A value for cooperative games with a coalition

structure�

Emilio Calvoy Esther Gutiérrezz

January 18, 2011

Abstract

A value for games with a coalition structure is introduced, where the rules guid-

ing the cooperation among the members of the same coalition are di¤erent from the

interaction rules among coalitions. In particular, players inside a coalition exhibit

a greater degree of solidarity than they are willing to use with players outside their

coalition. The Shapley value [Shapley, 1953] is therefore used to compute the ag-

gregate payo¤s of the coalitions, and the Solidarity value [Nowak and Radzik, 1994]

to obtain the payo¤s of the players inside each coalition.

Keywords: Coalitional value; Shapley value; Owen value; Solidarity value.

JEL Classi�cation: C71

1 Introduction

There are many settings in cooperative games where players naturally organize themselves

into groups for the purpose of negotiating payo¤s. This action can be modeled by including

�Acknowledgments. This research has been supported by the Spanish Ministry of Science and

Technology and the European Regional Development Fund for �nancial support under projects SEJ2007-

66581 and ECO2009-11213. Partial support from "Subvención a grupos consolidados del Gobierno Vasco"

is also acknowledged.yDepartment of Economic Analysis and ERI-CES. Universitat de Valencia, Spain. E-mail:

emilio.calvo@uv.es.zDepartamento de Economía Aplicada IV. Universidad del País Vasco/E.H.U., Spain. E-mail: mari-

aester.gutierrez@ehu.es

1

a coalition structure into the game, which consists of an exogenous partition of players

into a set of groups or unions. These unions sometimes arise for natural reasons. One

example is the existence of a large number of agents. Players are joined into groups of

similar interests and characteristics in the case of trade unions, political parties, cartels,

lobbies, etc. Another typical reason is due to geographical location as in the case of cities,

states and countries.

When groups are formed, the agents interact at two levels: �rst, bargaining takes

place among the unions, and then, bargaining occurs inside each union in order to share

what the union has obtained at the �rst interaction level among their members. Owen

(1977) was the �rst to follow this approach. In his coalitional value, unions play a quotient

game among themselves. Each union receives a payo¤ that is shared among its players

in an internal game. The payo¤s at each level are given by the Shapley value (Shapley,

1953b). Thus, the same properties (axioms) that govern the interaction between groups

also operate among the players of each group. Nevertheless, it could be questioned whether

a coalitional value following the same behavior at both levels is a legitimate point of view.

A greater degree of solidarity existing between members of the same group than in the

interaction between players of di¤erent groups seems natural.

For example, take the principle of paying players according to their productivity. This

is one of the principles behind the Shapley value. It can be expressed formally by the

marginality axiom (see Young, 1985), i.e. if the marginal contributions of a player in two

games are the same then his value should be the same. Alternatively, it can be expressed

by the null player axiom, that is, if all the marginal contributions of a player in a game

are zero, then the player should obtain zero. What happens if this productivity principle

is imposed when sharing rewards at both levels: between unions and inside the unions as

is the case with the Owen value?

Let the game1 be de�ned by the set of players N = f1; 2; 3; 4g, and the coalition

structure B = ff1g ; f2; 3g ; f4gg, that is, players 1 and 4 are isolated and 2 and 3 form a

union, and uT , with T = f1; 2g, is a unanimity game de�ned by

uT (S) =

�1 if S � T0 otherwise

Players 3 and 4 are null players in uT , but player 3 is in the same union as player 2, that is1This is only an informal discussion. All the precise de�nitions of the following concepts will appear

in next Sections.

2

not a null player. In the quotient game, the unions f1g and f2; 3g are symmetric players,

i.e. both contribute the same, so they obtain 1=2, and union f4g is a null player in the

quotient game and then obtains zero. Inside the union f2; 3g, player 3 is a null player in

uT , hence he must obtain zero. Therefore, the payo¤s associated to the Owen value are

Ow1 =1

2; Ow2 =

1

2; Ow3 = 0; Ow4 = 0:

Here, there is no di¤erence for player 3 between belonging to the union f2; 3g or being

isolated.

Our purpose is to consider coalitional values where player 3 obtains a positive gain

from its membership of union f2; 3g. One proposal was presented in Kamijo (2009) and

is known as the two-step Shapley value. This value satis�es most of the properties that

support the Shapley value in the setting of games without coalition structure. Therefore,

it can be considered as an alternative value extension of the Shapley value to the coalition

structure setting. In Calvo and Gutiérrez (2010), this value has been characterized by

means of a population solidarity principle, which can be expressed informally as follows:

if we add or draw players out of the union, then all members of the union change their

value equally. In our example, the two-step Shapley value yields

K1 =1

2; K2 =

1

4; K3 =

1

4; K4 = 0:

Player 3 now obtains the same as 2. However, this seems rather unfair from the produc-

tivity point of view, as 3 is a null player that does not contribute to the rewards of the

union.

We have then two di¤erent principles for rewarding players: paying according to the

productivity of players (marginal contributions), which is a good incentive rule; and re-

distributing positive rewards to all members of a union, which is a natural cohesion group

property. Can we make both principles compatible? In this paper, we propose a new

coalitional value inspired in the Solidarity value introduced by Nowak and Radzik (1994).

This value takes into account the productivity principle as well, as the players�marginal

contributions are used in the calculation. However, it also exhibits a redistribution e¤ect,

as it not only takes into account his own marginal contribution to the coalition that player

belongs to, but also the marginal contributions of the remaining players, in such a way

that the own marginal contribution is replaced by the average of the marginal contribu-

tions of all players in the coalition. According to this approach, the value is obtained in

3

two steps. First, unions play a quotient game among themselves and each union receives

a payo¤ given by the Shapley value; and second, the outcome obtained by the union is

shared among its members by paying the solidarity value in the internal game. In this

way, the payo¤s obtained in our unanimity example are

�1 =1

2; �2 =

3

8; �3 =

1

8; �4 = 0;

which are more in keeping with the approach of joining productivity and cohesion princi-

ples simultaneously to reward the players inside each union.

