On the Core of a Collection of Coalitions

J�J�M� Derks �� J�H� Reijnierse y

December� ����

Abstract

For a collection � of subsets of a �nite set N we de�ne its core

to be equal to the polyhedral cone fx � IRN �

Pi�N xi � � and

Pi�S xi � � for all S � �g� This note describes several applications

of this concept in the �eld of Cooperative Game Theory� Especially

collections � are considered with core only consisting of the origin�

This property of a one�point core is proved to be equivalent to the

non�degeneracy and balancedness of �� Further� the notion of exact

cover is discussed and used in a second characterization of collections

� with Core�� � f�g�

�Department of Mathematics� University of Maastricht� P�O�Box ���� ���� MD Maas�

tricht� The Netherlands �mailing address��yDepartment of Econometrics� Tilburg University� P�O�Box ���� ��� LE Tilburg�

The Netherlands�

�

� Introduction

Let � denote a collection of subsets of a �nite set N � In this paper we will

consider the core of �� which is de�ned to be equal to

Core��� � fx � IRN �P

i�N xi � andP

i�S xi � for each S � �g�

The core is a notion fromCooperative Game Theory� whereN denotes a set of

players and where the subsets of N are called coalitions� The jN jdimensional

vector space IRN is called the allocation space� Now Core��� � IRN is in fact

the set of side payments x �thus�P

i�N xi � � which are not unfavourable

for the coalitions S in �� i�e�P

i�S xi � �

Within Cooperative Game Theory the central notion is the game in char

acteristic function form� de�ned as a realvalued function v on the power set

�N of the player set N � with v��� � � The subsets S � N are called coali�

tions� and the value v�S� expresses the gain arising from the cooperation by

the members of S� For a collection � of coalitions we de�ne the ��restricted�

core Core��� v� of the game v to be the allocation set

Core��� v� � fx � IRN �X

i�N

xi � v�N� andX

i�S

xi � v�S� for each S � �g�

In case � equals �N this notion agrees with the standard core concept� Notice

that the core for a collection of coalitions equals the restricted core of the

allzero game�

The restricted core of a game is treated �rst in Faigle ��� �� where its

nonemptiness is proved to be equivalent to a similar balancedness property

as in the general case shown by Bondareva ������ and Shapley ������� In

�

Derks�Gilles ������ the restricted core is considered with respect to collec

tions � � �N � with the property S � T� S � T � � for each S� T � �� i�e�� �

is a lattice� The main result in the latter paper is the characterization of

the extreme rays of Core��� considered as a polyhedral cone� This result is

discussed brie�y in the next section�

The interest in subcollections of the power set �N of N is inspired by the

fact that in case of a restricted communication or cooperation structure� the

standard model of a cooperative game is not su�cient� As a consequence well

known solution concepts like the Shapley value� the nucleolus� and the core

are reconsidered in the context of the existence of unfeasible or unformable

coalitions� For the Shapley value we refer to Myerson ������� Owen ��� ���

Faigle�Kern ������� and Derks�Peters ������� and for the nucleolus we refer

to Maschler�Potters�Tijs �������

Although a game may have an empty core the restriction to a subcollec

tion may be nonempty� It may occur� however� that for two core allocations

x� y � Core��� v�� x �� y� the allocation x can be more desirable than y in the

sense thatP

i�S xi �P

i�S yi for all coalitions S � �� and strict inequality

for at least one of these coalitions� Observe that the di�erence x� y has to

be an element of Core���� so that the collection � should have the property

Core��� � fg ���

whenever we want to exclude the possibility of having restricted core alloca

tions which are more desirable than others� This is the main motivation of

considering necessary and su�cient conditions on � in order to obtain ����

Condition ��� is also used in the study of the socalled �pre�nucleolus

�

concept of a game v� coincides with the unique allocation x � IRN with the

property that all collections

�t � fS � N �X

i�S

xi � v�S� � t�X

i�N

xi � v�N�g

ful�ll ���� with t � IR such that �t �� ��

The core of a collection of coalitions is also of interest in the characteriza

tion of the extreme elements of the core of a game �Cf� Kuipers�Derks ������

If x is a core allocation of the game v� x � Core��� v�� the allocation set

C��� v� x� � fy � IRN �X

i�N

yi � � x� �y � Core��� v� x� for an � � g

expresses the possibilities of the players in N to deviate from the distribution

