Cooperation Tendencies and Evaluation of Games

Francesc Carreras

and Maria Albina Puente

ETSEIAT, Technical University of Catalonia, Colom 11, Terrassa, Spain

EPSEM, Technical University of Catalonia, Manresa, Spain

Keywords:

Cooperative Game, Shapley Value, Probabilistic Value, Binomial Semivalue.

Abstract:

Multinomial probabilistic values were ﬁrst introduced by one of us in reliability and later on by the other,

independently, as power indices. Here we study them on cooperative games from several viewpoints, and es-

pecially as a powerful generalization of binomial semivalues. We establish a dimensional comparison between

multinomial values and binomial semivalues and provide two characterizations within the class of probabilis-

tic values: one for each multinomial value and another for the whole family. An example illustrates their use

in practice as power indices.

1 INTRODUCTION

Weber’s general model for assessing cooperative

games (Weber, 1988) is based on probabilistic val-

ues, a family of values axiomatically characterized by

means of linearity, positivity, and the dummy player

property. Every probabilistic value allocates, to each

player in each game of its domain, a weighted (con-

vex) sum of the marginal contributions of the player

in the game. These allocations can be interpreted as a

measure of players’ bargaining relative strength. The

most conspicuous member of this family (in fact, the

inspiring one) is the Shapley value (Shapley, 1953).

In the present paper we study a subfamily of proba-

bilistic values that we call multinomial (probabilistic)

values.

Technically, their main characteristic is the

systematic generation of the weighting coefﬁcients in

terms of a few parameters (one parameter per player).

Our research group has been studying semival-

ues (Dubey et al., 1981), a subfamily of probabilis-

tic values characterized by anonymity and including

the Shapley value as the only efﬁcient member. In

the analysis of certain cooperative problems we have

successfully used binomial semivalues (Freixas and

Puente, 2002), a monoparametric subfamily that in-

cludes the Banzhaf value (Owen, 1975).

From this experience, we feel that multinomial

They were introduced in reliability (Freixas and

Puente, 2002) with the name of “multibinary probabilistic

values” and independently deﬁned for simple games only

—i.e. as power indices— in a work on decisiveness (Car-

reras, 2004), where they were called “Banzhaf α–indices.”

values (n parameters, n being the number of players)

offer a deal of ﬂexibility clearly greater than binomial

semivalues (one parameter), and hence many more

possibilities to introduce additional information when

evaluating a game. Fig. 1 describes the relationships

between the above values and families of values and

the main characteristics of each one of them.

Probabilistic values provide tools to study not

only games in abstracto (i.e. from a merely structural

viewpoint) but also the inﬂuence of players’ person-

ality on the issue. They are assessment techniques

that do not modify the game but only the criteria by

which payoffs are allocated. Parameters will be ad-

dressed here to cope with different attitudes the play-

ers may hold when playing a given game, even if they

are not individuals but countries, enterprises, parties,

trade unions, or collectivities of any other kind. We

will attach to parameter p

the meaning of generical

tendency of player i to form coalitions, assuming p

and p

independent of each other if i 6= j.

Multinomial values are a consistent alternative or

complement to classical values (Shapley, Banzhaf).

They represent a wide generalization of binomial

semivalues, whose monoparametric condition implies

a quite limited capability of analysis of cooperation

tendencies. Of course, these tendencies can neither

be analyzed, without modifying the game, by means

of the classical values, which can be concerned only

with the structure of the game.

The organization of the paper is as follows. Sec-

tion 2 includes a minimum of preliminaries. In Sec-

tion 3, we introduce multinomial values. Section 4

415

Carreras F. and Puente M..

Cooperation Tendencies and Evaluation of Games.

DOI: 10.5220/0004414104150422

In Proceedings of the 15th International Conference on Enterprise Information Systems (ICEIS-2013), pages 415-422

ISBN: 978-989-8565-59-4

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

linearity

positivity

dummy player property + anonymity + efﬁciency

Probabilistic values ← Semivalues ← Shapley value

↑ ↑

Multinomial values ← Binomial semivalues ← Banzhaf value

n parameters (one per player) 1 parameter parameter = 1/2

Figure 1: Inclusion relationships between values and families of values.

contains a result on the dimension of the subspace

spanned by multinomial values and two characteriza-

tions: one, individual, for each multinomial value; an-

other, collective, for the whole subfamily they form.

For space reasons, proofs have been omitted. Section

5 is devoted to analyze a political problem by using

these values.