A paradigmatic example is the case of simple games, which has been important in

applications to political sciences. Our value seems an interesting alternative to the Owen

value for the computation of the power that political parties have in parliaments under

di¤erent coalition con�gurations. In particular, this coalitional value gives support to em-

pirical evidence that sometimes government coalitions are not minimal winning coalitions

(as might be expected according to Riker�s (1962) size principle or the minimal winning

coalition principle), but rather include more parties than would be necessary to have a

majority of the votes in parliaments of proportional representation (see Strom, 1990; and

Scho�eld, 1993).

The rest of the paper is organized as follows. Section 2 is devoted to de�nitions

and notation. We also show how to compute the solidarity value by the random order

approach. Section 3 introduces the new coalitional value. We provide the axiomatic

characterization of this value in Section 4. The independence of this axiom system is

proved in Section 5 and a new characterization of the Owen value is also put forward as

a by-product.

2 De�nitions

2.1 Shapley value

Let U = f1; 2; :::g be the (in�nite) set of potential players. A cooperative game with

transferable utility (TU-game) is a pair (N; v) where N � U is a nonempty and �nite set

and v : 2N ! R is a characteristic function, de�ned on the power set of N , satisfying

v(;) = 0. An element i of N is called a player and every nonempty subset S of N a

coalition. The real number v(S) is called the worth of coalition S, and it is interpreted

4

as the total payo¤ that the coalition S, if it forms, can obtain for its members. Let GN

denote the set of all cooperative TU-games with player set N .

For all S � N , we denote the restriction of (N; v) to S as (S; v). For simplicity, we

write S [ i instead of S [ fig, N n i instead of N n fig, and v(i) instead of v (fig).

A value is a function which assigns to every TU-game (N; v) and every player

i 2 N , a real number i (N; v), which represents an assessment made by i of his gains

from participating in the game.

Let (N; v) be a game. For all S � N and all i 2 S, de�ne

@i(v; S) := v(S)� v(Sni):

We call @i(v; S) the marginal contribution of player i to coalition S in the TU-game

(N; v). The Shapley value (Shapley, 1953b) of the game (N; v) is the payo¤ vector

Sh(N; v) 2 RN de�ned by

Shi(N; v) =X

S�N :i�S

(n� s)! (s� 1)!n!

@i(v; S); for all i 2 N:

where s = jSj and n = jN j :

Two players i; j 2 N are symmetric in (N; v) if v (S [ i) = v (S [ j) for all S � N n

fi; jg. Player i 2 N is a null player in (N; v) if v (S [ i) = v(S) for all S � N n i. For any

two games (N; v) and (N;w), the game (N; v + w) is de�ned by (v+w)(S) = v(S)+w(S)

for all S � N .

Consider the following properties of a value in GN :

E¢ ciency: For all (N; v) ;Pi2N

i (N; v) = v(N).

Additivity: For all (N; v) and (N 0; w) with N = N 0, (N; v + w) = (N; v)+ (N;w).

Symmetry: For all (N; v) and all fi; jg � N , if i and j are symmetric players in (N; v),

then i (N; v) = j (N; v).

Null player axiom: For all (N; v) and all i 2 N , if i is a null player in (N; v), then

i (N; v) = 0.

The following theorem is due to Shapley (1953b).

Theorem 1 (Shapley, 1953b) A value on GN satis�es e¢ ciency, additivity, symmetry

and null player axiom if, and only if, is the Shapley value.

5

Let (N) be the set of all orders on N . Each ! 2 (N) is a bijection from N to

f1; 2; ::; ng and !(i), i 2 N , denotes the position of player i in the order !. We denote

by P !i the set of all predecessors of i in !, that is, P!i := fj 2 N : !(j) < !(i)g. Let

(N; v) be a game, for all i 2 N and all ! 2 (N), the marginal contribution that player

i receives in ! is de�ned by

m!i (N; v) := v (P

!i [ i)� v (P !i ) :

The Shapley value of the game (N; v) can also be computed by the formula

Shi(N; v) =1

n!

X!2(N)

m!i (N; v) ; for all i 2 N: (1)

For all T � N , the unanimity game of the coalition T , (N; uT ), is de�ned by

uT (S) =

�1 if S � T;0 otherwise:

It is well known that the family of games f(N; uT )gT�N is a basis for GN .

A weighted value w is a function that assigns to all (N; v) and all w 2 RN++ a vector

w(N; v) in RN : For each i 2 N , wi is the weight of player i: We will say that a weighted

value w extends a value if w(N; v) = (N; v) for any weight vector w with wi = wj

for all i; j 2 N: The most important weighted generalization of the Shapley value is the

weighted Shapley value Shw (Shapley (1953a), Kalai and Samet (1987)). For all w 2 RN++;

the weighted Shapley value Shw is the linear map Shw : GN �! RN , which is de�ned for

each unanimity game (N; uT ) as follows

Shwi (N; uT ) =

8>><>>:wiX

j2T

wj

for all i 2 T;

0 otherwise:

The weighted Shapley value Shw satis�es e¢ ciency, additivity and the null player

axiom, but not symmetry.

2.2 Solidarity value

Let (N; v) be a game. For all S � N , de�ne

@av(v; S) :=1

s

Xi2S@i(v; S):

6

@av(v; S) is the average of the marginal contributions of players within coalition S in

the game (N; v). The solidarity value of the game (N; v) is the payo¤ vector Sl(N; v) 2

RN de�ned by

Sli(N; v) =X

S�N :i�S

(n� s)! (s� 1)!n!

