A SURVEY OF CONSISTENCY PROPERTIES IN COOPERATIVE ...
? 1991 SocietyforIndustnaland AppliedMathematics 003
SIAM REVIEW Vol. 33, No. 1, pp 43-59, March 1991
A SURVEY OF CONSISTENCY PROPERTIES IN COOPERATIVE GAME THEORY* THEO S. H. DRIESSENt Abstract.The mainpurposeofthissurveypaperis to reviewtheaxiomaticcharacterizations oftheShapley value,theprekernel, theprenucleolus, and thecoreby meansofa consistency property in termsofthereduced games.Wheneverpossible,new resultsand new proofsare added. Key words. cooperativegame in characteristic functionform,reducedgame, Shapleyvalue, prekernel, prenucleolus,core,consistency AMS(MOS) subjectclassification.90D 12
1. Introduction.Sincetheintroduction ofthenotionofa cooperativegamein characteristicfunctionformin Von Neumann and Morgenstern[30], manysolutionsfor thesegames have been proposed.In the contextof cooperativegames in characteristic functionform,the solutionaims at describingone or more equitabledivisionsof the in thegame(or, equivalently, totalsavingsamongtheparticipants fairallocationsofthe totalcostto theusersin a joint project).The coreis one oftheveryfirst proposed"set" solutionsand was introducedand named in game theoryin Gillies [11], althoughthe idea of the core was alreadyforeshadowed by theTennesseeValleyAuthority cost alloin Ransmeier[22]. Shapley[24] producedan important"one-point" cationtreatment solution,theso-calledShapleyvalue. Moreover,we mentionthesetsolutionconceptof theprekerneland the one-pointsolutionconceptoftheprenucleolus. The mathematicalapproachto a proposedsolutionis to examinea numberof its (elementary)propertiesand, if possible,to providea minimalnumberof properties whichfullycharacterizethe solution.Shapley[24] listedthreepropertiesin his characterizationof the Shapleyvalue on the class of gameswitha fixedplayerset. The main oftheShapleyvalue,theprekernel, purposeofthispaperis to reviewthecharacterizations theprenucleolus, and thecore,on theclassof(almost)all gamesbymeansofa consistency oftheconsistency property. Generallyspeaking,the formulation property fora solution is in termsofthesolutionitselfand theso-calledreducedgames.This paperreviewsthe use of reducedgames throughout the literature on cooperativegames in characteristic functionform.Informally, thenotionof a reducedgame can be elucidatedas follows. A cooperativegame is alwaysdescribedbya finiteplayersetas wellas a real-valued "characteristic function"on thecollectionofsubsetsoftheplayerset.A so-calledreduced game is deduciblefroma givencooperativegame by removingone or moreplayerson theunderstanding thattheremovedplayerswillbe paid accordingto a specificprinciple (e.g., a proposedpayoff vector).The remainingplayersformtheplayersetofthereduced functionof which is composed of the originalcharacteristic game; the characteristic function,the proposedpayoffvector,and/orthe solutionin question.The consistency propertyforthe solutionstatesthatifall the playersare supposedto be paid according to a payoffvectorin thesolutionsetoftheoriginalgame,thentheplayersofthereduced vectorin thesolutionsetofthereducedgame. gamecan achievethecorresponding payoff in whatthe playersof the reducedgame can In otherwords,thereis no inconsistency achieve,in eitherthe originalgame or the reducedgame. * ReceivedbytheeditorsOctober24, 1988; acceptedforpublication(in revisedform)February20, 1990.