2 PRELIMINARIES

Let N be a ﬁnite set of players, usually denoted as

N = {1, 2,... ,n}. A (cooperative) game in N is a

function v that assigns a real number v(S) to each

coalition S ⊆ N, with v(

0) = 0. This number is un-

derstood as the utility that coalition S can obtain by

itself, that is, independently of the remaining players’

behaviour.

Game v is monotonic if v(S) ≤ v(T ) when S ⊂

T ⊆ N. Player i ∈ N is a dummy in v if v(S ∪ {i}) =

v(S) + v({i}) for all S ⊆ N\{i}. Players i, j ∈ N

are symmetric in v if v(S ∪ {i}) = v(S ∪ { j}) for all

S ⊆ N\{i, j}.

Endowed with the natural operations for real–

valued functions, v+v

and λv for all λ ∈ R, the set of

all cooperative games in N is a vector space G

. For

every nonempty T ⊆ N, the unanimity game u

in N

is deﬁned by u

(S) = 1 if T ⊆ S and u

(S) = 0 oth-

erwise, and it is easily checked that the set of all una-

nimity games is a basis for G

, so dimG

= 2

− 1.

By a value on G

we mean a map g : G

→ R

which assigns to every game v a vector g[v] with com-

ponents g

[v] for all i ∈ N. The total power of value g

in v is

(v) =

∑

i∈N

[v]. (1)

Following Weber’s axiomatic deﬁnition (Weber,

1988), φ : G

→ R

is a (group) probabilistic value if

it satisﬁes the following properties:

(i) linearity: φ[v + v

] = φ[v] + φ[v

] and φ[λv] =

λφ[v] for all v, v

∈ G

and λ ∈ R;

(ii) positivity

: if v is monotonic, then φ[v] ≥ 0;

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

in game v, then φ

[v] = v({i}).

There is an interesting characterization of the

probabilistic values (Weber, 1988): (a) given a set

P = {p

: i ∈ N, S ⊆ N\{i}} of n2

n−1

weighting co-

efﬁcients, such that

all p

≥ 0 and

∑

S⊆N\{i}

= 1 for each i, (2)

the expression

[v] =

∑

S⊆N\{i}

[v(S ∪ {i}) − v(S)] (3)

for all i ∈ N and v ∈ G

deﬁnes a probabilistic value

φ on G

; (b) conversely, every probabilistic value can

be obtained in this way; (c) the correspondence given

by P 7→ φ is one–to–one. Thus, the payoff that a prob-

abilistic value allocates to each player in any game is

a weighted sum of the marginal contributions of the

player in the game. We quote (Weber, 1988):

“Let player i view his participation in a

game v as consisting merely of joining some

coalition S and then receiving as a reward his

marginal contribution to the coalition. If p

the probability that he joins coalition S, then

[v] is his expected payoff from the game.”

The action of a probabilistic value φ on the basis

of unanimity games is as follows: if

0 6= T ⊆ N then

] =

∑

S⊆N\{i}:

T \{i}⊆S

if i ∈ T (4)

and φ

] = 0 otherwise.

Weber calls monotonicity to this property, but we prefer

to call to it positivity (Dubey et al., 1981).

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416

Among probabilistic values, semivalues (Dubey et

al., 1981) are characterized by the anonymity prop-

erty: there is a vector {p

}

n−1

s=0

such that p

= p

for

all i ∈ N and all S ⊆ N\{i}, where s = |S|, so that all

coalitions of a given size share a common weight and

Eq. (3) reduces therefore to

[v] =

∑

S⊆N\{i}

[v(S ∪ {i}) − v(S)]

for all i ∈ N and v ∈ G

. The weighting coefﬁcients

}

n−1

s=0

of any semivalue φ satisfy two characteris-

tic conditions, derived from Eq. (2): each p

≥ 0 and

∑

n−1

s=0



n−1



= 1.

Among semivalues, the Shapley value (Shapley,

1953), denoted here by ϕ and deﬁned by p



n−1



n for all s, is the only efﬁcient semivalue, in

the sense that

∑

i∈N

[v] = v(N) for every v ∈ G

. The

Banzhaf value (Owen, 1975), denoted here by β and

deﬁned by p

= 1/2

n−1

for all s, is the only semivalue

satisfying the total power property: for every v ∈ G

∑

i∈N

[v] =

n−1

∑

S⊆N

∑

i/∈S

[v(S ∪ {i}) − v(S)]. (5)

The multilinear extension (Owen, 1972) of a game

v ∈ G

is the real–valued function deﬁned in R

f (x

,. .. ,x

) =

∑

S⊆N

∏

i∈S

∏

j∈N\S

(1 − x

)v(S).