@av(v; S); for all i 2 N:

This value was introduced by Nowak and Radzik (1994) and it satis�es some solidarity

principle, since null players can obtain positive payo¤s (See example 1.1 in Nowak and

Radzik (1994)). They introduced a variation of the null player axiom in order to charac-

terize Sl on GN . Player i 2 N is an A-null player in (N; v) if @av(v; S) = 0 for all coalition

S � N containing i. There is clearly no relation between the null player and the A-null

player concepts. The solidarity value satis�es the following axiom:

A-Null player axiom: For all (N; v) and all i 2 N , if i is an A-null player, then

i (N; v) = 0.

The following theorem is due to Nowak and Radzik (1994).

Theorem 2 (Nowak and Radzik, 1994) A value on GN satis�es e¢ ciency, additivity,

symmetry and A-null player axiom if, and only if, is the solidarity value.

We can de�ne a weighted generalization of the solidarity value in the same way as the

weighted Shapley value. Nowak and Radzik (1994) de�ned a new basis for GN , denoted

by f(N; bT )gT�N : For all T � N , (N; bT ) is de�ned by

bT (S) =

��jSjjT j��1

if S � T0 otherwise

They proved that every player i 2 N�T is A-null in the game (N; bT ):

For all w 2 RN++; the weighted solidarity value Slw is the linear map Slw : GN �! RN ,

de�ned for all (N; bT ) as follows

Slwi (N; bT ) =

� wiXj2T

wj

bT (N) for all i 2 T ,

0 otherwise.

The weighted solidarity value Slw satis�es e¢ ciency, additivity and the A-null player

axiom, but not symmetry.

7

Remark 1 The solidarity value can be obtained recursively (see Calvo, 2008) by

Sli(S; v) =1

s@av(v; S) +

Xj2Sni

1

sSli(Snj; v); for all S � N and all i 2 S;

starting with

Sli(fig; v) = v(i); for all i 2 N:

The solidarity value was thus introduced by Sprumont (1990; Section 5) in order to

show that, in the class of increasing average marginal contributions (IAMC) games, i.e.

@av(v; S) � @av(v; T ), whenever S � T , it is possible to �nd a population monotonic

allocation scheme (PMAS), i.e. a payo¤ con�guration�xS�S�N 2

�RS�S�N such that

(i) For all S � N ,P

i2S xSi = v(S),

(ii) For all S; T � N and all i 2 S, S � T ) xSi � xTi .

Sprumont (1990, Proposition 4) shows that the solidarity value is a PMAS in the class of

IAMC games.

2.3 Random order approach

In the same way that the Shapley value can be computed by means of orders, we also

provide a formula to calculate the solidarity value by orders. Let (N; v) be a game, for

all ! 2 (N) and all i 2 N , we de�ne

&!i (N; v) :=Xj2N

!(j)�!(i)

1��P !j [ j��@j �v; P !j [ j� ; for all i 2 N:

For example, if N = fa; b; cg and ! is such that !(a) = 1, !(b) = 2 and !(c) = 3,

this means that the sequence of coalition formation S1 � S2 � S3 in ! is: at step r = 1,

S1 = fag, at step r = 2, S2 = fa; bg, and at step r = 3, S3 = fa; b; cg. Then the payo¤s

&!(N; v) are

&!a (N; v) = v(fag) + v(fa; bg)� v(fag)2

+v(fa; b; cg)� v(fa; bg)

3;

&!b (N; v) =v(fa; bg)� v(fag)

2+v(fa; b; cg)� v(fa; bg)

3;

&!c (N; v) =v(fa; b; cg)� v(fa; bg)

3:

Each player here shares his own marginal contribution with the players in front of him

in the order !.

We have the following theorem.

8

Theorem 3 For all (N; v) 2 GN , the solidarity value is obtained by

Sli(N; v) =1

n!

X!2(N)

&!i (N; v); for all i 2 N:

Proof. Let (N; v) be a game, for all j 2 N and all S � N such that j 2 S, de�ne

(S; j) :=�! 2 (N) : P !j [ j = S

:

That is, (S; j) is the set of all orders for which player j is the last player which completes

coalition S. Note that j(S; j)j = (s� 1)! (n� s)!. Then, for all ! 2 (N) and all i 2 N ,

player i obtains the sum of 1s@j(v; S) for all j such that !(j) � !(i), with S = P !j [ j.

Therefore, for all i 2 N;

1

n!

X!2(N)

&!i (N; v) =1

n!

X!2(N)

Xj2N

!(j)�!(i)

1��P !j [ j��@j �v; P !j [ j� =1

n!

XS�NS3i

Xj2S

X!2(S;j)

1

s@j(v; S) =

XS�NS3i

(n� s)! (s� 1)!n!

@av(v; S) = Sli(N; v):

Example 1 Consider the game with player set N = fa; b; cg and characteristic function

de�ned by v(a) = v(b) = v(c) = 0, v(a; b) = v(a; c) = 1, v(b; c) = 0, and v(a; b; c) = 2.

The following table sets out the marginal contributions &!i for every order:

orders players

! &!a &!b &!c

(a; b; c) 56

56

26

(a; c; b) 56

26

56

(b; a; c) 56

56

26

(b; c; a) 46

46

46

(c; a; b) 56

26

56

(c; b; a) 46

46

46

Therefore, the payo¤s are

Sla(N; v) =14

18; Slb(N; v) = Slc(N; v) =

11

18:

Note that @av(v; fa; bg) = @av(v; fa; cg) = 1, @av(v; fb; cg) = 0, and @av(v; fa; b; cg) = 43.

