Consistent restricted Shapley values

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Transcript of Consistent restricted Shapley values

Consistent Restricted Shapley Values

Jean Derks� Hans Petersy

Revised version� March ����z

Abstract

Transferable utility games with an additional power structure on thecoalitions are considered� This power structure is not explicitly given�but only implicitly via a value� a value is a map assigning an N �vectorto every game with player set N � The implicit power structure isdescribed by the concept of e�ectiveness of a coalition for a given value�E�ectiveness of coalitions is constrained by axioms� in particular� thecollection of e�ective coalitions is assumed to be closed under takingunions� Other axioms concern e�ciency� and consistency in a senserelated to the consistency axiom of Hart and Mas�Colell ����� Themain result of the paper is an axiomatic characterization of a classof restricted Shapley values� with the e�ective coalitions forming alattice�

� Introduction

Consider the situation where a number of �rms is connected in a network�for instance a network of cables connecting the �rms with a power supplier�or a network of pipelines connecting the �rms with an oil reservoir� or an

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

tricht� The Netherlands� E�mail address� jean�derks�math�rulimburg�nlyDepartment of Economics� University of Maastricht� P�O� Box ���� ���� MD Maas�

tricht� The Netherlands� E�mail address� h�peters�ke�rulimburg�nlzThe authors thank the referee and the associate editor for their comments� which led

to an improvement of the paper� The usual disclaimer applies�

infrastructure of roads or railways connecting the �rms with a harbour�� Bycooperating� coalitions of �rms may make additional pro�ts�or save costs�but only if all �rms in the coalition can be connected with the power supplier�Observe that such a situation is characterized by two structures� First� thereis a function assigning to each coalition its potential pro�ts� i�e�� pro�ts thatcan be realized given that all �rms in the coalition can be connected withthe power supplier� Second� there is the structure�described for instanceby a graph or network�determining which coalitions can actually realizethose pro�ts� While a situation like this may often be described by just onecharacteristic function by incorporating in some way or another the coalitionsthat are capable of realizing their pro�ts into the function describing thosepro�ts� it is not obvious that this is the best way of modeling the situation�In particular� the infrastructure �of cables� pipelines� roads� or railways� maybe more stable than the potential pro�ts� which may depend on prices ininput and output markets� and other unstable factors�

To be more speci�c� we consider the following numerical example� Thereare four �rms and one power station connected in a network as follows� �rm� and �rm � are both connected with the power station and also with �rm� and �rm is only connected with �rm � The potential pro�ts of thecoalitions� which can be realized only if the �rms can be connected with thepower station� are as follows� One��rm coalitions make pro�ts of �� two��rmcoalitions of � three��rm coalitions of �� and the grand coalition of all four�rms makes a pro�t of � by convention� the empty coalition makes zeropro�ts� The coalitions fg� fg� and f� g� however� are not connected withthe power station� and thus make the same pro�ts as the empty coalition�namely zero� The coalition f�� g makes the same pro�t as the coalition f�g�since �rm cannot be connected� Similarly� f� �g makes the same pro�ts asf�g� and f�� � �g as f�� �g� If we assume that the grand coalition forms andwe apply a solution concept like the Shapley value �Shapley� ����in thispaper the focus is on the Shapley value� to this modi�ed pro�ts game� weobtain the payo� vector �

����� � ��� ��� This clearly re�ects the structure

of the network without this additional structure� the Shapley value wouldjust be the vector �

����� �� �� ��� Note that� if the pro�ts would change

but the network would remain unaltered� a similar qualitative e�ect of thenetwork structure on the Shapley value would obtain� Therefore� it seemsnatural to model the network structure by means of the value� and that iswhat we will do in this paper�

As a second example� suppose the set of players is a parliament� parti�tioned in political parties� Suppose that each political party is subject to thestrict party discipline that only unanimous votes may be cast� In this case�only cooperation of complete political parties �fractions� is useful� becauseindividual members are not expected to vote on their own� In a similar formalmodel� one may think of the players as divisions partitioned into �rms � whichmay only cooperate�for instance� exchange technological information�withthe consent of the other divisions within the same �rm�

More formally� we try to capture this idea of a �xed power structure inthe framework of games with transferable utility� We assume that this powerstructure is re�ected by a value� i�e�� a map assigning to every game withtransferable utility a distribution among the players� More precisely� we willassume that every coalition is either e�ective� i�e�� a�ects the distribution inevery game where it is present� or ine�ective� i�e�� never a�ects the distribu�tion� Consequently� e�ectiveness of a coalition is re�ected by the value anddoes not depend on the particular game �characteristic function� at hand�As a minimal condition� we assume that the union of e�ective coalitions isagain e�ective�as is the case in the examples above� Both assumptions areformalized by axioms�

We will propose three other axioms� The �rst one of these is an axiom ofe�ciency� The second axiom requires the value to be consistent across gameswith varying numbers of players� and is closely related to the consistency re�quirement of Hart and Mas�Colell ������� Finally� there is an axiom� �rst�determining the value for players who belong to zero or one e�ective coali�tions and for who all other coalitions are ine�ective� and� second� establishingthe basic distribution between players in two disjoint minimal e�ective coali�tions� The main result of the paper is that these axioms determine a classof coalitionally restricted Shapley values� A consequence of the axioms turnsout to be that also the intersection of e�ective coalitions is again e�ective�i�e�� the class of e�ective coalitions forms a lattice� In the second exampleabove� the e�fective coalitions form a lattice� but in the �rst example theyclearly do not�

Our main result generalizes the Hart�Mas�Colell characterization of theShapley value� The uniqueness part of the proof also implies a new proofof the uniqueness part of the Hart�Mas�Colell theorem� while the necessitypart uses the fact� proved by Hart and Mas�Colell� that the Shapley value isconsistent� A completely self�contained proof can be based on the concept

of a restricted potential�generalizing the Hart�Mas�Colell potential�at thecost of brevity see Derks and Peters �������

Coalitionally restricted Shapley values are obtained by considering onlydividends � or marginal contributions in the probabilistic representation �of e�ective coalitions� The characterized class forms a subclass of the class ofrestricted Shapley values introduced in Derks and Peters ������ Recently�restrictions on coalitions in games have been discussed by several other au�thors� In fact� the paper by Derks and Peters ����� arose as an attemptto generalize some of these approaches� It presents an axiomatic character�ization of a class of restricted Shapley values for which the set of e�ectivecoalitions is closed under taking unions� not necessarily under taking inter�sections� This characterization is based on an axiom of monotonicity relatedto the condition used by Young ������ to characterize the Shapley value�Related works are Gilles and Owen ������� Gilles et al� ������ Faigle andKern ������ Hsiao and Raghavan ������� Feldman ������ Earlier contri�butions include Myerson ������ and Owen ������� The lattice structure inthe context of cooperative game theory is also studied by Derks and Gilles������ and by Gilboa and Lehrer �������

The organization of the paper is as follows� In Section � after somepreliminary de�nitions� we introduce the axioms brie�y described above� InSection we introduce restricted Shapley values by means of dividends� andwe also present a corresponding probabilistic formula� Next� some prepara�tions are made for the main result� which is stated and proved in Section ��Section � concludes with some remarks on other characterizations�

� Preliminaries and Axioms

Let Z � IN denote the set of potential players� �IN is the set of naturalnumbers� � denotes inclusion and � denotes strict inclusion�� The reasonfor considering a �possibly large� set of potential players is that we wish tobe able to vary the number of players� A coalition is a �nite subset of Z� For��person coalitions we often omit braces� i�e�� we write i instead of fig�

A �transferable utility� game is a pair �N� v�� where N is a coalition andv � N � IR is a function with v��� � �� The function v is the characteristicfunction� The number v�S� is called the worth of coalition S� By G we denotethe set of all games� For �N� v� � G and M � N � the characteristic function

of the game �M�v� � G is the map v restricted to M � A value is a map� assigning a vector in IRN to every game �N� v� � G� to be interpreted asa solution of the game� A well�known value is the Shapley value �Shapley���� see Section ��

For a given value � a nonempty coalition S is called ��e�ective if for allgames �N� v�� �N�w� � G with S � N � v�T � � w�T � for all T � N withT �� S� and with v�S� �� w�S�� we have ��N� v� �� ��N�w�� A coalitionS is called ��ine�ective if for all games �N� v�� �N�w� � G with S � Nand v�T � � w�T � for all T � N with T �� S� we have ��N� v� � ��N�w��Observe that the empty coalition is ��ine�ective� If it is clear which valueis meant� we will often use the expressions e�ective and ine�ective� omittingthe pre�x �� A coalition S is a maximal ��e�ective coalition in N if it is ��e�ective� S � N � and every S � T � N is not ��e�ective� For k � IN � f�g�a player in N is called a k�player �in N� if� of all coalitions in N containingthat player� exactly k coalitions are e�ective while the other coalitions areine�ective� Note that also this concept depends on a given value ��

Observe that the classi�cation in e�ective and ine�ective coalitions isnot exhaustive a coalition might be in�uential and a�ect the value in onegame �or� more precisely� pair of games�� but not exert any in�uence inanother game� Such an occurrence� however� would be inconsistent with theassumption of a �xed power structure� independent of the particular gameat hand� This consideration is formalized by our �rst axiom� �Axioms areformulated for an arbitrary value ���

Axiom � Every nonempty coalition S is either ��e�ective or ��ine�ective�

Also the second axiom concerns the power structure among coalitions� Itrequires the collection of e�ective coalitions to be closed under taking unions�

Axiom � For all coalitions S and T � if S and T are ��e�ective� then so isS � T �

We will assume that in every game �N� v� some maximally ��e�ective coali�tion is formed and that the worth of this coalition is distributed among theplayers of N � This is formalized by the following e�ciency requirement�

Axiom � For all �N� v� � G such that N contains a ��e�ective coalition�Xi�N

�i�N� v� � v�S�

for some maximal ��e�ective coalition S in N �

Obviously� under Axiom � such a maximal ��e�ective coalition is unique�The next axiom is a consistency property closely related to that intro�

duced by Hart and Mas�Colell ������ for the Shapley value� Let �N� v� � G�and U � N � The reduced game �NnU� v��U� is de�ned as follows� For every� �� S � NnU �

v��U�S� �Xi�S

�i�S � U� v��

Thus� in the reduced game� the worth of coalition S is equal to what theplayers in S receive together if they would play the �restriction of the� originalgame with the coalition U � If � is e�cient in the usual sense� implying thatP

i�S�U �i�S � U� v� � v�S � U�� then the reduced game can be written as

v��U�S� � v�S � U��Xi�U

�i�S � U� v��

which is the de�nition of the Hart�Mas�Colell reduced game�The following axiom of consistency �or reduced game property� requires

that in a reduced game for an e�ective coalition the players do not redis�tribute the total of what they receive in the original game� In this sense� thevalue is consistent in the distributions if the number of players varies�

Axiom � For every �N� v� � G and every U � N with NnU ��e�ective� wehave

�i�NnU� v��U� � �i�N� v�

for every i � NnU �

It might not be obvious that consistency is a natural requirement in ourframework with an additional power structure� since this structure couldbe torn up by considering reduced games for arbitrary smaller coalitions�For this reason� in Axiom � only reduced games for e�ective coalitions areconsidered� In this way we avoid� for instance� players becoming ��players ina reduced game who were not ��players in the original game �cf� Lemma ����Actually� it will follow from our main result that if a value satis�es Axioms��� and Axiom � below� then the corresponding e�ective coalitions form a

lattice� This implies that in the reduced game the original power structureis preserved�

The consistency axiom will be used to relate distributions assigned by� across games with distinct numbers of e�ective coalitions� In order toapply this procedure we need to assume a few basic distribution codes� Thisis done in our �nal axiom� which may be regarded as an extension of thestandardness axiom of Hart and Mas�Colell �������

Axiom � Let S and T be coalitions�

�a� If T � S and T consists exactly of the ��players in S� then �i�S� v� � �for every �S� v� � G and i � T �

�b� If S is ��e�ective� T � S and T consists exactly of the ��players in S�then �i�S� v� � ��jSj�v�S�� v�SnT �� for every �S� v� � G and i � T �

�c� If N � S � T with S � T � � and such that N � S� and T are the only��e�ective coalitions in N � then

�i�N� v� � ���jT j�v�T � � ���jN j��v�N�� v�S�� v�T ��

for every �N� v� � G and i � T �

The cases �a�� �b�� and �c� of Axiom � take care of ��players� ��players� and�players� respectively� A ��player obtains �� A ��player obtains his fractionof the marginal contribution to the grand coalition observe that� since in theformulation of this axiom the grand coalition is e�ective by assumption� it isthe unique e�ective coalition to which the members of T belong� In �c� thebasic divisional proportion between players in two minimal disjoint e�ectivecoalitions is determined�

Note that Axiom � is a generalization of the standardness axiom of Hart�Mas�Colell as formulated for the Shapley value in that case� all coalitionsare e�ective� implying that only part �c� of Axiom � applies� and only for thecase S � fjg� T � fig� Then� �i�N� v� � v�i�� �����v�fi� jg�� v�i�� v�j���which is the requirement of the standardness axiom�

The main result of the paper is the characterization of a class of coali�tionally restricted Shapley values by Axioms ���� Before we can formulateand prove this result� we �rst need to introduce and study these solutions this is the subject of the next section�

� Restricted Shapley values

As in the preceding section� let Z be the set of potential players� Let � bea ��nite or in�nite� possibly empty� collection of coalitions which is closedunder taking unions� Such a collection is called a power structure� An ele�ment S of � is called minimal if it contains no proper subset which is in ��The collection � is a lattice if for every pair of its elements with nonemptyintersection� this intersection is in �� For an arbitrary coalition S� let

S ��

T �T�S� T��

T

where the union over an empty collection is by de�nition the empty set� Notethat� since � is union�closed by de�nition� if S �� �� then S is the �unique�maximal subset of S which is in �� Thus� for S �� �� S � S if and only ifS � ��

For �N� v� � G and a power structure � we de�ne dividends ��

v �S� forcoalitions S � � recursively� as follows�

��

v �S� �

�v�S� if S is a minimal element of �v�S��

PT �T�S�T���

v �T � otherwise�

Then a value �� on G is de�ned as follows� For every �N� v� � G and everyi � N �

��

i �N� v� �X

S�i�S�N�S��

��

v �S��jSj�

where summation over the empty set yields zero by de�nition� If � is thelattice of all nonempty coalitions� then �� is the Shapley value� denoted by� see Shapley ����� and Harsanyi ������ In general� we call �� a restrictedShapley value�

Restricted Shapley values were introduced earlier in Derks and Peters����� by their probabilistic formula� The equivalence of both formulationsis proved in Section of Derks and Peters ������ and stated next as a lemma�For a coalition S� jSj denotes the cardinality of S�

Lemma ��� Let � be a power structure� Then for every �N� v� � G andevery i � N

��

i �N� v� �X

S�S�N�i ��S

jSj��jN j � jSj � ���

jN j��v�S � fig�� v�S���

For the Shapley value� where � consists of all nonempty coalitions� Lemma ��just gives the familiar equivalence between Shapley�s probabilistic formulaand the Harsanyi dividends�

Note that� from the de�nition of �� or from Lemma ����

���v� � ��v� where v�S� �� v�S� for all S � N� ���

The following lemma will imply e�ciency �Axiom � of restricted Shapleyvalues�

Lemma ��� Let � be a power structure� Let �N� v� � G� ThenXi�N

��

i �N� v� � v�N��

ProofXi�N

��

i �N� v� �Xi�N

�i�N� �v�

� �v�N�

� v�N��

where the �rst equality follows from ��� and the second from e�ciency of theShapley value� �

The next lemma will be used to prove consistency �Axiom �� of restrictedShapley values when � is a lattice�

Lemma ��� Let � be a lattice� Then� for coalitions N and U � N withNnU � �� we have v���U � v���U �

In the proof of this lemma we need the following property of a lattice ofe�ective coalitions�

Lemma ��� Let � be a lattice� Let N be a coalition� U � N with NnU � ��Let S � NnU � Then