ofTwente,Enschede,theNetherlands. t DepartmentofAppliedMathematics,University 43
This content downloaded from 130.89.45.231 on Thu, 17 Dec 2015 08:41:44 UTC All use subject to JSTOR Terms and Conditions
44
THEO S. H. DRIESSEN
We advance threemotivesforthe studyof the consistencyproperty in the fieldof cooperativegame theory.First,a distinctionbetweengame theoreticsolutionscan be made by distinguishing betweenthe corresponding consistencyproperties.Second,the consistencypropertyfora fixedsolutionmay be of importanceforthedevelopmentof thetheoryofthatsolution.Third,theconsistency in thegametheoreticframeproperty workmaybe usefulin orderto determinethe"consistent"solutionofa realisticproblem (see ? 6 foran application). In ? 2 we bringthe framework of cooperativegame theoryto the forefront and discussthepropertieswhichwillbe used in the axiomatizationsof the solutionsas presentedin thenextsections.In ? 3 we reviewtheaxiomaticcharacterization oftheShapley value by means of a consistencypropertydue to Hart and Mas-Colell (Theorem 3.1 ) and due to Sobolev(Theorem 3.2). Section4 considerscharacterization oftheprekernel due to Peleg (Theorem 4.1 ) and of the prenucleolusdue to Sobolev (Theorem 4.2), whiletheaxiomatizationofthecoreofbalancedand totallybalancedgamesdue to Peleg (Theorems 5.1 and 5.2) is reviewedin ? 5. In summary,thispaper unifiessix related theoremsthatare scatteredthroughoutthe literatureon mathematics.Whenevernew proofsof theexistenceor uniquenesspartoftheaxiomatizationscan be given,theyare also included.An additionalpurposeof thispaper is to provethe uniquenesspartsof Theorems3.1, 3.2, and 4.2 by means of a singleline of argument.In othercases, we referto theproofsin the originalpapers. 2. Propertiesforsolutionsof cooperativegames. The mathematicalmodel of a cooperativegame in characteristic function formis describedby a finitenonemptysetN and a real-valuedfunctionv on the family2 N of subsetsof theplayerset N, satisfying v( 0 ): = 0. We referto a nonemptysubsetS ofN as a coalitionand to v(S) as the worth of coalitionS. The functionv itselfis called the characteristic functionof the game (N, v). The set of gameswithplayersetN is denotedby GN, whileG denotestheclass of all games. Suppose the grandcoalitionN is formed,thenthe playersmustdivide the total earningsv(N) ofthegrandcoalition.A payoffvectorfortheplayersetN is a real-valued function x on N, denotedbyx E p N. Herex( i), whichis usuallydenotedbyxi, represents thepayoff to playeri E N accordingto x E P N. A payoff vectorthatdistributes theamount v(N) among the playersis said to be Pareto-optimal(or efficient).A Pareto-optimal payoffvectoris also called a pre-imputation and the set of pre-imputations fora game (N, v) is denotedby I*(N, v). Thus, I* (N v):= XEpN
E x = v(N)}
For the sake of convenience,if xE pN and S c N, then we writex(S) instead of := 0. >JESxj, where x(0) A solutionis a functiona whichassociateswithany game (N, v) a subsetU(N, v) of its pre-imputation set I* (N, v). Note thatthe solutionset c(N, v) is allowed to be empty.A solutiona is called a value ifforany game (N, v) thecorresponding solution set u(N, v) is a singleton. Given any game (N, v), any coalitionT c N, and any payoffvectorx E pN, there are variouswaysto definea reducedgame ( T, vX) withrespectto x, whichis givenin termsof the originalcharacteristic functionv and the payoffvectorx. Note thatthe playerset T of the reducedgame is obtainedhereby removingthe nonmembersof T fromtheoriginalplayersetN. The reducedgame (T, vX)shoulddescribethe following situation.Suppose thatall theplayersin N agreethatthenonmembersof Twill be paid
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CONSISTENCY
PROPERTIES
IN COOPERATIVE
GAME THEORY
45
accordingto a specificprinciple(e.g., the payoffvectorx). Moreover,supposethatthe nonmembersof T continueto cooperatewiththemembersof T. Then theworthvx(S) ofcoalitionS c T in thereducedgame represents thetotalsavingsthatthemembersof S may achievesubjectto theforegoing two suppositions. A solutiona is said to possessthe reducedgame property (RGP) (withrespectto a specifiedtypeof reducedgames) wheneverthe nextconditionis satisfied: RGP. If (N, v) is a game, 0 # T c N and x E u(N, v), thenxTE o(T, vX). xT E T denotestherestriction ofx E pN to T c N. Here thereducedgameproperty saysthat if a payoffvectorx is a point in the solutionset u(N, v) of the originalgame (N, v) (accordingto thesolutiona ), thentherestriction XTofX to anycoalitionT c N belongs to the solutionset ( T, vX) of the corresponding reducedgame (T, vX) (accordingto thesolutiona). In case thesolutionis a value,thereducedgameconditionrequiresthatiftheplayers are supposedto be paid accordingto the value,thenthereis no inconsistency in what theplayersofthereducedgamewillget,in eithertheoriginalgameor thereducedgame. Thus, theRGP can be seen as a property of consistency. The firstversionof a reducedgame and the corresponding RGP can be foundin Davis and Maschler[ 6 ]. A systematic studyofRGP is presentedin Aumannand Dreze [1]. However,theirstudyrefersto solutionsof games withcoalitionstructures (i.e., partitionsof the playerset). Usually,solutionsof games are definedwithreference to the all-playercoalitionand not withreference to an arbitrary coalitionstructure. Here we considerthe usual treatment of solutions.The core consistsof pre-imputations that cannotbe improvedupon by any coalition.That is,thecoreof a game (N, v) is defined to be C(N, v) := {xEI* (N, v) I x(S)
v(S) forall ScN}l.