9

3 Coalitional solidarity value

3.1 The Owen and the two-step Shapley values

For all �nite set N � U , a coalition structure over N is a partition of N , i.e., B =

fB1; B2; :::; Bmg is a coalition structure if it satis�es that[

1�k�m

Bk = N and Bk \Bl = ?

when k 6= l. We also assume Bk 6= ; for all k. The sets Bk 2 B are called �unions�

or �blocks�. There are two trivial coalition structures: The �rst, which we denote by

BN , where only the grand coalition forms, that is, BN = fNg; and the second is the

coalition structure where each union is a singleton and it is denoted by Bn, that is,

Bn = ff1g ; f2g ; :::; fngg. Denote by B(N) the set of all coalition structures over N . A

game (N; v) with coalition structure B 2 B(N) is denoted by (B; v). Let CSGN denote

the family of all TU-games with coalition structure with player set N , and let CSG denote

the set of all TU-games with coalition structure.

For all game (B; v) 2 CSGN , with B = fB1; B2; :::; Bmg, the quotient game is the TU -

game (M; vB) 2 GM where M = f1; 2; :::;mg and vB(T ) := v

[i2TBi

!for all T � M .

That is, (M; vB) is the game induced by (B; v) by considering the coalitions of B as

players. Notice that for the trivial coalition structure Bn we have (M; vBn) � (N; v). For

all fk; lg �M; we say that Bk and Bl are symmetric in (B; v) if k and l are symmetric in

the game (M; vB). For all k 2 M , we say that Bk is a null coalition if k is a null player

in the quotient game (M; vB).

Let B 2 B(N). For all k 2 M and all S � Bk, denote by B jS the new coalition

structure de�ned on ([j 6=kBj) [ S, which appears when the complementary of S in Bkleaves the game. That is,

B jS= fB1; :::; Bk�1; S; Bk+1; :::; Bmg:

A coalitional value is a function � that assigns a vector in RN to each game with

coalition structure (B; v) 2 CSGN . One of the most important coalitional values is the

Owen value (Owen, 1977). Let (B; v) 2 CSGN and k 2 M , for all S � Bk, denote

S 0 = Bk nS. Owen (1977) de�ned a game (M; vBjS) that describes what would happen in

the quotient game if union Bk were replaced by S, i.e.,

vBjS(T ) = v ([j2TBj n S 0) for all T �M:

10

Next, he de�ned an internal game (Bk; vk) by setting vk(S) = Shk(M; vBjS) for all S � Bk.

Thus, vk(S) is the payo¤ to S in vBjS . The Owen value of the game (B; v) is the payo¤

vector Ow(B; v) 2 RN de�ned by

Owi(B; v) := Shi (Bk; vk) ; for all k 2M and all i 2 Bk: (2)

Thus, �rst union k plays the quotient game (M; vB) among the unions, and the payo¤

obtained is shared among its members by playing the internal game (Bk; vk). At both

levels of bargaining, the payo¤s are obtained using the Shapley value Sh. In that sense,

we can denote the Owen value as Ow � �(Sh;Sh).

The Owen value satis�es the quotient game property:Xi2Bk

Owi(B; v) = Shk (M; vB) ; for all k 2M:

Note that for the trivial coalition structures Bn and BN , Ow(BN ; v) = Ow(Bn; v) =

Sh(N; v).

The Owen value can also be de�ned by orders. Let B be a coalition structure over N

and ! 2 (N). We say that ! is admissible with respect to B if for all fi; j; kg � N and

l 2 M such that fi; kg � Bl, if !(i) < !(j) < !(k); then j 2 Bl. In other words, ! is

admissible with respect to B if players in the same component of B appear successively

in !. We denote by (B;N) the set of all admissible orders (on N) with respect to B.

The Owen value is given by the formula

Owi(B; v) =1

j(B;N)jX

!2(B;N)

m!i (N; v); for all i 2 N:

We now present the axioms that characterize Ow in CSGN . Let � be a coalitional

value, de�ne

� (B; v) [S] =Xi2S�i (B; v) , for all S � N:

(E) E¢ ciency: For all (B; v) 2 CSGN , � (B; v) [N ] = v(N).

(A) Additivity: For all (B; v) ; (B0; w) 2 CSGN with B = B0, � (B; v + w) = � (B; v) +

� (B;w).

(ISy) Intracoalitional symmetry: For all (B; v) 2 CSGN , all k 2M and all fi; jg � Bk, if

i and j are symmetric players in (N; v), then �i (B; v) = �j (B; v).

11

(CSy) Coalitional symmetry: For all (B; v) 2 CSGN and all fk; lg � M , if Bk and Bl are

symmetric in (B; v), then � (B; v) [Bk] = � (B; v) [Bl].

(NP) Null player axiom: For all (B; v) 2 CSGN and all i 2 N , if i is a null player in

(N; v), then �i (B; v) = 0.

This theorem is due to Owen (1977).

Theorem 4 (Owen, 1977) A value � on CSGN satis�es e¢ ciency, additivity, intracoali-

tional symmetry, coalitional symmetry and null player axiom if, and only if, � is the

Owen value.

Let (B; uT ) be the game de�ned by the set of players N = f1; 2; 3; 4g, the coalition

structure B = ff1g ; f2; 3g ; f4gg, and the unanimity game uT , with T = f1; 2g. Players

3 and 4 are null players in (N; uT ), but player 3 is in the same union as player 2, which

is not a null player. In the quotient game, unions f1g and f2; 3g are symmetric players,

i.e. both contribute the same, so they obtain 1=2, and union f4g is a null player in the

quotient game and then obtain zero. Inside union f2; 3g, player 3 is a null player in

(N; uT ), hence he must obtain zero. Therefore the payo¤s associated to the Owen value

are

Ow1 (B; uT ) =1

2; Ow2 (B; uT ) =

1

2; Ow3 (B; uT ) = 0; Ow4 (B; uT ) = 0:

Here, for player 3, there is no di¤erence between belonging to the union f2; 3g or being

isolated.