As is well known, both the Shapley and Banzhaf val-

ues of any game v can be obtained from its multilinear

extension. Indeed, ϕ[v] can be calculated by integrat-

ing the partial derivatives of the multilinear extension

of the game along the main diagonal x

= x

= ··· =

of the cube [0,1]

(Owen, 1972) while the partial

derivatives of that multilinear extension, evaluated at

point (1/2,1/2, .. ., 1/2), give β[v] (Owen, 1975).

3 MULTINOMIAL VALUES

Deﬁnition 3.1. Set N = {1,2, .. ., n} and let a proﬁle

p ∈ [0,1]

, that is, p = (p

, p

,. .. , p

) with 0 ≤ p

≤ 1

for i = 1, 2,. .. ,n, be given. Then the coefﬁcients

∏

j∈S

∏

k∈N\S

k6=i

(1 − p

) (6)

for all i ∈ N and S ⊆ N\{i} (where the empty product,

arising if S =

0 or S = N\{i}, is taken to be 1) deﬁne

(Freixas and Puente, 2002) a probabilistic value on

that we call the p–multinomial value λ

. Thus,

[v] =

∑

S⊆N\{i}

∏

j∈S

∏

k∈N\S

k6=i

(1 − p

)[v(S ∪ {i}) − v(S)]

for all i ∈ N and v ∈ G

As was announced in Section 1, we will attach

to p

the meaning of generical tendency of player i

to form coalitions, and thus we will say that p is a

tendency proﬁle on N. According to Eq. (6), coef-

ﬁcient p

, the probability of i to join S, will depend

on the positive tendencies of the members of S to

form coalitions and also on the negative tendencies

in this sense of the outside players, i.e. the members

of N\(S ∪ {i}).

Remarks 3.2. (a) For example, for n = 2 we have

p = (p

, p

) and, if i 6= j,

[v] = (1 − p

)[v({i}) − v(

0)] + p

[v(N) − v({ j})].

Hence, the allocation given by λ

to player i does

not depend on p

but only on p

. If player j is not

greatly interested in cooperating (say, p

tends to 0),

player i’s allocation will tend to his individual utility

v({i}). Instead, if player j is highly interested in co-

operating (say, p

tends to 1), player i’s allocation will

tend to his marginal contribution to the grand coali-

tion v(N) − v({ j}).

(b) It is easy to check that the action of λ

on any

unanimity game u

is given by:

] =

∏

j∈T

j6=i

if i ∈ T (7)

and λ

] = 0 otherwise. Using Eq. (7), it readily

follows that, for n ≥ 2, p 6= p

implies λ

6= λ

(if

n = 1 all proﬁles give rise to a unique multinomial

value).

= p

= ·· · = p

q for some q ∈ [0,1], coefﬁcients p

reduce, for all

i ∈ N, to

= p

= q

(1 − q)

n−s−1

for s = 0,1, .. ., n − 1,

where s = |S| and 0

= 1 by convention in cases q =

0 and q = 1. These coefﬁcients {p

}

n−1

s=0

deﬁne the

q–binomial semivalue ψ

(Freixas and Puente, 2002)

and, obviously, λ

= ψ

. If, moreover, q = 1/2 then

we obtain ψ

1/2

= β, the Banzhaf value.

(d) The multilinear extension procedure extends

well to all binomial semivalues and even to any multi-

nomial value λ

(Freixas and Puente, 2002): if f is the

multilinear extension of game v ∈ G

then

[v] =

∂ f

∂x

, p

,. .. , p

) for all i ∈ N.

4 THEORETICAL RESULTS

We devote this section to extending three results

stated in the previous literature on binomial semival-

ues. In all cases the extension is not straightforward

CooperationTendenciesandEvaluationofGames

417

and reveals new features of multinomial values. We

assume n ≥ 2 because for n = 1 all is trivial.

4.1 About Dimensions

Let L(G

) denote the space of all linear maps

from G

to R

, which includes most values studied

in the literature. It is clear that dimL(G

) =

n(2

− 1). Let BS(G

) denote the subspace

spanned by binomial semivalues. It is known that

dimBS(G

) = n (Freixas and Puente, 2002).