The familiarformulafortheShapleyvalue 4(N, v) E (Shapley [ 24]): forall i E N
pN
of a game (N, v) is as follows
(2.1)
{i
)V(S)],
4),(N,v)=
E
ScN-
YN(S) [V(SU {i}
where 'YN(S):= (INIJ!)-IS!(INI -I SI-1
(2.2) Here
)! forall Sc N, S# N.
IA I denotes the number of elements in the finite set A. Note that
formulaforthe -YN(S) = 1 forall i E N. We also mentionan alternative Shapleyvalue due to Driessen [7], [9]. The Shapleyvalue 4(N, v) E PN of a game (N, v) is equal to ISc
(2.3)
N- {}
4i(N,v)=
ScN-
{i}
'YN(S)[V(N-S)-v(S)]
foralliEN.
We postponethedefinitions oftwoothermajorsolutions,theprekernel and theprenucleolus. In the remainderof thissectionwe listand discussseveralelementaryproperties forsolutions.In thenextsectionsitwillbe shownthatsuitablychosenproperties together in termsofthereducedgamesfullycharacterize witha consistency property a particular solution.We beginby listingthe well-knownnonemptinessand individualrationality propertiesfora solutiona. NE. A solutiona is said to possessthe nonemptiness (NE) property if o(N, v) # 0 foranygame (N, v).
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THEO S. H. DRIESSEN
46
rational(IR) ifforanygame(N, v), any IR. A solutiona is said to be individually x, _ v({i}). RISE. A solution a is said to be relativelyinvariantunderstrategicequivalence (RISE) wheneverthe nextconditionis satisfied:if(N, v) is a game withU(N, v) # 0, a > 0, and : E pN, then u(N, atv+ 3) = au(N, v) + f. Here thegame (N, atv+ f) is givenby (atv+ :)(S) := av(S) + j-JES3#forall S c N. The relativeinvarianceunderS-equivalencerequiresthatthesolutionbehavein a naturalway withrespectto changesin scale thatare comparablewithpositiveaffine The followingtwo properties referto payoffs to "identical"playersactransformations. iftheworthof cordingto a pointin the solutionset. Two playersare called substitutes any coalitioncontainingexactlyone of thetwo playersis not affected by interchanging in a game (N, v) if the two players.That is, the playersi, j E N, i #j, are substitutes v(SU {i}) = v(SU {j})forallScN{i,j}. SYM. A value a is said to be symmetric (SYM) wheneverthe nextconditionis satisfied:if(N, v) is a game, i E N, x = u(N, v) and Ha permutationon N, thenOx = a(N, Ov). Here thegame (N, Hv)and thepayoffvectorOxE pNare givenby(Ov)(OS):= v(S) forall S c Nand (Ox)(Hi) := x(i) forall i E N. (ETP) ifforany ETP. A solutiona is said to possessthe equal treatment property i,jEN, i #j, in a game (N, v) and anyx E u(N, v): xi = xJ. substitutes oftheplayers, The symmetry requiresthatthevalueis notaffected bya renumbering whilethe equal treatmentpropertymeans thatsubstitutesreceivethe same payoffby to verifythat a symmetricvalue any point in the solution set. It is straightforward possessesETP. to concludethatthesolutionis stanUsually,NE, RISE, and ETP mustbe satisfied dard fortwo-persongames.Here a solutiona is called standardfortwo-person games if forall games({i, j}, v) withi #j xE U(N, v) and any iEN:
J({i,j}, v)
v({ i})
2
[(f
i,J}) - v({ i}) -
j})],
i.e.,the"surplus"v( { i, j v( { i} -v( { j}) is equallydividedamongthetwoplayers accordingto thesolution. LEMMA2.1. If a solutiona satisfiesNE, RISE, and ETP, thena is standardfor two-person games. Proof. Let thesolutiona satisfy NE, RISE, and ETP and let(N, v) be a two-person game,whereN = { i, j}, i j. Definethetwo-persongame (N, w) by w(S) := v(S) k-keS v( { k}) for all Sc N. Because the playersi and j are substitutesin the game (N, w), we obtain that xi =X = Iw({i,j}) for any xEE(N, w) = a(N, v)As such, (v({i}), v({j})). -
o1(N,v)=v({i})?+o(N,w)=v({i})?+
[v({i,j})-v({i})-v({j})].
