A New Procedure to Calculate the Owen Value

Jos

e Miguel Gim

enez and Mar

ıa Albina Puente

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

Keywords:

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

Abstract:

In this paper we focus on games with a coalition structure. Particularly, we deal with the Owen value, the

coalitional value of the Shapley value, and we provide a computational procedure to calculate this coalitional

value in terms of the multilinear extension of the original game.

1 INTRODUCTION

Shapley (Shapley, 1953) (see also (Roth, 1988) and

(Owen, 1995)) initiated the value theory for coopera-

tive games. The Shapley value applies without restric-

tions and provides, for every game, a single payoff

vector to the players. The restriction of the value to

simple games gives rise to the Shapley–Shubik power

index (Shapley and Shubik, 1954), that was axioma-

tized in (Dubey, 1975) introducing the transfer prop-

erty. As a sort of reaction, Banzhaf (Banzhaf, 1965)

proposed a different power index that Owen (Owen,

1975) extended to a dummy–independent and some-

how “normalized” Banzhaf value for all coopera-

tive games. A nice almost common characterization

of the Shapley and Banzhaf values would be given

in (Feltkamp, 1995).

Games with a coalition structure were introduced

in (Aumann and Dr

eze, 1974), who extended the

Shapley value to this new framework in such a man-

ner that the game really splits into subgames played

by the unions isolatedly from each other, and every

player receives the payoff allocated by the restric-

tion of the Shapley value to the subgame he is play-

ing within his union. A second approach was used

in (Owen, 1977), when introducing and axiomatically

characterizing his coalitional value (Owen value).

The Owen value is the result of a two–step proce-

dure: ﬁrst, the unions play a quotient 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 ap-

plying the Shapley value. Further axiomatizations of

the Owen value have been given in e.g. (Hart and

Kurz, 1983), (Peleg, 1989), (Winter, 1992), (Amer

and Carreras, 1995) and (Amer and Carreras, 2001),

azquez et al., 1997), (V

azquez, 1998), (Hamiache,

1999), (Hamiache, 2001) and (Albizuri, 2002).

Owen applied the same procedure to the Banzhaf

value and obtained the modiﬁed Banzhaf value or

Owen–Banzhaf value (Owen, 1982). In this case

the payoffs at both levels (unions in the quotient

game and players within each union) are given by the

Banzhaf value.

Alonso and Fiestras suggested 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

(Alonso and Fiestras, 2002). That same year, Car-

reras et al. 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 (Amer et al., 2002). Thus, the pos-

sibilities to deﬁne a coalitional value by combining

the Shapley and Banzhaf values were complete at that

moment.

In 1972 Owen introduced the multilinear exten-

sion (Owen, 1972) and applied it to the calculus of

the Shapley value. The computing technique based

on the multilinear extension has been applied to

many values: in 1975 to the Banzhaf value (Owen,

1975); in 1992 to the Owen value (Owen and Win-

ter, 1992); in 1994 to the Owen–Banzhaf value (Car-

reras and Maga

na, 1994); in 1997 to the quotient

game (Carreras and Maga

na, 1997); in 2000 to bino-

mial semivalues and to multinomial probabilistic in-

dices (Puente, 2000); in 2004 to the α–decisiveness

and Banzhaf α–indices (Carreras, 2004); in 2005 to

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

2011 to symmetric coalitional binomial semivalues

228

GimÃl’nez J. and Puente M.

A New Procedure to Calculate the Owen Value.

DOI: 10.5220/0006113702280233

In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 228-233

ISBN: 978-989-758-218-9

(Carreras and Puente, 2011); in 2011 to semival-

ues (Carreras and Gim

enez, 2011); in 2015 to coali-

tional multinomial probabilistic values (Carreras and

Puente, 2015).

The present paper focus on giving a new computa-

tional procedure for the Owen value by means of the

multilinear extension of the game.

The organization of the paper is as follows. In

Section 2, a minimum of preliminaries is provided.

Section 3 is devoted to give a procedure to compute

the Owen value.