�a� S � U � S � U �

�b� ��players of S are ��players of S � U �

�The idea to consider the game v was suggested by a referee�

Proof Using the intersection�closedness of � and NnU�S � U � �� weobtain S � U � NnU � �� Therefore�

S � U � �S � U �NnU� � �S � U � U�

� S � U�

This proves statement �a��Statement �b� follows from

SnS � Sn�S � U� � SnS � U � �S � U�n�S � U �� �

Proof of Lemma ��� From the de�nition of �� it is evident that

��

i �S� v� � ��

i �T� v� � ��

i �S� v� ��

for all coalitions S� T � with S � T � S� and players i � S� Now let S � NnUbe arbitrary� Then

v���U �S� � v���U�S�

�Xi�S

��

i �S � U� v�

�Xi�S

��

i �S � U� v�

�Xi�S

��

i �S � U� v�

� v���U�S��

where the last and �rst two equalities follow by de�nition� the third equalityfollows from Lemma ���a� and ��� and the fourth equality follows fromLemma ���b� and the observation that ��players are assigned � by ��� �

� The Characterization Result

In this section we prove the main result of the paper� yielding the class ofall values that satisfy Axioms ���� Independence of the axioms follows fromexamples collected in the Appendix�

Theorem ��� A value � on G satis�es Axioms � if� and only if� � � ��

for a lattice ��

��

We �rst show that every �� with � a lattice satis�es the �ve axioms�

Proposition ��� Let � be a lattice� Then �� satis�es Axioms ��

Proof Let S be a nonempty coalition� If S �� � then S is ���ine�ectiveby de�nition of ��� If S � � then let i � S and let �N� v� be a game withS � N � By Lemma ��

��

i �N� v� �X

T �T�N�i ��T

jT j��jN j � jT j � ���

jN j��v�T � fig�� v�T���

Observe that the numbers v�T � never change if we vary v�S�� since i � Sbut i �� T � On the other hand� v�T � fig� changes if we vary v�S� wheneverT � fig � S and in particular for T � Sni� Therefore� ��

i �N� v� changeswhenever v�S� is varied� We conclude that S is ���e�ective� So �� satis�esAxiom �� Also� since � is the collection of all ���e�ective coalitions� Axiom is satis�ed since � is a lattice by assumption� Axiom follows by Lemma ��Axiom � follows by the de�nition of ��� as is easily established �alternatively�Lemma �� may be applied��

In order to show that �� satis�es Axiom �� let �N� v� � G� U � N � NnU ��� Observe that v���U � v��U since� for arbitrary coalitions S � NnU � wehave

v���U �S� �Xi�S

��

i �S � U� v�

�Xi�S

�i�S � U� v�

� v��U�S�� ��

Therefore�

��

i �NnU� v���U� � �i�NnU� v���U�

� �i�NnU� v���U�

� �i�NnU� v��U�

� �i�N� v�

� ��

i �N� v��

where the last and the �rst equalities are by de�nition� the second by Lemma�� the third by ��� and the fourth equality follows from the consistency of

��

the Shapley value �Hart and Mas�Colell� ������ We conclude that �� satis�esAxiom �� �

Observe that intersection�closedness of � is only used to prove that �� sat�is�es Axiom �� Indeed� if � is just a power structure but not a lattice� then�� does not satisfy this axiom� This follows from Theorem ��� and the factthat for a power structure � the value �� satis�es Axioms �� and Axiom ��

We next prove that Axioms ��� determine the class of restricted Shapleyvalues corresponding to lattices�

Until further notice we make the following assumptions� By � we denotea value on G satisfying Axioms ���� � is the collection of coalitions that are��e�ective� �In�e�ectivity always refers to the value �� and every referenceto a power structure refers to �� In particular� for a coalition N � if N �� �then N is the unique maximal e�ective coalition in N � and if N � � thereare no e�ective coalitions in N �

We will prove that � � ��� We start with a lemma concerning ��players�

Lemma ��� Let �N� v� � G��a� �i�N� v� � � for every i � NnN ��b� �i�N� v� � �i�N� v� for every i � N �

Proof Part �a� is just Axiom ��a�� For part �b� assume that U �� NnN �� ��Observe that� for every S � N � we have S � U � S� Hence� by Axioms and ��a�� for every such S

v��U�S� �Xi�S

�i�S � U� v�

� v�S��Xi�U

�i�S � U� v�

� v�S�� ���

This implies that v��U and v coincide on the e�ective coalitions in N � so that��N� v��U� � ��N� v� according to Axiom �� By Axiom �� for all i � N ��i�N� v� � �i�N� v��U�� Therefore �i�N� v� � �i�N� v� for every i � N � �

Our next objective is to prove that � is in fact a lattice� In order to provethis we need to know the distribution ��N� v� on games �N� v� in which allcoalitions except possibly N have worth �� We call such a game a plain game�

Lemma ��� In a plain game the value � assigns to each player the sameamount�

Proof The proof is by induction on the number of players� The lemmais obviously true for ��player games� Assume that the lemma is true forplain games with less than n players �n � and let �N� v� be a game withjN j � n�

If N contains ��players then� because the game �N� v� is also plain� ac�cording to the induction hypothesis all players in N obtain the same amountunder �� Axiom implies that the sum of these amounts equals v�N� � ��Therefore� �i�N� v� � � for all i � N � By Lemma ���a� and �b� we concludethat �i�N� v� � � for all i � N �

If N possesses ��players� say U �� � is the set of ��players in N � thenconsider the set M � NnU � A player in M but not in M would be a ��playerand therefore belong to U � an obvious impossibility� So M � M � i�e��M � ��By Axiom ��b� we have

�i�N� v� ��

n�v�N�� v�M�� �

nv�N� for all i � U� ���

So by Axiom

v��U�M� �Xj�M

�j�N� v� � v�N��Xj�U

�j�N� v� �jM j

nv�N��

For S � M we have v��U�S� �P

j�S �j�S � U� v� � � by the inductionhypothesis and Axiom applied to the plain game �S � U� v�� which has allcoalition worths equal to �� Thus� v��U is a plain game and by the inductionhypothesis and Axiom � �i�M�v��U � � ���n�v�N� for every i � M � This�together with Axiom � and ���� proves that �i�N� v� � �j�N�V � for alli� j � N �

Finally� suppose that N is e�ective without ��players� By the same argu�ment as above �M�v��NnM� is a plain game for everyM � �� So the inductionhypothesis and Axiom � imply

�i�N� v� � �j�N� v� for all i� j �M� for all M � �� ���

Since there are no ��players we can write N as a union of e�ective coalitionsunequal to N � Since � is union�closed we can �nd S� T �� N in � withN � S � T �

If S � T �� � let i � S � T � By ���� �i�N� v� � �j�N� v� for all j � S andj � T � so we are done�

If S�T � � and there is an e�ective coalitionM inN withM �� fS� T�Ng�then without loss of generality assume M � S �� � and let i �M � S� Againby ��� �i�N� v� � �j�N� v� for all j � S and j � M � T � so also in this casewe are done�

The only case left to consider is the situation of Axiom ��c�� In thatcase Axiom ��c� can be applied� since �N� v� is plain� this axiom impliesthat �i�N� v� � ���n�v�N� for every i � N � This completes the proof of thelemma� �

Proposition ��� If � satis�es Axioms �� then � is a lattice�

Proof Suppose � is not a lattice� Then there are e�ective coalitionsS� T with nonempty intersection but not in �� Let S� T be such coalitionswith N �� S � T of minimal cardinality� i�e�� within N � there is no smallere�ective coalition that is the union of two e�ective coalitions with nonemptyintersection not in ��

We claim that there is a player i in S � T who is a ��player in S andalso in T � If not� then for every i � S � T there is an Si � S with i � Siand Si � �� hence Si � T � �� or a Ti � T with i � Ti and Ti � �� henceTi�S � �� �Observe that the minimality of N is used to conclude Si�T � �or Ti � S � ��� Hence� every i � S � T is a member of an e�ective coalitioncontained in S � T � But then� by union�closedness� S � T � �� which is acontradiction� This proves our claim�

Let i be a player as in the previous paragraph� Let �N� v� be a gamewith for all W � N � if i �� W � then v�W � � �� Consider the reduced game�S� v��TnS�� For W � S� W � �� we have

v��TnS�W � �Xj�W

�j�W � �TnS�� v� � ��

where the second equality follows from Lemma ���� Axiom � and the factthat �W � �TnS�� v� is a zero�game� observe that i �� W � �TnS�� Hence��S� v��TnS� is a plain game� Again using Lemma ���� Axiom � and S � ��we obtain �j�S� v��TnS� � v��TnS�S��jSj for every j � S� In particular�by Axiom �� �k�N� v� � �j�N� v� for all k� j � S� Similarly� one proves�k�N� v� � �j�N� v� for all k� j � T � Because S � T �� � it follows that

��

�k�N� v� � �j�N� v� for all k� j � N � Hence� by Axiom � �j�N� v� �v�N��jN j for all j � N � This contradicts the e�ectivity of S and T �

We have proved that the restriction of � to N is a lattice� �

Before we can prove Theorem ��� we need one more lemma�

Lemma ��� Let N � Z� Suppose that for all M � N and all �M�v� � Gwe have

��M�v� � ���M�v�� ���

Then ��N� v� � ���N� v� for all �N� v� � G�

Proof If N is not e�ective the desired result follows from Lemma ��applied to � and ��� So assume that N is e�ective� Let M � N � M � ��For S �M we have by ���

v��NnM�S� �Xi�S

�i�S � �NnM�� v�

�Xi�S

��

i �S � �NnM�� v�

� v���NnM�S��

Let uM denote the unanimity game on M � i�e�� uM �S� � � if M � S anduM�S� � � otherwise� Then we can write

v���NnM � v��NnM � �vMuM

where �vM � v���NnM�M��v��NnM�M�� By de�nition of �� and ��� it followsthat for all i �M

��

i �M�v���NnM� � ��

i �M�v��NnM� � �vM�jM j

� �i�M�v��NnM� � �vM�jM j�

hence

��

i �N� v� � �i�N� v� � �vM�jM j� ���

where we use Axiom � for both � and ���If N contains ��players� say U �� � is the coalition of ��players of N �

then M � NnU is an e�ective coalition �see the proof of Lemma ����� The

��

��players in N obtain the same amounts from both �� and � according toAxiom ��b�� So by Axiom the total amounts for the players in M from ��

and � are also the same� implying with ��� that �vM � � and also the playersin M receive the same from �� and �� This proves the lemma for the casewhere N contains ��players�

Now suppose there are also no ��players in N � Then we may write N ��i�NMi where� for each i � N � Mi is an e�ective coalition with i �Mi � N �

Suppose� without loss of generality� that �vMi� � for some i � N � By ���

for M �Mi this implies that ��

j �N� v� � �j�N� v� for all j �Mi� Hence� forall e�ective M � N with M �Mi �� N we have� also by ����

��

j �N� v� � �j�N� v� for all j �M �Mi� ���

From ��� and the fact that it is impossible that ���N� v� � ��N� v� by Ax�iom � there is a player j � N withMi�Mj � N andMi�Mj � �� Note thatthere is no M �Mj with M � �� otherwise we would get a contradiction byconsidering M �Mi and applying ���� Similarly� there is no M � Mi withM � �� Since� by Proposition ���� � is a lattice� it follows that Mi� Mj�and N are the only e�ective coalitions in N � Therefore� the conditions inAxiom ��c� are satis�ed� implying �vMi

� �� a contradiction� This completesthe proof� �

Proof of Theorem ��� That �� satis�es all axioms if � is a lattice isProposition ��� The converse follows from Lemma ���� noting that condition��� holds for coalitions consisting of one player� in view of Axiom ��a�b�� �

Independence of the axioms is proved in the Appendix�

� Concluding Remarks

The present paper is related to Hart and Mas�Colell ������� which presents acharacterization of the Shapley value by means of two axioms� Consistency�and Standardness� The latter axiom �xes the value for two�player games our Axiom � reduces to this axiom if all coalitions are e�ective� as is the casewith the Shapley value� Besides� we use an additional e�ciency axiom Hartand Mas�Colell implicitly assume e�ciency� not only in their Standardnessaxiom� but also in their de�nition of the consistency property� since they have