DG
Since SYM is a stronger fromLemma 2.1 requirement thanETP, itfollowsimmediately thata valueis standardfortwo-person RISE and SYM. gameswhenever thevaluesatisfies We concludethissectionwithsome notationand definitions. Let (N, v) and (N, w) be two gameswiththesame playerset.Then we write(N, v) = (N, w), wheneverv(S) = w(S) forall S c N. Further,forany S c N and any i E N, we usuallywriteSC,S U i, and S- i, respectively, insteadof N- S, S U { i }, and S - i}. Finally,forany i, j E N, i #j, we denote the set of coalitionscontainingplayeri but not playerj, by FIJ. That is, F1J:=
{Sl
ScN,iA S,jT
S}.
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CONSISTENCY
IN COOPERATIVE
PROPERTIES
GAME THEORY
47
3. On thefunctional equationsdetermining theShapleyvalue. Sincetheintroduction of the Shapleyvalue in Shapley[24], severalaxiomaticcharacterizations of thisvalue havebeengiven,e.g.,in [3 ], [7 ], [10], [24], and [31]. The axiomatization oftheShapley value 0 on the class GN of games witha fixedplayerset N (e.g., in [3]) makes use of fourproperties: thePareto-optimality, theso-calleddummyplayerproperty, thesymmetry (the equal treatment property, respectively), and theadditivity property, i.e., + 4(N, v + w) = 4(N, v)?(N,
w) forall games(N, v) and (N, w).
Here thegame (N, v + w) is givenby (v + w)(S) := v(S) + w(S) forall S c N. We recallthatthe additivityaxiom is the mostessentialone in the above characterizationof the Shapleyvalue on GN since the firstthreeaxioms are requiredto be satisfiedin orderto determinethe value on the so-calledunanimitygameswhichform an additivebasis forthelinearspace GN (see [ 24 ] ). However,theadditivity axiom is no longerusefulforcharacterizing the Shapleyvalue on the class G of all games. First,we pay attentionto the axiomatizationof the Shapleyvalue on G withthe in termsofthereducedgamesas definedin Hartand Masaid ofa consistency property Colell [ 13]. They considera typeof reducedgamesdealingwiththeso-calledsubgames of a fixedgame. Given any game (N, v) and any coalitionT c N, the subgame(T, v) of(N, v) on T is obtainedbyrestricting thecharacteristic functionv to subsetsof T only, i.e., to 2T
Given any value a on G, any game (N, v) with INI ' 2 and any playeri E N, the v withrespectto the value a is as follows:for reducedgame (N - { i }, D) corresponding allSc:N- {i} (3.1)
v-(S):=
v(SU
{ i})-1i(SU
{i}
, V).
Note thattheplayersetofthereducedgame is obtainedherebyremovingonlyone the worthof any coalitionin the above playerof the originalplayerset. Furthermore, withthesingleplayer reducedgameis equal to theoriginalworthofthecoalitiontogether minusthe payoffto the singleplayer,accordingto the value in the subgamewiththe withthesingleplayeras playerset.The reducedgameproperty coalitiontogether requires to theplayersin thereducedgame in comparison thatthevalue yieldsthesame payoffs withtheoriginalgame,i.e., oj(N-{i},v')=oj(N,v)
foralljEN-{i}.