2 PRELIMINARIES

2.1 Cooperative Games

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|.

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.

Well known example of value is the Shapley value

ϕ (Shapley (Shapley, 1953)), deﬁned as

[v] =

∑

S⊆N\{i}

[v(S ∪ {i}) − v(S)

for all i ∈ N, v ∈ G

, where s = |S| and p

= 1/n



n−1



Notice that this value is deﬁned for each N. In fact,

it is deﬁned on cardinalities rather than on speciﬁc

player sets: this means the weighting vector {p

}

n−1

s=0

deﬁnes the Shapley value on all N such that n = |N|.

When necessary, we shall write ϕ

(n)

for the Shapley

value on cardinality n and p

for its weighting coef-

ﬁcients. ϕ

(n)

induces values ϕ

(t)

for all cardinalities

t < n, recurrently deﬁned by the Pascal triangle (in-

verse) formula given by Dragan (Dragan, 1997). That

= p

t+1

+ p

t+1

s+1

for 0 ≤ s < t, (1)

The multilinear extension (Owen, 1972) of a

game v ∈ G

is the real–valued function deﬁned on

) =

∑

S⊆N

∏

i∈S

∏

j∈N\S

(1 − x

)v(S). (2)

where X

denotes the set of variables x

for i ∈ N.

As 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).

2.2 Games with Coalition Structure

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

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

A New Procedure to Calculate the Owen Value

229

2.2.1 The Owen Value

The Owen value (Owen (Owen, 1977)) is the coali-

tional value Φ deﬁned by

[v;P] =

∑

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

, Q =

r∈R

and

m−1



m−1



, p

−1



−1



This coalitional value was axiomatically charac-

terized by Owen (Owen, 1977) as the only coalitional

value that satisﬁes the following properties: the natu-

ral extensions to this framework of

• efﬁciency

• additivity

• the dummy player property

and also

• symmetry within unions: if i, j ∈ B

are symmet-

ric in v then

[v;B] = Φ

[v;B]

• symmetry in the quotient game: if B

∈ P are

symmetric in [v;B] then

∑

i∈B

[v;B] =

∑

j∈B

[v;B].

Finally, as Φ is deﬁned 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 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].

3 A COMPUTATIONAL

PROCEDURE TO CALCULATE

THE OWEN VALUE

In this section we present a new computational proce-

dure to calculate this coalitional value. Before that,

we need two previous results that will be given in

Lemma 3.1 and Proposition 3.2.

Lemma 3.1. Let [v; B] ∈ G

, B = {B

,.. .,B

} a

coalition structure in N. The allocations given by Φ

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).

By linearity, for all i ∈ B

[v;B] =

∑

T ⊆N: T 6=

]

and it sufﬁces consider unanimity games u

with

T = V ∪ A

∪ A

∪ ...∪ A

V ⊆ B

, {i

,...,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]. 

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

, the allocation

to a player i belonging to B

in a unanimity game

, T = V ∪ B

∪ ··· ∪ B

, V ⊆ B

and {i

,...,i

} ⊆

M \ { j} is given by

;B] =



ψ/ϕ



;B] =







h+1

v−1

i ∈ T

0 i /∈ T

where (p

h+1

)

s=0

and (p

)

v−1

s=0

are the weighting coef-

ﬁcients of the induced Shapley value and p

h+1

and p

v−1

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

230

Proof For i ∈ T we have

;B] =

∑

R⊆M\{ j}

∑

S⊆B

\{i}

(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] = p

h+1

v−1

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

(Q∪S ∪

{i}) − u

(Q ∪ S) vanish. 

Example 3.1 On the players set N = {1,2,3,4,5,6},

let B = {{1,2,3}, {4,5},{6}} be a coalition structure

on N. We will obtain the allocations to players i ∈ B

according to Φ for the unanimity games u

{1,2,4,6}

and

{1,2,4,5,6}

. They are

{1,2,4,6}

;B] = p

, for i = 1,2 and

{1,2,4,6}

;B] = 0,

where p

and p

are the corresponding

weighting coefﬁcient of the induced Shapley value.