Moreover, it coincides with the subspace spanned by

all semivalues, and any n different binomial semival-

ues ψ

,ψ

,. .. ,ψ

form a basis.

Now, let M V (G

) denote the subspace

spanned by multinomial values. We shall determine

its dimension. To this end, the following auxiliar no-

tion is useful (and a basis for this subspace is found

during the proof).

Deﬁnition 4.1. A value g on G

satisﬁes the property

of neutrality (for unanimity games) if, for each T ⊆ N

with 0 ≤ |T | ≤ n − 2,

T ∪{i}

] = g

T ∪{ j}

] for any i, j /∈ T .

This property is satisﬁed by any multinomial value

since, by Remark 3.2(b), we have

T ∪{i}

] =

∏

k∈T

= λ

T ∪{ j}

Theorem 4.2. Let M V (G

) be the subspace

spanned by multinomial values within the space

L(G

) of linear maps. Then dimM V (G

) =

− 1.

The difference between n = dim BS (G

) and

− 1 = dim M V (G

) reﬂects the much greater

versatility of multinomial values.

4.2 Individual Characterization of each

Multinomial Value

The notion of total power given by Eq. (1) has been

proven to be fruitful in absence of efﬁciency. For ex-

ample, when applying a normalization process to a

value. The total power property of the Banzhaf value

given by Eq. (5) was the natural substitute of efﬁ-

ciency in well–known axiomatic characterizations of

this value (e.g. Feltkamp, 1995). It was extended to

all binomial semivalues (Carreras and Puente, 2012),

giving rise to the q–binomial total power property:

∑

i∈N

[v] =

∑

S⊆N

(1 − q)

n−s−1

∑

i/∈S

[v(S ∪ {i}) − v(S)]

In particular, all binomial semivalues, but also the

Shapley value, satisfy this property.

for every v ∈ G

For each q ∈ [0,1], this property characterizes the

q–binomial semivalue ψ

among semivalues, and this

characterization can be alternatively stated as follows:

if ψ is a semivalue such that

∑

i∈N

[v] =

∑

i∈N

[v]

for all v ∈ G

then ψ = ψ

. The natural extension

of the property to probabilistic values must be carried

out in the following terms.

Deﬁnition 4.3. Let p ∈ [0,1]

be a proﬁle on N. A

(probabilistic or not) value g on G

satisﬁes the p–

multinomial total power property if, for all v ∈ G

∑

i∈N

[v] =

∑

S⊆N

∑

i/∈S

∏

j∈S

∏

k∈N\S

k6=i

(1 − p

)[v(S ∪ {i}) − v(S)].

(8)

However, this property, clearly equivalent to

∑

i∈N

[v] =

∑

i∈N

[v] and hence obviously satisﬁed

by the p–multinomial value λ

, does not characterize

it within the class of probabilistic values. Indeed, it is

easy to see, e.g. for n = 2 and using Eqs. (4) and (7),

that in general not only λ

but also inﬁnitely many

probabilistic values satisfy Eq. (8) for a given p.

Therefore, we need to introduce a second prop-

erty in order to characterize each λ

within the class

of probabilistic values. The reader will notice that,

due to anonymity, this property holds for all binomial

semivalues and hence it was irrelevant for them.

Deﬁnition 4.4. Let p ∈ [0,1]

be a proﬁle on N. A

value g on G

satisﬁes the property of p–weighted

payoffs for unanimity games if, for every nonempty

T ⊆ N,

] = p

] for all i, j ∈ T .

By Eq. (7) it is clear that λ

satisﬁes this property.

Theorem 4.5. (Characterization of each p–multino-

-mial value). Let p be a proﬁle on N. Then the

unique probabilistic value on G

that satisﬁes the p–

multinomial total power property and the property of

p–weighted payoffs for unanimity games is the multi-

nomial value λ

4.3 Collective Characterization

of Multinomial Values

Among semivalues, the binomial family is character-

ized by the monotonicity of the weighting coefﬁcients

(Alonso et al., 2007): a semivalue ψ on G

is bino-

mial if and only if its weighting coefﬁcients {p

}

n−1

s=0

are in geometric progression, i.e. satisfy, for some

µ, the condition p

s+1

= µp

for s = 0,1,2,...,n − 2

ICEIS2013-15thInternationalConferenceonEnterpriseInformationSystems

418

(maybe the simplest form of monotonicity).

The ex-

tension, not completely straightforward, will be given

by Theorem 4.9. To this end, we need to consider two

special types of players with regard to the weighting

coefﬁcients of a probabilistic value.