THEOREM 3.1 (Hart and Mas-Colell[ 13]). The Shapleyvalue0 on G is theunique value on G whichpossessesRISE, ETP, and RGP withrespectto the reducedgame of(3.1). Second,we mentionanothertypeof reducedgamesconsideredby Sobolev [ 27 ] in orderto axiomatizetheShapleyvalue on G. Given any game (N, v) with INI _ 2, any playeri E N, and any payoffvectorx E reducedgame (N - { i }, vX)withrespectto x is as follows:for RN, the corresponding allSc(N- {i}
(3.2) vx(S):= ( INJ-1)I
I SI [v(SU { i} )-x] + (INI -1 )( INI
-I
SI -1 )v(S).
Note thatthe worthof any coalitionin the above reducedgame is obtainedas a convexcombinationof the worthof the coalitionin the originalgame and the original worthof the coalitiontogetherwiththe singleplayerminusthe payoffxl to the single playeri forhis participation. THEOREM 3.2 (Sobolev [27]). The Shapleyvalue 0 on G is theuniquevalueon G whichpossessesRISE, SYM, and RGP withrespectto thereducedgame of(3.2).
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THEO S. H. DRIESSEN
48
Next,we lista thirdtypeofreducedgamesin orderto statea thirdand lastconsistency fortheShapleyvalue.Thistypeofreducedgamesis derivedfromthealternative property formula(2.3) forthe Shapleyvalue. Given any game (N, v) with INI _ 2, any playeri E N, and any payoffvectorx E reducedgame (N - { i }, vx) withrespectto x is as follows: [RN, the corresponding vX(0):= 0, (3.3)
forallotherSScN-{i},
vx(N-{i}):=v(N)-xi, S-1
vx(S):=(INI-l)'(INI-I
)[v(S)-v(N-S)]-(INI-1)-'ISlxl.
THEOREM 3.3 (Driessen). The Shapleyvalue0 on G possessestheRGP withrespect to thereducedgame of(3.3). We concludethissectionby provingthe existencepartof Theorem3.2 as well as theuniquenesspartof Theorems3.1, 3.2. For a proofof theexistencepartof Theorem 3.1 (i.e., theproofoftheRGP ofthe Shapleyvalue withrespectto the reducedgame of (3. 1)), we referto Hart and Mas-Colell [ 13]. Our purposeis to provethe uniqueness Theorem4.2) by means of one partsof Theorems3.1, 3.2 (as well as the forthcoming byplacing In otherwords,theuniquenessproofsarehighlighted specifiedargumentation. theproofsin thecontextof a singleline of argument. Proofoftheexistencepartof Theorem3.2. It is well knownthattheShapleyvalue 0 on G possessesSYM and RISE. Thus, it remainsto proveRGP for0 withrespectto thereducedgame of (3.2). Let (N, v) be a game, i E N, j E N- i, x = 4(N, v), and let (N- i, vx) be the corresponding reduced game of (3.2). We must show that 4j(N - i, vx) = 41(N, v). Put n := INI. It followsfrom(2.3) that
41(N,v)-
(3.4)
= 4j(N,v)-
)-v(S)]
'YN(S)[V(S
, SErj,
'YN(S)[V(S) )v(S)].