x without leaving the core� It is immediate that C��� v� x� equals the core

of �v�x � �� with

�v�x � fS � N �X

i�S

xi � v�S�g�

Of course� the extremality of x in Core��� v� is equivalent to the condition

that C��� v� x� does not have a nontrivial linear subspace as a subset� and

this is again equivalent to the condition that the core of the collection �v�x �

� constitutes a pointed cone� The necessary and su�cient condition on

a collection �� such that Core��� is a pointed cone� is the socalled non

degeneracy �see Rosenm�uller ������ for several applications of this notion in

Game Theory��

The contents of the paper is now as follows� Section � discusses the sit

uation where the collection � is a lattice� The extremal rays of the core

are described� and with this result characterizations are given of collections

�

having a pointed core� and a core equal to fg� The results are applied to

the core of convex games� Section � discusses the nondegeneracy property

of collections� We show its equivalence with the pointedness of the corre

sponding core� Section � treats the notion of balancedness� We prove that

the core of a collection is a linear subspace if and only if the collection is

balanced� Together with our characterization of collections having a pointed

core we derive necessary and su�cient conditions for a collection such that

its core equals fg�

Another characterization of collections with core equal to fg is given in sec

tion � using the notion of exactness� Several links with former introduced

notions and results are described� and we will go brie�y into a relation with

the notion of exact game as studied in Schmeidler �������

In the sequel the summationP

i�S xi is abbreviated to x�S�� the allocation

ei� i � N � distributes � unit to player i and to the other players� Further�

we assume N � � for any collection � of coalitions� This does not a�ect

generality where the core of � is considered because x�N� � for all x �

Core����

� Lattices

In Derks�Gilles ������ the extreme rays� i�e�� the �dimensional faces� of the

�polyhedral� cone Core��� are characterized in case � is a lattice of coalitions�

These extreme rays correspond to the side payments ej � ei� with i� j � N

and player j chosen such that j � S for each S � � containing player i� i�e��

the side payment in which a player i pays one unit of utility to player j in

�

case j is a member of all coalitions in � for which i is a member� Let us

denote the set of these players j by D�

i �

D�

i � �fS � � � i � Sg�

In case � is intersectionclosed it is the smallest coalition in � containing

player i�

In general� the allocations ej � ei� with j � D�

i � are elements of the core

of �� So� Core��� contains the cone generated by these side payments� In

the lattice case we have equality�

Theorem � �Derks�Gilles� Let � � �N be a lattice� The core of � equals

the cone generated by the allocations ej � ei� with i � N and j � D�

i �

The core of a lattice � is pointed whenever the setsD�

i � i � N � are unequal

to eachother� This follows from the property that D�

j � D�

i whenever j is

an element of D�

i � A lattice is called discerning if D�

j �� D�

i for di�erent

players i and j�

Corollary � The core of a lattice � is a pointed cone if and only if � is

discerning�

Examples of discerning collections are those collections � for which the sets

D�

i � i � N � are minimal� D�

i � fig for all i � N � We will call these collections

strictly discerning� From Theorem � it is clear that

Corollary � The power set �N is the only lattice with a core equal to fg�

�

Lattices of coalitions appear in the study of the core of a convex game�

A game v is convex if it ful�lls the inequalities

v�S� � v�T � v�S � T � � v�S � T � for all S� T � N �

For a core allocation x of a convex game v and S� T � �v�x we have

v�S � T � � v�S � T � x�S � T � � x�S � T � � x�S� � x�T � � v�S� � v�T ��

This impliesS�T� S�T � �v�x� i�e� �v�x is a lattice� We observed already that

the extremality of x in the �standard� core Core��N � v� is equivalent to the

pointedness of the core of �v�x� According to Corollary � this is equivalent to

the discerning property of �v�x as far as convex games are concerned� Observe

that Theorem � implies that the �dimensional faces of the standard core of a

convex game connects those extreme elementswhich arise from side payments

between precisely two players�

� The collections with a pointed core

Corollaries � and � are not valid for arbitrary collections �� Consider the

following example of a collection of subsets of a �player set N � f�� �� �� �g�

let � consist of the empty set �� N � and the �player sets f�� �g� f�� �g� f�� �g�

and f�� �g� � is clearly �strictly� discerning and is not a lattice� Furthermore�

the side payments ���������� �� with � � IR are all contained in the core

of �� this shows that Core��� is not pointed� and certainly not equal to fg�

For a collection � � �N the span of � is de�ned to be the set� denoted

by Span���� consisting of the coalitions R whose indicator vectors eR � IRN

�

are contained in the linear subspace spanned by the indicator vectors of the

coalitions in �� i�e��

Span��� � fR � N � eR �X

S��

�SeS for some �S � IR� S � �g�

We call � non�degenerate if the indicator vectors eS� S � �� span the whole

allocation space IRN � i�e�� Span��� � �N �

It is clear that nondegeneracy of a collection � implies that it is dis

cerning� The reverse holds if � is a lattice� To show this we prove that the

indicator vectors of the sets D�

i � i � N � form a basis of IRN � Suppose this is

not the case� thus there exist scalars �i� i � N � not all zero� such that

X

i�N

�ieD�i

� �

Let j � N be such that �j �� and jD�

j j as large as possible� Using that

D�

j � D�

i whenever j � D�

i � i � N � and the assumption that � is discerning�

we conclude that

�X

i�N

�ieD�i

�j �X

i�j�D�i

�i � �j �

implying �j � � a contradiction�

Thus� nondegeneracy plays a similar role in the general case as the dis

cerning property does in the lattice case� The following theorem shows this

in more detail when compared with Corollary ��

Theorem � The core of a collection � � �N is a pointed cone if and only

if � is non�degenerate�

Proof � Core��� is a pointed cone if and only if for an x � IRN with x�S� �

for all S � � we must have x � � And this is equivalent to the assertion

that feS � S � �g is a basis� i�e�� � is nondegenerate� �

� Balanced collections

In the characterization of games with a nonempty core the notion of balanced

collections of coalitions plays the leading role �cf� Bondareva ����� Shapley

������ A collection � is said to be balanced if there exist positive scalars �S

for all S � � such that eN �P

S�� �SeS�

The following theorem shows the relation of the balancedness notion with

our subject�

Theorem � The core of a collection � � �N is a linear subspace if and only

if � is balanced�

Proof � Let � be balanced� say with positive scalars �S � S � �� such that

eN �P

S�� �SeS� Then� for x � Core��� and arbitrary T � � we have

�x�T � � x �

�T�X

S���S ��T

�SeS � eN� ��

�T

X

S���S ��T

�Sx�S� � �

This proves that �x is an element of the core of �� i�e�� Core��� is a linear

subspace�

Now suppose Core��� is a linear subspace� The inequalities x�N�

and x�S� � � S � �� therefore imply x�T � � with T � � arbitrary �recall

that N � ��� Using a well known theorem of Farkas �cf� Weyl ���� Theorem

�� we may conclude that �eT is in the cone generated by the allocations eS�

�

S � �� and �eN � say

� eT �X

S��

�TS eS � �T eN � ���

where the coe�cients in ��� are supposed to be nonnegative� Notice that

we in fact used a duality correspondence here� Adding these equalities we

obtain

�eN �X

S��

�� �X

T��

�TS �eS�

with � �P

S�� �T � Without loss of generality we assume �N � �� and� thus�

� � � This shows that � is balanced� �

In Ray ��� �� coalitions S � N are discussed� given a game v on N � with

the property that the core of the game v on the player set S is nonempty� i�e��

fx � IRS � x�S� � v�S�� and x�T � � v�T � for all T � Sg �� �� Coalitions

with this property are called credible� and Ray showed that the core of v on

N is equal to the restricted core Core�R�v�� v�� where R�v� is the collection

of credible coalitions of v�

From this it follows that Core�R�v�� � fg so that R�v� is a balanced

collection according to Theorem ��

By combining the last two theorems we obtain

Corollary � The core of a collection � of coalitions equals fg if and only

if � is both balanced and non�degenerate�

The example at the beginning of section � shows that balancedness and

the discerning condition� which is weaker than nondegeneracy� is not suf

�cient in order to have a core equal to fg� One easily checks that for a

�

balanced collection the discerning property is equivalent to the strict dis

cerning property�

� Exact collections

Another characterization of collections with one point cores is obtained by

examining under which condition a coalition T � N yields Core��� �

Core�� � fTg�� Trivially we have Core��� � Core�� � fTg�� The reverse

inclusion holds in case the inequality x�T � � is implied by the inequalities

x�S� � � S � �� and x�N� � By the same reasoning as in the proof of

Theorem � we conclude that eT is a nonnegative weighted sum of the vectors

eS� S � �� and �eN �

Let us denote the set of coalitions with this property by E����

E��� � fT � N � eT �X

S��nfNg

�SeS � �NeN � with �S � for S � �g�

If T � E��� and x � Core��� then� for eT �P

S�� �SeS��NeN with �S � �

S � �� we have

x�T � � x eT � x �X

S��

�SeS � �NeN� �X

S��

�Sx�S�� �Nx�N� � �

Therefore�

Core��� � Core�� � fTg� if and only if T � E���� ���

In case � equals E��� we call the collection � exact � According to the above

reasoning the core of a collection � shrinks if � is enlarged with an arbitrary

coalition outside � if and only if the collection is exact� It is immediate that

��

E��� is contained in any exact collection containing � as a subset� E��� is

therefore called the exact cover of ��

Simple examples of exact collections are the sets �x � fS � N � x�S� �

g� with x � IRN � x�N� � � Observe that Core��� � fx � IRN � x�N� �

� � � �xg�

Theorem The cores of a collection � of coalitions and its exact cover

coincide� Further� it is equal to fg if and only if its exact cover E��� equals

�N �

Proof � Of course� � � E���� Furthermore� E��� � E�� � fTg� for each

T � E���� as can easily be checked� Using ��� we conclude that

Core��� � Core�E�����

It is now evident that for collections with exact cover equal to �N the

core is equal to fg� Furthermore� according to ��� we have� for T �� E����

Core���fTg� � Core���� This proves that for collections � with E��� �� �N �

the core unequals fg� �

From Corollary � and Theorem � it follows that the equality E��� � �N

is equivalent to the balancedness and nondegeneracy of �� Another link

between �properties of� the exact cover and balancedness is the following

characterization of a balanced collection�

Theorem The collection � � �N is balanced if and only if its exact cover

equals the span of ��

��

Proof � Obviously we generally have the inclusion E��� � Span���� Now

suppose that � is balanced� with positive weights �S � S � �� such that

eN �P

S�N �SeS� For T � Span���� say eT �P

S�� �SeS� and scalar � we

have

eT � �eN �X

S��

���S � �S�eS�

so that for � large enough all coe�cients are nonnegative� implying that T

is an element of the exact cover of ��

The proof of the reverse implication is based on the fact that N � Span���

�since we assume N � ��� We thus obtain that the complements NnS of the

elementsS of � are contained in Span��� and� thus� in E���� In combination

with ��� this implies that the core of � must be a linear subspace� According

to Theorem � we conclude that � is balanced� �

One easily shows that collections � for which � � Span��� are exact� so

that Theorem implies that for exact collections the balancedness property

is equivalent to � � Span����

The notion of exactness is also used in the context of a game �cf� Schmei

dler ���� and Faigle �� ��� a game v is exact if it has a nonempty core

and

v�S� � min�fx�S� � x � Core��N � v�g� for each nonempty coalition S�

The following characterization of an exact game explains the relationship

with the previous notion of exactness�

��

Theorem � �Schmeidler� A game v is exact if and only if for each equality

eT � �NeN �P

S�N �SeS with non�negative coe�cients we have

v�T � � �Nv�N� �X

S�N

�Sv�S��

It is evident that exactness of a game v implies that the collections �v�x

are exact for each x � Core��N � v�� Using Schmeidler�s characterization one

easily proves the reverse�

Corollary �� Let v be a game with Core��N � v� �� �� Then v is exact if and

only if the collections �v�x are exact for each x � Core��N � v��

��

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