In a similar way and according to Lemma 3.1, for

{1,2,4,5,6}

we obtain

{1,2,4,5,6}

;B] = p

, for i = 1,2 and

{1,2,4,5,6}

;B] = 0,

Notice that the allocations in both games are the

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

intersect the same unions B

and B

In next theorem we present a new method to com-

pute the Owen value by means of the multilinear ex-

tension of the game.

Theorem 3.3. Let [v; B] ∈ G

, B = {B

,.. .,B

}

a coalition structure in N.

Then 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.

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 p

j,v

v−1

and each

product

∏

r∈W

by p

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 many well

known works to obtain the modiﬁed multilinear ex-

tension of union B

. Step 4 shows the modiﬁed mul-

tilinear extension as a linear combination of multilin-

ear extensions of unanimity games. Step 5 weights

each unanimity game according to Proposition 3.2 so

that step 6 gives as usual the marginal contribution of

player i and his allocation Φ

[v;B] is obtained. 

Example 3.2 Let v ≡ [68; 50,21,20,19,13,9,3] be the

7–person weighted majority game and the coalition

structure B = {{1}, {2,3,5}, {4},{6},{7}}. We will

compute Φ[v;B].

The set of minimal winning coalitions of the game

(v) = {{1,2},{1, 3},{1,4}, {1,5,6}},

so that players 2, 3 and 4 on one hand, and 5 and 6 on

the other, are symmetric in v. Moreover, player 7 is

null and the multilinear extension of v is

f (X

) =x

+ x

− x

+ x

− x

−x

+ x

− x

−x

+ x

− x

The coalition structure is

B = {{1},{2,3, 5},{4},{6}, {7}}

and steps 1–4 in Theorem 3.3 give the modiﬁed mul-

tilinear extension of each union B

, for j = 1,2,3, 4

(notice that player 7 is null in v and it is not necessary

to compute g

) = x

+ x

− 2x

+ y

A New Procedure to Calculate the Owen Value

231

) = x

+ x

+ y

− x

+ x

− x

+ x

− x

+ x

− x

) = y

+ x

− 2x

) = y

+ y

− 2y

Step 5 leads to g

for each j = 1,2,3, 4.

) = p

1,1

+ p

1,1

− 2p

1,1

+ p

) =

+ p

− p

+ p

− p

+ p

− p

+ p

− p

+ p

− p

) = p

+ p

− 2p

) = p

+ p

− 2p

Finally, step 6 yields

[v;B] = 2p

− 2p

[v;B] = p

− p

+ p

− p

, for i = 2,3,

[v;B] = 2p

− 2p

[v;B] = p

− 2p

− p

+ p

− p

+ 2p

[v;B] = 0 and

[v;B] = 0.

4 CONCLUSIONS

As we have said before, the present work is focussed

on the calculus of the Owen value. More precisely, the

computation of players’ allocations are obtained from

the multilinear extension of the game. In the con-

text of games with a coalition structure, the multilin-

ear extension technique has been also applied to com-

puting the Owen value in (Owen and Winter, 1992);

as well as the Owen–Banzhaf value in (Carreras and

Maga

na, 1994); in 1997 to the quotient game (Car-

reras and Maga

na, 1997); the Alonso–Fiestras value

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

nomial semivalues in (Carreras and Puente, 2011);

and coalitional multinomial probabilistic values in

(Carreras and Puente, 2015). In all these cases, the

ﬁrst three steps of the procedure are the same.

Instead, the consideration of the modiﬁed MLE

for the union B

obtained from the initial one has

changed the procedure: ﬁrst, we weight the terms of

multiplying each product

∏

k∈V

by p

v−1

and each

product

∏

r∈W

by q

w+1

obtaining a new multilin-

ear function called g

. Second, we obtain players’

marginal contributions by partial differentiation of g

This new procedure has an advantage with respect to

the traditional method: the allocations given by the

Owen value are available since the weighting coefﬁ-

cients p

k−1

and q

k+1

can be always easily obtained.

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