Deﬁnition 4.6. Let φ be a probabilistic value on G

with weighting coefﬁcients {p

• A player h ∈ N is a φ–ordinary player

if there

is µ

≥ 0 such that, for all i ∈ N, p

= µ

S\{h}

whenever h ∈ S ⊆ N\{i}.

• A player h ∈ N is a φ–magnetic player if p

S\{h}

0 whenever h ∈ S ⊆ N\{i}. This condition is

equivalent to saying that p

= 0 for all S ⊆

N\{i,h}.

Examples 4.7. (a) For the Banzhaf value β, all play-

ers are ordinary, with µ

= 1 for all of them. The

same happens for every binomial semivalue ψ

, with

= q/(1 − q), unless q = 1 (marginal index), in

which case all players are magnetic.

(b) The Shapley value ϕ does not admit magnetic

players. For n = 2 both players are ordinary, with

= 1. For n > 2 there are not ordinary players.

abilistic value φ, players 1 and 2 are ordinary and

player 3 is magnetic. Then we have, for some µ

,µ

the links given by Table 1. Imposing Eq. (2) yields

the relevant weighting coefﬁcients in terms of µ

,µ

{3}

1 + µ

, p

{3}

1 + µ

+ µ

Choosing, for example, µ

= 1/2 and µ

= 1, we ob-

tain all weighting coefﬁcients and hence a probabilis-

tic value.

Remarks 4.8. (a) The conditions of Deﬁnition 4.6 are

incompatible. If there were a simultaneously ordinary

and magnetic player h then, for any other i ∈ N, we

would have p

= 0 for all S ⊆ N\{i}, contradicting

that these coefﬁcients sum up to 1.

Strictly speaking, the condition is as follows: (i)

s+1

= µp

for all s or (ii) p

= µ

s+1

for all s. The dic-

tatorial index ψ

satisﬁes (i) only, with p

= 1 and µ = 0.

The marginal index ψ

satisﬁes (ii) only, with p

n−1

= 1 and

= 0. Any other binomial semivalue, with q 6= 0,1, satis-

ﬁes (i) and (ii) because µ =

1 − q

6= 0; thus, q =

1 + µ

and

(1 + µ)

n−1

We use this term to emphasize that exceptionality cor-

responds to the next option, that of magnetic player.

(b) The condition of ordinary player means that

the relation between p

and p

S\{h}

follows a pattern

common to all i ∈ N and very similar to the mono-

tonicity in the binomial semivalue case, although the

proportionality factor depends here on player h.

implies that none of the other players would join a

coalition excluding h.

(d) Let φ = λ

, a multinomial value. Then player

h ∈ N is φ–ordinary if p

< 1, and φ–magnetic if p

1. This follows from the proof of the next result.

Theorem 4.9. (Collective characterization of all

multinomial values). A probabilistic value φ on G

is a multinomial value if and only if all players h ∈ N

are φ–ordinary or φ–magnetic. In this case, φ = λ

where p = (p

, p

,. .. , p

) is given by











1 + µ

if h is a φ–ordinary player,

1 if h is a φ–magnetic player.

The difference between monotonicity at individ-

ual level established in Theorem 4.9 and the uniform

monotonicity that characterizes binomial semivalues

is a new good sample of the higher versatility of the

multinomial values.

Examples 4.10. (a) The Shapley value is multinomial

only for n = 2. In fact, in this case ϕ and β coincide.

(b) According to Theorem 4.9, the value obtained

in Example 4.7(c) is multinomial. From µ

and µ

get the proﬁle that deﬁnes it: p = (1/3, 1/2, 1).

ﬁned by the weighting coefﬁcients

= 0, p

{2}

= 0, p

{3}

= 0.2,

{2,3}

= 0.8, p

= 0, p

{1}

= 0,

{3}

= 0.8, p

{1,3}

= 0.2, p

= 0.4,

{1}

= 0.1, p

{2}

= 0.4, p

{1,2}

= 0.1.

It is easy to check that, with regard to φ, player 1 is

ordinary (with µ

= 1/4) and player 3 is magnetic, but

player 2 is neither ordinary nor magnetic. Then, using

again Theorem 4.9, φ is not a multinomial value.

5 A POLITICAL EXAMPLE:

IDEOLOGICAL CONSTRAINTS

The model based on multinomial values is able to en-

compass additional information due to ideological re-

CooperationTendenciesandEvaluationofGames

419

Table 1: Links between weighting coefﬁcients in Example 4.7(c).