��

v��U�S� � v�S � U� �P

i�U �i�S � U� v�! in their de�nition of the reducedgame� Finally� our Axioms � and are speci�c for our restricted coalitionsmodel� As remarked in the Introduction� a new proof of the uniqueness partof the Hart�Mas�Colell characterization of the Shapley value can be derivedfrom the proof of our main result this new proof is based on the second partof the proof of Lemma ����

Of the existing characterizations of the Shapley value� some can be extendedto restricted Shapley values in a rather direct manner� This is true� forinstance� for characterizations involving linearity or additivity axioms �seeB"acker� ������ Another characterization can be based on the following axiomof balanced contributions!� cf� Myerson ������ and Mas�Colell ��������

Axiom � For all �N� v� � G and all i� j � N �

�i�N� v�� �i�Nnfjg� v� � �j�N� v�� �j�Nnfig� v��

Instead of the whole Axiom � it will su�ce to impose only part �a�� dealingwith ��players� We then have the following characterization�

Theorem ��� A value � on G satis�es Axioms ��� �a�� and � if� and onlyif� � � �� for a power structure ��

We omit the proof of this theorem� which is straightforward and can be basedon induction� It can also be shown that the axioms are independent� Notethat in this characterization � does not have to be a lattice� Actually� onecould even relax Axiom � union�closedness of the class of e�ective coalitions�to obtain a larger class of restricted Shapley values�

Appendix� Independence of the Axioms

We present seven examples of values� satisfying all axioms except for Ax�ioms �� � � �� ��a�� ��b�� and ��c�� respectively that is� Example i satis�esall axioms except for Axiom i� Proofs are left to the reader�

�� Let Z � f�� g� � � ffg� f�� gg� and let � be given by ��f�� g� v� ����f�� g� v� for all �f�� g� v� � G� ���� v� � v���� ��� v� � v�� forall ��� v�� �� v� � G� Coalition f�g is neither e�ective nor ine�ective�

�This characterization has been sugggested to us by a referee of the paper�

��

� Let Z � f�� � g� and for every �N� v� � G let ��N� v� � ��N� #v�� where� is the Shapley value and #v�S� �� v�S� if S �� Z� #v�Z� �� v�f�� g��All coalitions are e�ective except for Z�

� Let Z � f�� g� ���f�� g� v� � � and ���f�� g� v� � �����v�f�� g� �v���� for all �f�� g� v� � G� ���� v� � v��� for all ��� v� � G� ��� v� � �for all �� v� � G� E�ciency for f�� g is violated�

�� Let Z � f�� � g� ��N� v� � ��N� v� for all �N� v� � G with jN j � �where � is again the Shapley value�� and

���f�� � g� v� � v��� � v�f�� g� � v�f�� g� � ��

���f�� � g� v� � v�� � v�f�� g� � v�f� g� � ��

���f�� � g� v� � v�� � v�f�� g� � v�f� g� � ��

for all �f�� � g� v� � G� where � � v�N� � �v��� � v�� � v�� �v�f�� g� � v�f�� g� � v�f� g��� This value � violates Axiom ��

��a�� Let Z � f�� � g� � � ff�g� fg� f�� gg� and for all �Z� v� � G�

���Z� v� � ��

��Z� v��

���Z� v� � ��

��Z� v��

���Z� v� � v�f�� g���

Let further ��N� v� � ���N� v� for all �N� v� � G with jN j � � Thisvalue � violates Axiom ��a�� but not �b� or �c��

��b�� Let Z � f�� g� let for all �f�� g� v� � G�

���f�� g� v� � ���v�f�� g� � ����v����

���f�� g� v� � ����v�f�� g� � ����v����

Let ���� v� � v��� and ��� v� � � for all ��� v�� �� v� � G� This value� violates Axiom ��b�� but not �a� or �c��

��c�� Let Z � f�� g� let for all �f�� g� v� � G�

���f�� g� v� � ����v�f�� g� � �����v��� � v����

���f�� g� v� � ���v�f�� g� � �����v���� v����

Let ���� v� � v��� and ��� v� � v�� for all ��� v�� �� v� � G� Thisvalue � violates Axiom ��c�� but not �a� or �b��

��

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��

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