E SErY
By (2.3) and (3.2), we also have (3.5)
,
4j(N-i,vX)=
(3.6)
Sc {ji,j}C
'YN-i(S)[vx((N-i)-S)-vx(S)]
ISI
(n - 1)-'
=
SC
(3.7)
+(n-1)-1
(3.8)
+(n-l)-
SIAM REVIEW Vol. 33, No. 1, pp 43-59, March 1991
A SURVEY OF CONSISTENCY PROPERTIES IN COOPERATIVE GAME THEORY* THEO S. H. DRIESSENt Abstract.The mainpurposeofthissurveypaperis to reviewtheaxiomaticcharacterizations oftheShapley value,theprekernel, theprenucleolus, and thecoreby meansofa consistency property in termsofthereduced games.Wheneverpossible,new resultsand new proofsare added. Key words. cooperativegame in characteristic functionform,reducedgame, Shapleyvalue, prekernel, prenucleolus,core,consistency AMS(MOS) subjectclassification.90D 12
1. Introduction.Sincetheintroduction ofthenotionofa cooperativegamein characteristicfunctionformin Von Neumann and Morgenstern[30], manysolutionsfor thesegames have been proposed.In the contextof cooperativegames in characteristic functionform,the solutionaims at describingone or more equitabledivisionsof the in thegame(or, equivalently, totalsavingsamongtheparticipants fairallocationsofthe totalcostto theusersin a joint project).The coreis one oftheveryfirst proposed"set" solutionsand was introducedand named in game theoryin Gillies [11], althoughthe idea of the core was alreadyforeshadowed by theTennesseeValleyAuthority cost alloin Ransmeier[22]. Shapley[24] producedan important"one-point" cationtreatment solution,theso-calledShapleyvalue. Moreover,we mentionthesetsolutionconceptof theprekerneland the one-pointsolutionconceptoftheprenucleolus. The mathematicalapproachto a proposedsolutionis to examinea numberof its (elementary)propertiesand, if possible,to providea minimalnumberof properties whichfullycharacterizethe solution.Shapley[24] listedthreepropertiesin his characterizationof the Shapleyvalue on the class of gameswitha fixedplayerset. The main oftheShapleyvalue,theprekernel, purposeofthispaperis to reviewthecharacterizations theprenucleolus, and thecore,on theclassof(almost)all gamesbymeansofa consistency oftheconsistency property. Generallyspeaking,the formulation property fora solution is in termsofthesolutionitselfand theso-calledreducedgames.This paperreviewsthe use of reducedgames throughout the literature on cooperativegames in characteristic functionform.Informally, thenotionof a reducedgame can be elucidatedas follows. A cooperativegame is alwaysdescribedbya finiteplayersetas wellas a real-valued "characteristic function"on thecollectionofsubsetsoftheplayerset.A so-calledreduced game is deduciblefroma givencooperativegame by removingone or moreplayerson theunderstanding thattheremovedplayerswillbe paid accordingto a specificprinciple (e.g., a proposedpayoff vector).The remainingplayersformtheplayersetofthereduced functionof which is composed of the originalcharacteristic game; the characteristic function,the proposedpayoffvector,and/orthe solutionin question.The consistency propertyforthe solutionstatesthatifall the playersare supposedto be paid according to a payoffvectorin thesolutionsetoftheoriginalgame,thentheplayersofthereduced vectorin thesolutionsetofthereducedgame. gamecan achievethecorresponding payoff in whatthe playersof the reducedgame can In otherwords,thereis no inconsistency achieve,in eitherthe originalgame or the reducedgame. * ReceivedbytheeditorsOctober24, 1988; acceptedforpublication(in revisedform)February20, 1990.
ofTwente,Enschede,theNetherlands. t DepartmentofAppliedMathematics,University 43
This content downloaded from 130.89.45.231 on Thu, 17 Dec 2015 08:41:44 UTC All use subject to JSTOR Terms and Conditions
44
THEO S. H. DRIESSEN
We advance threemotivesforthe studyof the consistencyproperty in the fieldof cooperativegame theory.First,a distinctionbetweengame theoreticsolutionscan be made by distinguishing betweenthe corresponding consistencyproperties.Second,the consistencypropertyfora fixedsolutionmay be of importanceforthedevelopmentof thetheoryofthatsolution.Third,theconsistency in thegametheoreticframeproperty workmaybe usefulin orderto determinethe"consistent"solutionofa realisticproblem (see ? 6 foran application). In ? 2 we bringthe framework of cooperativegame theoryto the forefront and discussthepropertieswhichwillbe used in the axiomatizationsof the solutionsas presentedin thenextsections.In ? 