= 0

−→ p

{2}

= µ

= 0, p

{3}

−→ p

{2,3}

= µ

{3}

= 0

−→ p

{1}

= µ

= 0, p

{3}

−→ p

{1,3}

= µ

{3}

−→ p

{1}

= µ

, p

{1}

−→ p

{1,2}

= µ

{1}

−→ p

{2}

= µ

, p

{2}

−→ p

{1,2}

= µ

{2}

strictions. We will discuss here a political problem

described by a simple game.

We recall that v is a simple game if it is monotonic

and v(S) = 0 or 1 for all S ⊆ N. It is determined by

the set W (v) = {S ⊆ N : v(S) = 1} of winning coali-

tions and even by the subset W

(v) = {S ∈ W (v) :

T /∈ W (v) if T ⊂ S} of minimal winning coalitions. In

particular, v is a weighted majority game if there ex-

ist a quota q > 0 and weights w

,. .. ,w

≥ 0 such

that S ∈ W (v) if and only if

∑

i∈S

≥ q. We denote

this fact by v ≡ [q; w

,. .. ,w

Example 5.1. We consider a 50–member parliamen-

tary body with n = 4 parties and a seat distribution

of 21, 18, 7 and 4 seats, respectively. Assuming that

voting is ruled by absolute majority and voting disci-

pline holds within each party, its structure is described

by the weighted majority game v ≡ [26; 21,18,7,4].

The family of minimal winning coalitions is

(v) = {{1, 2},{1, 3},{2, 3,4}}, so that W (v) =

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

{1,2, 3,4}} is the family of all winning coalitions.

The expression of game v in terms of unanimity

games is

v = u

{1,2}

{1,3}

−u

{1,2,3}

{2,3,4}

−u

{1,2,3,4}

. (9)

Let us assume that the basic ideological feature is

deﬁned by a classical left–to–right axis

where parties

can be precisely located as for example in Fig. 2.

As to the additional information given by ideological

constraints in politics, it is worthy of mention, at least inci-

dentally, a singular example. In the general elections held in

Greece in May 7 and June 17, 2012, the willingness of the

parties to form any coalition was being, due to Greek econ-

omy’s dramatic situation, much more decisive than the ide-

ological constraints. Our model might well apply to study

this situation. The proﬁle components after May 7 were

very low and led to an impasse, whereas they increased af-

ter June 17 and gave rise, ﬁnally, to a coalition government.

A similar scheme could be applied if the relevant notion

were nationalism (vs. centralism), as for example in regions

like Catalonia or the Basque Country. Higher–dimensional

ideological spaces might be treated in a similar but more

complicated way.

The Shapley value yields the following evaluation

of the game:

ϕ[v] = (5/12, 3/12, 3/12, 1/12).

It is clear that the Shapley value strictly represents the

relative strength of each party in the game, disregard-

ing the effect, in the coalition formation process, due

to the ideological positions of the involved parties.

We wish to incorporate this exogenous information to

the evaluation of the game by using a suitable proba-

bilistic value.

Any probabilistic value φ is deﬁned by a set {p

}

of weighting coefﬁcients for all i ∈ N and all S ⊆

N\{i}. For each i ∈ N, coefﬁcients {p

} must pro-

vide a probability distribution on the family of coali-

tions S ⊆ N\{i}. In our case (n = 4), 32 coefﬁcients

are needed in principle. However, since the game

is simple, we only have to deﬁne p

when i is crucial

for S ∪{i} in v, i.e. when S /∈ W (v) but S ∪{i} ∈ W (v)

(we will set S ∈ C

(i) to denote this fact). This reduces

the set to 12 coefﬁcients,

{2}

, p

{3}

, p

{2,3}

, p

{2,4}

, p

{3,4}

{1}

, p

{1,4}

, p

{3,4}

, p

{1}

, p

{1,4}

, p

{2,4}

, p

{2,3}

and there are restrictions in choosing these coefﬁ-

cients for each S ∈ C

(i):

all p

≥ 0 and

∑

S∈C

(i)

≤ 1 for each i.

Once the coefﬁcients are chosen, we will simply have,

from Eq. (3),

[v] =

∑

S∈C

(i)

. (10)

Note that: (a) φ

[v] ≤ 1 for all i, and (b) the total power

∑

i∈N

[v] ≤ n.