Kamijo (2009) de�ned a new coalitional value, named the two-step Shapley value. For

all game (B; v) 2 CSGN , the two-step Shapley value of (B; v) is given by the formula:

Ki (B; v) = Shi (Bk; v) +1

jBkj[Shk (M; vB)� v (Bk)] ; for all k 2M and all i 2 Bk (3)

In Calvo and Gutiérrez (2010) this value has been characterized by means of a solidar-

ity principle, that can be expressed informally as follows: if the data of the game change

due to factors external to the union, then all members of the union change their value

equally. For all l 2M and all h 2 Bl, de�ne B�h := (B1; :::; Blnh; :::; Bm).

(PS) Population solidarity within unions: For all (B; v) 2 CSGN ; all fk; lg � M; with

k 6= l; and all fi; j; hg � N , where fi; jg � Bk and h 2 Bl,

�i (B; v)� �i (B�h; v) = �j (B; v)� �j (B�h; v) :

12

De�ne two additional axioms:

(NC) Null coalition axiom: For all (B; v) 2 CSGN and all k 2M; if Bk is a null coalition

in (B; v), then � (B; v) [Bk] = 0.

(Coh) Coherence2: For all (B; v) 2 CSGN , ��BN ; v

�= �(Bn; v).

Coherence means that games where all players belong to only one union and where all

of them act as singletons are indistinguishable.

This theorem is in Calvo and Gutiérrez (2010).

Theorem 5 A value � on CSG satis�es e¢ ciency, additivity, coalitional symmetry, null

coalition axiom, coherence and population solidarity within unions if, and only if, � � K.

The two-step Shapley value in our previous example (B; uT ) yields

K1 (B; uT ) =1

2; K2 (B; uT ) =

1

4; K3 (B; uT ) =

1

4; K4 (B; uT ) = 0:

Player 3 now obtains the same as 2, which appears rather unfair from the productivity

point of view, as 3 is a null player that does not contribute the rewards of the union.

3.2 De�nition of �

A new coalitional value is now de�ned, which can be considered as an extension of the

solidarity value to the coalition structure setting.

De�nition 1 For all game (B; v) 2 CSGN , for all k 2M and all i 2 Bk we de�ne:

�i(B; v) := Sli (Bk; vk) : (4)

First union k now plays the quotient game (M; vB) among the unions, and the payo¤

obtained (by the Shapley value) is shared among its members by computing the solidarity

value in the internal game (Bk; vk). Thus, the value � can be denoted as �(Sh;Sl).

As the solidarity value satis�es e¢ ciency in the internal game (Bk; vk) ; it follows that

� also satis�es the quotient game property:Xi2Bk

�i(B; v) = Shk (M; vB) ; for all k 2M:

2This axiom was called the Coalitional structure equivalence in Albizuri (2008).

13

Moreover, we have that �(BN ; v) = Sl(N; v) and �(Bn; v) = Sh(N; v).

The structure of the value � as a solution of the form �(Sh;Sl) can also be seen in the

next proposition.

Proposition 1 For all (B; v) 2 CSGN ,

�i(B; v) =XT�Mk2T

(m� t)!(t� 1)!m!

�Sli(Bk; v

T )�; for all k 2M and all i 2 Bk;

where

vT (S) = v

0B@S [[r2Tr 6=k

Br

1CA� v0B@[r2Tr 6=k

Br

1CA ; for all S � Bk and all T �M;

m = jM j and t = jT j.

Proof. Let (B; v) 2 CSGN be a game. For all k 2M , denote bk = jBkj. Then, for all

i 2 Bk,

�i(B; v) = Sli (Bk; vk) =

=XS�Bki2S

(bk � s)!(s� 1)!bk!

1

s

Xj2S

(vk(S)� vk(Snj)) =

=XS�Bki2S

(bk � s)!(s� 1)!bk!

1

s

Xj2S

�Shk(M; vBjS)� Shk(M; vBjS�j)

�=

=XS�Bki2S

(bk � s)!(s� 1)!bk!

1

s

Xj2S

0B@XT�Mk2T

(m� t)!(t� 1)!m!

��vBjS(T )� vBjS(Tnk)

���vBjS�j(T )� vBjS�j(Tnk)

��1CA =

=XS�Bki2S

(bk � s)!(s� 1)!bk!

XT�Mk2T

(m� t)!(t� 1)!m!

1

s

Xj2S

�vBjS(T )� vBjS�j(T )

�!=

=XT�Mk2T

(m� t)!(t� 1)!m!

XS�Bki2S

(bk � s)!(s� 1)!bk!

1

s

Xj2S

�vT (S)� vT (Snj)

�!=

=XT�Mk2T

(m� t)!(t� 1)!m!

�Sli(Bk; v

T )�:

Remark 2 The structure of the Owen value as �(Sh;Sh) can be proven in a similar way:

Owi(B; v) =XT�Mk2T

(m� t)!(t� 1)!m!