3 we reviewtheaxiomaticcharacterization oftheShapley value by means of a consistencypropertydue to Hart and Mas-Colell (Theorem 3.1 ) and due to Sobolev(Theorem 3.2). Section4 considerscharacterization oftheprekernel due to Peleg (Theorem 4.1 ) and of the prenucleolusdue to Sobolev (Theorem 4.2), whiletheaxiomatizationofthecoreofbalancedand totallybalancedgamesdue to Peleg (Theorems 5.1 and 5.2) is reviewedin ? 5. In summary,thispaper unifiessix related theoremsthatare scatteredthroughoutthe literatureon mathematics.Whenevernew proofsof theexistenceor uniquenesspartoftheaxiomatizationscan be given,theyare also included.An additionalpurposeof thispaper is to provethe uniquenesspartsof Theorems3.1, 3.2, and 4.2 by means of a singleline of argument.In othercases, we referto theproofsin the originalpapers. 2. Propertiesforsolutionsof cooperativegames. The mathematicalmodel of a cooperativegame in characteristic function formis describedby a finitenonemptysetN and a real-valuedfunctionv on the family2 N of subsetsof theplayerset N, satisfying v( 0 ): = 0. We referto a nonemptysubsetS ofN as a coalitionand to v(S) as the worth of coalitionS. The functionv itselfis called the characteristic functionof the game (N, v). The set of gameswithplayersetN is denotedby GN, whileG denotestheclass of all games. Suppose the grandcoalitionN is formed,thenthe playersmustdivide the total earningsv(N) ofthegrandcoalition.A payoffvectorfortheplayersetN is a real-valued function x on N, denotedbyx E p N. Herex( i), whichis usuallydenotedbyxi, represents thepayoff to playeri E N accordingto x E P N. A payoff vectorthatdistributes theamount v(N) among the playersis said to be Pareto-optimal(or efficient).A Pareto-optimal payoffvectoris also called a pre-imputation and the set of pre-imputations fora game (N, v) is denotedby I*(N, v). Thus, I* (N v):= XEpN
E x = v(N)}
For the sake of convenience,if xE pN and S c N, then we writex(S) instead of := 0. >JESxj, where x(0) A solutionis a functiona whichassociateswithany game (N, v) a subsetU(N, v) of its pre-imputation set I* (N, v). Note thatthe solutionset c(N, v) is allowed to be empty.A solutiona is called a value ifforany game (N, v) thecorresponding solution set u(N, v) is a singleton. Given any game (N, v), any coalitionT c N, and any payoffvectorx E pN, there are variouswaysto definea reducedgame ( T, vX) withrespectto x, whichis givenin termsof the originalcharacteristic functionv and the payoffvectorx. Note thatthe playerset T of the reducedgame is obtainedhereby removingthe nonmembersof T fromtheoriginalplayersetN. The reducedgame (T, vX)shoulddescribethe following situation.Suppose thatall theplayersin N agreethatthenonmembersof Twill be paid
This content downloaded from 130.89.45.231 on Thu, 17 Dec 2015 08:41:44 UTC All use subject to JSTOR Terms and Conditions
CONSISTENCY
PROPERTIES
IN COOPERATIVE
GAME THEORY
45
accordingto a specificprinciple(e.g., the payoffvectorx). Moreover,supposethatthe nonmembersof T continueto cooperatewiththemembersof T. Then theworthvx(S) ofcoalitionS c T in thereducedgame represents thetotalsavingsthatthemembersof S may achievesubjectto theforegoing two suppositions. A solutiona is said to possessthe reducedgame property (RGP) (withrespectto a specifiedtypeof reducedgames) wheneverthe nextconditionis satisfied: RGP. If (N, v) is a game, 0 # T c N and x E u(N, v), thenxTE o(T, vX). xT E T denotestherestriction ofx E pN to T c N. Here thereducedgameproperty saysthat if a payoffvectorx is a point in the solutionset u(N, v) of the originalgame (N, v) (accordingto thesolutiona ), thentherestriction XTofX to anycoalitionT c N belongs to the solutionset ( T, vX) of the corresponding reducedgame (T, vX) (accordingto thesolutiona). In case thesolutionis a value,thereducedgameconditionrequiresthatiftheplayers are supposedto be paid accordingto the value,thenthereis no inconsistency in what theplayersofthereducedgamewillget,in eithertheoriginalgameor thereducedgame. Thus, theRGP can be seen as a property of consistency. The firstversionof a reducedgame and the corresponding RGP can be foundin Davis and Maschler[ 6 ]. A systematic studyofRGP is presentedin Aumannand Dreze [1]. However,theirstudyrefersto solutionsof games withcoalitionstructures (i.e., partitionsof the playerset). Usually,solutionsof games are definedwithreference to the all-playercoalitionand not withreference to an arbitrary coalitionstructure. Here we considerthe usual treatment of solutions.The core consistsof pre-imputations that cannotbe improvedupon by any coalition.That is,thecoreof a game (N, v) is defined to be C(N, v) := {xEI* (N, v) I x(S)
v(S) forall ScN}l.