Given {p

}, let q

(v) be the probability that i joins

any coalition S /∈ C

(i), i.e. such that i is not crucial

in S ∪{i}. This is the amount of irrelevant probability

that we may leave undeﬁned. Then, from Eq. (10)

it follows that φ

[v] = 1 − q

(v). Thus, the more is

probability q

(v) the less is the allocation that player

ICEIS2013-15thInternationalConferenceonEnterpriseInformationSystems

420

• • • •

4 1 2

0.1 0.4 0.6 0.80 1

(left)

(right)

Figure 2: Party–distribution on a left–to–right axis.

i will get according to the corresponding probabilistic

value.

How should we take into account the ideological

constraints? At this point, it is worthy of mention that,

in Weber’s general model, p

may well depend not

only on i’s interest in forming coalition S ∪ {i} but

also on the opinion of the members of S as to join-

ing (accepting) i. In other words, coefﬁcient p

needs

not being only a choice of i himself. The multinomial

values offer a reasonable solution to this since, given

a proﬁle p = (p

, p

,. .. , p

), the corresponding value

is deﬁned by means of Eq. (6).

Thus, we will use multinomial values. It remains

only to choose the proﬁle p = (p

, p

,. .. , p

) in terms

of Fig. 2. An alternative interpretation of the proﬁle in

simple games is as follows (Carreras, 2004). There is

a status quo Q and a proposal P to modify it. The ac-

tion of the parliamentary members reduces to vote for

or against P. Then each p

can be viewed as the prob-

ability that player i votes for P. Since the result of a

voting is essentially equivalent to forming a coalition

(the coalition of players that vote for P), this interpre-

tation of p

agrees with that of “tendency to form a

coalition” that we are using in this paper.

Step 1. Additional Assumption. Initially, we will

assume that the coalition to form will also have an

ideological degree µ, such that 0 ≤ µ ≤ 1. Then, it is

natural to take p

as the level of agreement of party i

with this “coalitional” degree, i.e.

= 1 − |µ − µ

|, (11)

where µ

is the position of party i in Fig.2. This is a

simple but not too radical assumption. If µ

≤ µ then

can vary between 1 − µ and 1, whereas if µ ≤ µ

then p

can vary between µ and 1. As extreme cases,

= 0 if and only if either µ = 0 and µ

= 1 or µ

= 0

and µ = 1, and p

= 1 if and only if µ = µ

Step 2. A Particular Case. E.g., let us take µ = 0.5.

Then, by Eq. (11),

= 0.9, p

= 0.7, p

= 0.6.

The weighting coefﬁcients are given by Eq. (6).

To compute λ

[v] we can use Eq. (10) or, directly,

Eq. (9), linearity, and the action of a multinomial

value on unanimity games, given by Eq. (7). Then

we obtain

[v] = 0.592, λ

[v] = 0.312,

[v] = 0.144, λ

[v] = 0.063.

Table 2: Parameters p

, p

as functions of µ.

µ p

[0,0.1] 0.6 + µ 0.4 + µ 0.2 + µ 0.9 + µ

[0.1,0.4] 0.6 + µ 0.4 + µ 0.2 + µ 1.1 − µ

[0.4,0.6] 1.4 − µ 0.4 + µ 0.2 + µ 1.1 − µ

[0.6,0.8] 1.4 − µ 1.6 − µ 0.2 + µ 1.1 − µ

[0.8,1] 1.4 − µ 1.6 − µ 1.8 − µ 1.1 − µ

These allocations are the result of combining both the

strategic position of each party in the game and its ide-

ological relevance in forming a “politically balanced”

coalition (µ = 0.5). Notice that the symmetry of 2

and 3 in the game, reﬂected by the Shapley value, has

been broken. The total power is π

(v) = 1.111.

Looking at q

(v) we ﬁnd

(v) = 0.408, q

(v) = 0.688,

(v) = 0.856, q

(v) = 0.937.

These amounts represent the probability wasted by

each party in joining coalitions where it is not cru-

cial. For example, party 1 is not crucial in {1}, {1,4}

and {1,2, 3,4}, and q

(v) is therefore the probability

that party 1 joins

0, {4} or {2,3,4}. This waste of

probability is the effect of the choice of p

but also of

, p

Step 3. Arbitrary Ideological Degree. Now we pro-

ceed for a general µ. Then, from Eq. (11), we have

the results displayed in Table 2. So we get the multi-

nomial value λ

[v] in terms of µ:

[v] =











−µ

− 2.5µ

+ 0.78µ + 0.448,

− 1.5µ

+ 0.82µ + 0.432,

−µ

+ 3.5µ

− 2.62µ + 1.128,

− 5.5µ

+ 8.02µ − 2.648,

if, respectively,

0 ≤ µ ≤ 0.1, 0.1 ≤ µ ≤ 0.6,

0.6 ≤ µ ≤ 0.8, 0.8 ≤ µ ≤ 1,

and similar expressions for the remaining values λ

[v]

for i = 2, 3,4.