Shi(Bk; vT ); for all k 2M and all i 2 Bk:

14

Similarly to the Owen value, we can provide a formula to compute the coalitional value

� by means of orders. Let (B; v) 2 CSGN : For all ! 2 (B;N), all k 2M and all i 2 Bk,

de�ne

&!i (B; v) :=Xj2Bk

!(j)�!(i)

1���Bk \ P !j � [ j��@j �v; P !j [ j� :For example, if N = fa; b; c; dg, B = fB1; B2g, where B1 = fa; bg and B2 = fc; dg;

and ! 2 (B;N) is such that �! = (a; b; c; d); this means that the sequence of coalition

formation S1 � S2 � S3 � S4 in ! is: at step r = 1, S1 = fag, at step r = 2, S2 = fa; bg,

at step r = 3, S3 = fa; b; cg, and at step r = 4, S4 = fa; b; c; dg. Then the payo¤s &!(B; v)

are

&!a (B; v) = v(fag) + v(fa; bg)� v(fag)2

;

&!b (B; v) =v(fa; bg)� v(fag)

2;

&!c (B; v) = (v(fa; b; cg)� v(fa; bg)) + v(fa; b; c; dg)� v(fa; b; cg)2

;

&!d (B; v) =v(fa; b; c; dg)� v(fa; b; cg)

2:

Proposition 2 For all (B; v) 2 CSGN ,

�i (B; v) =1

j(B;N)jX

!2(B;N)

&!i (B; v); for all i 2 N:

Proof. Let (B; v) 2 CSGN be a game. For all k 2M and all i 2 Bk,

1

j(B;N)jX

!2(B;N)

&!i (B; v) =

=1

j(B;N)jXT�Mk2T

(m� t)!(t� 1)!Y

r2M�k

br!XS�Bki2S

(bk � s)!(s� 1)!Xj2S

1

s@j

0@v; S [ [r2T�k

Br

1A =

=XT�Mk2T

(m� t)!(t� 1)!m!

XS�Bki2S

(bk � s)!(s� 1)!bk!

Xj2S

1

s@j

0@v; S [ [r2T�k

Br

1A =

=XT�Mk2T

(m� t)!(t� 1)!m!

XS�Bki2S

(bk � s)!(s� 1)!bk!

Xj2S

1

s@j�vT ; S

�=

=XT�Mk2T

(m� t)!(t� 1)!m!

Sli(Bk; vT ) = �i(B; v):

15

In our unanimity example (B; uT ), the payo¤s obtained with our coalitional solidarity

value are

�1 (B; uT ) =1

2; �2 (B; uT ) =

3

8; �3 (B; uT ) =

1

8; �4 (B; uT ) = 0:

These are more in keeping with the approach of joining productivity and cohesion princi-

ples simultaneously to reward the players inside the union f2; 3g.

Remark 3 The solidarity value can be extended in a di¤erent way to games with a coali-

tion structure. For example, we can de�ne the values �(Sl;Sl) and �(Sl;Sh) as follows:

�(Sl;Sl)i (B; v) : = Sli (Bk; vk) ; for all k 2M and all i 2 Bk;

�(Sl;Sh)i (B; v) : = Shi (Bk; vk) ; for all k 2M and all i 2 Bk;

where vk(S) = Slk(M; vBjS) for all S � Bk. Note that

�(Sl;Sl)(BN ; v) = Sl(N; v) �(Sl;Sl)(Bn; v) = Sl(N; v)

�(Sl;Sh)(BN ; v) = Sh(N; v) �(Sl;Sh)(Bn; v) = Sl(N; v)�Ow � �(Sh;Sh)

�(BN ; v) = Sh(N; v)

�Ow � �(Sh;Sh)

�(Bn; v) = Sh(N; v)�

� � �(Sh;Sl)�(BN ; v) = Sl(N; v)

�� � �(Sh;Sl)

�(Bn; v) = Sh(N; v)

The next table summarizes the payo¤s given to the game (B; uT ) by all the aforementioned

solutions:(B; uT ) �1 �2 �3 �4

Ow � �(Sh;Sh) 1=2 1=2 0 0

K 1=2 1=4 1=4 0

�(Sl;Sl) 7=18 7=24 7=72 4=18

�(Sl;Sh) 7=18 7=18 0 4=18

� � �(Sh;Sl) 1=2 3=8 1=8 0

The Owen value yields nothing to player 3 as a null player, hence he has no bene�ts

from belonging to the union f2; 3g. The opposite case is the two-step Shapley value, as it

yields the same payo¤ 1=4 to both players in the union, without taking into account the

di¤erences in their productivity. The value � is an intermediate option between these two

opposite options. The coalitional values �(Sl;Sl) and �(Sl;Sh) do not seem reasonable as

union f4g is a null coalition in the quotient game and, yet, it obtains a positive payo¤ in

both values. Moreover, player 4 obtains a greater payo¤ than player 3.

16

4 Axiomatic approach

This section provides an axiomatic characterization of the coalitional value �.

Two new properties are de�ned in order to characterize �.

The �rst one is a weaker version of the A-null player axiom:

(A-NP�) A-Null player axiom in BN : For all (N; v) 2 GN and all i 2 N , if i is an A-null

player in (N; v), then �i�BN ; v

�= 0.

Hence, we only require an A-null player in (N; v) to receive zero payo¤ for the trivial

coalition structure BN = fNg.

The second is a consistency property. Consistency states the independence of a value

with respect to departure of some players with their assigned payo¤s. It says that the

recommendation made for any problem should always agree with the recommendation

made for any problem obtained by imagining some agents leaving with their payo¤s,

and "reassessing the situation" from the viewpoint of the remaining agents. It has been

introduced in di¤erent ways depending upon how the payo¤s of the agents that leave

the game are de�ned. As the value � satis�es the quotient game property (computing

the payo¤s by using the Shapley value) we follow the approach established by Hart and

Mas-Colell (1989). We brie�y recall their de�nitions of reduced game and consistency in

G. Let be a value, (N; v) be a game, and T � N . The reduced game (T; v T ) is de�ned

as follows:

v T (S) := v (S [ T c)�Xi2T c

i (S [ T c; v) ; for all S � T;

where T c = NnT . A value is consistent if, for all game (N; v) and all coalition T � N ,

it holds

i (T; v T ) = i(N; v), for all i 2 T:

A value is transferable-utility invariant if, for all pair of games (N; v) and (N; u);

and all real constants a > 0 and fbigi2N ,

u(S) = av(S) +Xi2Sbi; for all S � N;

implies

i (N; u) = a i (N; v) + bi; for all i 2 N:

Then the following result holds:

17

Theorem 6 (Hart and Mas-Colell, 1989) Let be a value on GN . Then:

(i) is consistent;

(ii) for two-person games: (a) is e¢ cient, (b) is transferable-utility invariant, (c)

satis�es symmetry;

if and only if is the Shapley value.

