,l by

means of a product of matrices as in (5), we ob-

tain the same result as in Proposition 3.3, that is,

i, j,k,l

,B] = 0 for p = i, j,k, l, ∀ψ ∈ Sem(G

), ∀B

coalition structure in N.

For the remaining players, ψ

i, j,k,l

,B] = 0 ∀q ∈

N \ {i, j, k,l}, since variable x

does not appear in the

MLE. 

Theorem 5.4. Let us consider vector spaces of coop-

erative games G

with ﬁve or more players, |N| ≥ 5.

Then,

(a) dim D





− n;

(b) the vector subspace D

is spanned by expanded

of commutation games with coalition size 2.

Proof. We can see in (Amer et al., 2003) that the

shared kernel C

for |N| ≥ 4 is spanned by 2

− n

n − 2 commutation games whose coalitions with non-

null value vary from cardinality s = 2 to n − 2. We

choose the





− n commutation games with coalition

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

248

size 2. As they are linearly independent in G

, its

expanded games are also linearly independent and, by

above Proposition, inseparable from the null game by

modiﬁed semivalues. The linear subspace spanned by

these expanded commutation games is contained in

subspace D

for |N| ≥ 5.

In addition, as D

⊆ C

, the freedom degrees in

by a consequence of conditions (7) for coalitions

with sizes s > n/2 disappear according to necessary

condition of inseparability from the null game in D

v(S) = v(N \ S) (Proposition 4.1). Also, the free-

dom degrees for coalitions with size from s = 3 to

s = n/2 disappear according to necessary condition

v(S) =

∑

T ⊂S, |T |=2

v(T ) ∀S ⊆ N with 3 ≤ |S| ≤ n/2

(Proposition 4.2).

Only the





−n freedom degrees for coalition size

s = 2 in C

remain in vector subspace D

. Then,

the vector subspace spanned by the





− n expanded

commutation games agrees with D

. 

6 CONCLUSION

It is known that every cooperative game with two or

three players is separable from the null game by semi-

values, so that dimension for the shared kernel C

zero in cases n = 2,3. Consequently, vector subspace

is only formed by the null game in cases n = 2,3.

For games with four players, Proposition 3.3 proves

that both separability concepts coincide: D

= C

for

n = 4.

Table 1 compares dimensions of C

and D

for

cooperative games with few players.

Table 1: Dimensions of kernels according to N.

|N| = n 2 3 4 5 6 7 8

dimG

3 7 15 31 63 127 255

dimC

0 0 2 10 32 84 198

dimD

0 0 2 5 9 14 20

For games with ﬁve or more players, the intro-

duction of modiﬁed semivalues for games with coali-

tion structure allows us to reduce in a signiﬁcant way

the dimension of the vector subspace of inseparable

games from the null game. According to the linearity

property, separability between two games is reduced

by both concepts of solution to separability of their

difference from the null game. The ability of sep-

aration by semivalues has considerably increased by

introduction of a priori coalition structures.

ACKNOWLEDGEMENTS

Research supported by grant MTM2015-66818-P

from the Spanish Ministry of Economy and FEDER.

REFERENCES

Albizuri, M. J. (2009). Generalized coalitional semivalues.

European Journal of Operational Research, 196:578–

584.

Amer, R., Derks, J., and Gim

enez, J. M. (2003). On cooper-

ative games, inseparable by semivalues. International

Journal of Game Theory, 32:181–188.

Amer, R. and Gim

enez, J. M. (2003). Modiﬁcation of semi-

values for games with coalition structures. Theory and

Decision, 54:185–205.

Banzhaf, J. F. (1965). Weighted voting doesn’t work: A

mathematical analysis. Rutgers Law Review, 19:317–

343.

Carreras, F., Freixas, J., and Puente, M. A. (2003). Semi-

values as power indices. European Journal of Opera-

tional Research, 149:676–687.

Carreras, F. and Gim

enez, J. M. (2011). Power and poten-

tial maps induced by any semivalue: Some algebraic

properties and computation by multilinear extensions.

European Journal of Operational Research, 211:148–

159.

Dragan, I. (1999). Potential and consistency for semivalues

of ﬁnite cooperative TU games. International Journal

of Mathematics, Game Theory and Algebra, 9:85–97.

Dubey, P., Neyman, A., and Weber, R. J. (1981). Value

theory without efﬁciency. Mathematics of Operations

Research, 6:122–128.

Gim

enez, J. M. and Puente, M. A. (2015). A method to

calculate generalized mixed modiﬁed semivalues: ap-

plication to the catalan parliament (legislature 2012-

2016). TOP, 23:669–684.

Owen, G. (1972). Multilinear extensions of games. Man-

agement Science, 18:64–79.

Owen, G. (1975). Multilinear extensions and the Banzhaf

value. Naval Research Logistics Quarterly, 22:741–

750.

Owen, G. (1977). Values of games with a priori unions. In

Essays in Mathematical Economics and Game The-

ory, pages 76–88. R. Henn and O. Moeschelin (Edi-

tors), Springer-Verlag.

Owen, G. (1981). Modiﬁcation of the Banzhaf-Coleman in-

dex for games with a priori unions. In Power, Voting

and Voting Power, pages 232–238. M.J. Holler (Edi-

tor), Physica-Verlag.

Shapley, L. S. (1953). A value for n-person games. In Con-

tributions to the Theory of Games II, pages 307–317.

H.W. Kuhn and A.W. Tucker (Editors), Princeton Uni-

versity Press.

Weber, R. J. (1988). Probabilistic values for games. In The

Shapley value: Essays in honor of L.S. Shapley, pages

101–119. A.E. Roth (Editor), Cambridge University

Press.

Ability to Separate Situations with a Priori Coalition Structures by Means of Symmetric Solutions

249