Finally, if we wish to aggregate these results and

obtain a single evaluation of the relative strength of

each party in the coalition formation process in ab-

stracto, i.e. without prescribing any ideological de-

gree µ to the coalition, it sufﬁces to integrate the

multinomial value of each party with respect to µ, thus

getting

[v] =

[v]dµ ≈ 0.6333

CooperationTendenciesandEvaluationofGames

421

and, similarly,

[v] ≈ 0.3365, ξ

[v] ≈ 0.2681, ξ

[v] ≈ 0.1393.

Remark 5.2 An important difference between the

Shapley value assessment and the result of applying

a (multinomial or not) probabilistic value is that the

former is efﬁcient whereas the latter, in general, is not

(Weber, 1988). For this reason we speak of relative

strength. If the results have to be applied to shar-

ing political responsibilities, they can be directly ap-

plied in the Shapley value case by efﬁciency, whereas

a normalization process, similar to that of the original

Banzhaf power index, is needed in the multinomial

case, by deﬁning

[v] =

[v]

(v)

[v]

∑

j∈N

[v]

for each i ∈ N and any v ∈ G

for which this nor-

malization makes sense. The normalization may of

course be applied also to the single evaluation ξ[v] ob-

tained in Step 3, giving normalized values

[v] ≈ 0.4598, ξ

[v] ≈ 0.2443,

[v] ≈ 0.1947, ξ

[v] ≈ 0.1012.

Which is therefore the meaning of the results ob-

tained in Step 3? In the same way as one accepts the

Shapley value of the game as an a priori evaluation

of the relative strength of each player in the coalition

formation bargaining, the values just obtained repre-

sent an analogous a priori evaluation of this relative

strength when the political relationships between the

parties are taken into account. The differences be-

tween our (normalized) assessment and the mere eval-

uation of the game provided by the Shapley value are

interesting: if ∆

[v] = ξ

[v] − ϕ

[v] for all i, then

∆

[v] = 0.0431, ∆

[v] = −0.0057,

∆

[v] = −0.0553, ∆

[v] = 0.0179.

This indicates that the political relationships in

this particular game improve party 1 strongly (around

10.34%) and party 4 very strongly (around 21.37%),

while they damage party 2 very slightly (around

2.28%) and party 3 very strongly (around 22.12%).

ACKNOWLEDGEMENTS

Research supported by Grant SGR 2009–01029 of the

Catalonia Government (Generalitat de Catalunya)

and Grant MTM 2012–34426 of the Economy and

Competitiveness Spanish Ministry.

REFERENCES

Alonso, J. M., Carreras, F. and Puente, M. A. (2007): “Ax-

iomatic characterizations of the symmetric coalitional

binomial semivalues.” Discrete Applied Mathematics

155, 2282–2293.

Carreras, F. (2004): “α–decisiveness in simple games.”

Theory and Decision 56, 77–91. Also in: Essays

on Cooperative Games (G. Gambarelli, ed.), Kluwer

Academic Publishers, 77–91.

Carreras, F. and Puente, M. A. (2012): “Symmetric coali-

tional binomial semivalues.” Group Decision and Ne-

gotiation 21, 637–662.

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

theory without efﬁciency.” Mathematics of Operations

Research 6, 122–128.

Feltkamp, V. (1995): “Alternative axiomatic characteri-

zations of the Shapley and Banzhaf values.” Interna-

tional Journal of Game Theory 24, 179-186.

Freixas, J. and Puente, M. A. (2002): “Reliability impor-

tance measures of the components in a system based

on semivalues and probabilistic values.” Annals of Op-

erations Research 109, 331–342.

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.

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

Contributions to the Theory of Games II (H.W. Kuhn

and A.W. Tucker, eds.), Princeton University Press,

Annals of Mathematical Studies 28, 307–317.

Weber, R. J. (1988): “Probabilistic values for games.”

In: The Shapley Value: Essays in Honor of Lloyd

S. Shapley (A.E. Roth, ed.), Cambridge University

Press, 101–119.

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