In Winter (1992), this property is extended to games with a coalition structure as fol-

lows: Let� be a coalitional value and (B; v) 2 CSGN be a game whereB = fB1; B2; :::; Bmg.

For all k 2M and all R � Bk, the reduced game (R; v�R;B) is de�ned by

v�R;B(S) := v (N�R [ S)�X

i2N�R

�i (B jRc[S; v) ; for all S � R,

where Rc = Bk�R.

(C) Consistency: For all (B; v) 2 CSGN , all k 2 M , and all R � Bk, �i (B; v) =

�i�BR; v�R;B

�, for all i 2 R.

The interpretation is as follows: The players in S cooperate with the complement of

R. The players of NnR get the anticipated payo¤ according to the value � in the game

(B jRc[S; v), and the players of S get the remaining payo¤ from v (N�R [ S). A value

� is then consistent if each player in R gets in v�R;B and the trivial coalition structure

BR the same amount he gets in the original game (B; v). This axiom makes sense for

coalitional values de�ned on CSG, given that the value in this axiom must be applied on

N and also on R, for all k 2M and all R � Bk.

Winter proved that the Owen value satis�es consistency and this property is used in

an axiom system that characterizes the value.

Nevertheless, consistency is a too stringent property in order to characterize the coali-

tional value �.

Proposition 3 The coalitional value � does not satisfy consistency.

Proof. Consider the game (B; uT ) 2 CSGN where N = f1; 2; 3; 4; 5g, T = f1; 2g and

B = fB1; B2; B3g with B1 = f1g, B2 = f2; 3; 4g and B3 = f5g. The coalitional value � is

�(B; uT ) =

�1

2;11

36;7

72;7

72; 0

�:

18

Now consider R = f2; 3g � B2. The reduced game�R; (uT )

�R;B

�is given by

(uT )�R;B (2) =

3

8;

(uT )�R;B (3) = 0;

(uT )�R;B (2; 3) =

29

72:

By de�nition, ��BR; (uT )

�R;B

�= Sl

�R; (uT )

�R;B

�. Thus,

�2

�BR; (uT )

�R;B

�=

85

2886= 11

36= �2(B; uT );

�3

�BR; (uT )

�R;B

�=

31

2886= 7

72= �3(B; uT ):

Nevertheless, a weaker version of this property can be used, where it would only hold

for the particular case where R = Bk, for all k 2M .

(CC) Coalitional consistency: For all (B; v) and all k 2 M , �i (B; v) = �i�BBk ; v�Bk;B

�,

for all i 2 Bk.

We are now ready to present the axiomatic characterization of the value � on CSG.

Theorem 7 A coalitional value � on CSG satis�es e¢ ciency, additivity, intracoalitional

symmetry, coalitional symmetry, null coalition axiom, A-null player axiom in BN and

coalitional consistency if, and only if, � � �:

Proof. Existence. Let (B; v) 2 CSGN be a game. Since the Shapley value and the

solidarity value satisfy e¢ ciency, for all k 2 M we have thatPi2Bk

�i (B; v) = vk(Bk) =

Shk (M; vB), and thenPi2N

�i (B; v) =Pk2M

Pi2Bk

�i (B; v) =Pk2M

Shk (M; vB) = v(N). Thus

� satis�es e¢ ciency. Moreover, since the Shapley and the solidarity values satisfy addi-

tivity, � also satis�es additivity.

The coalitional value � also satis�es intracoalitional symmetry because, for all k 2M;

if fi; jg � Bk are symmetric in (N; v), they are also symmetric in (Bk; vk), due to the

symmetry of the Shapley value. Thus, �i (B; v) = �j (B; v) since the solidarity value is

also symmetric. Moreover, � satis�es coalitional symmetry and the null coalition axiom

because the Shapley value satis�es symmetry and the null player axiom andPi2Bk

�i (B; v) =

Shk (M; vB), for all k 2M .

19

It is straightforward that � satis�es the A-Null player axiom in BN , because the

solidarity value satis�es the A-Null player axiom and, by de�nition, ��BN ; v

�= Sl(N; v).

In order to prove that � satis�es coalitional consistency, let k 2 M . Taking into

account the de�nition of �, and that � satis�es e¢ ciency, for all S � Bk,

v�Bk;B(S) =Xi2S�i (B jS; v) = Shk(M; vBjS) = vk(S):

Therefore,

�i

�BBk ; v�Bk;B

�= Sli(Bk; v

�Bk;B

) = Sli(Bk; vk) = �i(B; v); for all i 2 Bk:

Uniqueness. Let � be a coalitional value satisfying the above axioms, and let (B; v) 2

CSGN .

Suppose that jBj = 1, that is, B = BN . Since � satis�es e¢ ciency, additivity,

intracoalitional symmetry and the A-null player axiom in BN , then by Theorem 2, it

holds that ��BN ; v

�= Sl(N; v). Thus � is uniquely determined when jBj = 1.

Suppose that jBj � 2. To prove that � (B; v) is uniquely determined, due to the

additivity axiom, it is su¢ cient to show that � is uniquely de�ned on the set of unanimity

games. Let T � N and B(T ) = fBk 2 B j Bk \ T 6= ?g. For all h 2 M such that

Bh =2 B(T ); Bh is a null coalition, then, by the null coalition axiom and intracoalitional

symmetry, �i (B; uT ) = 0, for all i 2 Bh. Hence, only rests to show that �i (B; uT ) is

uniquely determined for players in the coalitions Bk 2 B(T ).

LetBk 2 B(T ). By coalitional consistency, �i (B; uT ) = �i�BBk ; (uT )

�Bk;B

�, for all i 2

Bk. We have just proved that ��BBk ; (uT )

�Bk;B

�= Sl(Bk; (uT )

�Bk;B

), then �i (B; uT ) =

Sli(Bk; (uT )�Bk;B

), for all i 2 Bk. By e¢ ciency and coalitional symmetry, for all S � Bk,

(uT )�Bk;B

(S) =Xi2S�i (B jS; uT ) =

�0 if S ! Bk \ T;1

jB(T )j if S � Bk \ T;

that is, (uT )�Bk;B

= 1jB(T )juBk\T . Therefore,

�i (B; uT ) = Sli(Bk; (uT )�Bk;B

) =1

jB(T )jSli(Bk; uBk\T ); for all i 2 Bk:

Thus, � (B; uT ) is uniquely determined.

Remark 4 In the proof of Theorem 7, the null coalition axiom is used only on the set of

unanimity games. Therefore, this axiom could be weakened in the following way:

(NC�) Null coalition on unanimity games: For all B 2 B(N), all T � N and all h 2 M

such that Bh \ T = ?, � (B; uT ) [Bh] = 0.

20

5 Complementary results

5.1 Independence of the axiomatic system in Theorem 7

The axiom system in Theorem 7 is independent. Indeed:

1. The Owen value satis�es all axioms, except A-null player axiom in BN .

2. The coalitional value �(Sl;Sl) satis�es all axioms, except null coalition axiom. See

the example of Section 3: f4g is a null coalition, but player 4 does not obtain zero.

3. Let the coalitional value F be de�ned as

Fi(B; v) = Sli (Bk; v) +1

jBkj[Shk (M; vB)� v (Bk)] ; for all k 2M and all i 2 Bk:

This value is de�ned in the same way as the two-step Shapley value, but the sol-

idarity value is applied within each union. Taking into account the properties of

the Shapley and the solidarity values, F satis�es E, A, ISy, CSy, NC and A-NP�

(F (BN ; v) = Sl(N; v)). Nevertheless, F does not satisfy weak coalitional consis-

tency. Consider the example of Section 3. The value is F (B; uT ) = (1=2; 1=4; 1=4; 0).

LetB2 = f2; 3g, the reduced game (B2; (uT )F�B2;B) is given by (uT )F�B2;B

= 12uf2g. Thus,

F (BB2 ; (uT )F�B2;B

) = Sl(B2; (uT )F�B2;B

) = (3=8; 1=8) 6= (1=4; 1=4).

4. The coalitional value �(Sh;Slw) satis�es all axioms, except intracoalitional symmetry.

5. The coalitional value �(Shw;Sl) satis�es all axioms, except coalitional symmetry.

6. Let the coalitional value G be de�ned as Gi(B; v) = 0 for all (B; v) 2 CSG and all

i 2 N: It satis�es all axioms except e¢ ciency.

7. De�ne the Bounded egalitarian value as

BEi(N; v) =

� v(N)jCarr(N;v)j ; if i 2 Carr(N; v);0; otherwise;

where Carr(N; v) denotes the carrier of (N; v), which is the set of non-null players

in (N; v). Then, �(BE;Sl) satis�es all axioms, except additivity.

21

5.2 An alternative axiomatic characterization of the Owen value

Following the same lines as in Theorem 7, we can provide a new axiomatic characterization

of the Owen value on CSG. We therefore also need a weaker version of the null player

axiom as we have done with the A-null player one.

(NP�) Null player axiom in BN : For all (N; v) 2 GN and all i 2 N , if i is a null player in

(N; v), then �i�BN ; v

�= 0.

Theorem 8 A coalitional value � on CSG satis�es e¢ ciency, additivity, intracoalitional

symmetry, coalitional symmetry, null coalition axiom, null player axiom in BN and coali-

tional consistency if, and only if, � � Ow.

The proof follows the same lines as in Theorem 7 and it is left to the reader.

It becomes clear from the comparison of Theorems 7 and 8 that the main di¤erence

between the Owen and � values is based only on the di¤erence between NP� and A-NP�:

when all the players form the grand coalition, null players obtain zero in the Owen value,

whereas the A-null players obtain zero in the coalitional value �.

The axiomatic system in Theorem 8 is independence. Indeed:

1. The coalitional value � satis�es all axioms, except null player axiom in BN .

2. The coalitional value �(Sl;Sh) satis�es all axioms, except null coalition axiom. See

the example of Section 3: f4g is a null coalition, but player 4 does not obtain zero.

3. The two-step Shapley value satis�es all axioms, except weak coalitional consistency.

Consider the example of Section 3. The value is K(B; uT ) = (1=2; 1=4; 1=4; 0) : Let

B2 = f2; 3g, the reduced game (B2; (uT )K�B2;B) is given by (uT )K�B2;B

= 12uf2g. Thus,

K(BB2 ; (uT )K�B2;B

) = Sh(B2; (uT )K�B2;B

) = (1=2; 0) 6= (1=4; 1=4).

4. The coalitional value �(Sh;Shw) satis�es all axioms, except intracoalitional symmetry.

5. The coalitional value �(Shw;Sh) satis�es all axioms, except coalitional symmetry.

6. The coalitional value G; de�ned as Gi(B; v) = 0 for all (B; v) 2 CSG and all i 2 N;

satis�es all axioms except e¢ ciency.

7. The coalitional value �(BE;Sh) satis�es all axioms, except additivity.

22

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24