The familiarformulafortheShapleyvalue 4(N, v) E (Shapley [ 24]): forall i E N
pN
of a game (N, v) is as follows
(2.1)
{i
)V(S)],
4),(N,v)=
E
ScN-
YN(S) [V(SU {i}
where 'YN(S):= (INIJ!)-IS!(INI -I SI-1
(2.2) Here
)! forall Sc N, S# N.
IA I denotes the number of elements in the finite set A. Note that
formulaforthe -YN(S) = 1 forall i E N. We also mentionan alternative Shapleyvalue due to Driessen [7], [9]. The Shapleyvalue 4(N, v) E PN of a game (N, v) is equal to ISc
(2.3)
N- {}
4i(N,v)=
ScN-
{i}
'YN(S)[V(N-S)-v(S)]
foralliEN.
We postponethedefinitions oftwoothermajorsolutions,theprekernel and theprenucleolus. In the remainderof thissectionwe listand discussseveralelementaryproperties forsolutions.In thenextsectionsitwillbe shownthatsuitablychosenproperties together in termsofthereducedgamesfullycharacterize witha consistency property a particular solution.We beginby listingthe well-knownnonemptinessand individualrationality propertiesfora solutiona. NE. A solutiona is said to possessthe nonemptiness (NE) property if o(N, v) # 0 foranygame (N, v).
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THEO S. H. DRIESSEN
46
rational(IR) ifforanygame(N, v), any IR. A solutiona is said to be individually x, _ v({i}). RISE. A solution a is said to be relativelyinvariantunderstrategicequivalence (RISE) wheneverthe nextconditionis satisfied:if(N, v) is a game withU(N, v) # 0, a > 0, and : E pN, then u(N, atv+ 3) = au(N, v) + f. Here thegame (N, atv+ f) is givenby (atv+ :)(S) := av(S) + j-JES3#forall S c N. The relativeinvarianceunderS-equivalencerequiresthatthesolutionbehavein a naturalway withrespectto changesin scale thatare comparablewithpositiveaffine The followingtwo properties referto payoffs to "identical"playersactransformations. iftheworthof cordingto a pointin the solutionset. Two playersare called substitutes any coalitioncontainingexactlyone of thetwo playersis not affected by interchanging in a game (N, v) if the two players.That is, the playersi, j E N, i #j, are substitutes v(SU {i}) = v(SU {j})forallScN{i,j}. SYM. A value a is said to be symmetric (SYM) wheneverthe nextconditionis satisfied:if(N, v) is a game, i E N, x = u(N, v) and Ha permutationon N, thenOx = a(N, Ov). Here thegame (N, Hv)and thepayoffvectorOxE pNare givenby(Ov)(OS):= v(S) forall S c Nand (Ox)(Hi) := x(i) forall i E N. (ETP) ifforany ETP. A solutiona is said to possessthe equal treatment property i,jEN, i #j, in a game (N, v) and anyx E u(N, v): xi = xJ. substitutes oftheplayers, The symmetry requiresthatthevalueis notaffected bya renumbering whilethe equal treatmentpropertymeans thatsubstitutesreceivethe same payoffby to verifythat a symmetricvalue any point in the solution set. It is straightforward possessesETP. to concludethatthesolutionis stanUsually,NE, RISE, and ETP mustbe satisfied dard fortwo-persongames.Here a solutiona is called standardfortwo-person games if forall games({i, j}, v) withi #j xE U(N, v) and any iEN:
J({i,j}, v)
v({ i})
2
[(f
i,J}) - v({ i}) -
j})],
i.e.,the"surplus"v( { i, j v( { i} -v( { j}) is equallydividedamongthetwoplayers accordingto thesolution. LEMMA2.1. If a solutiona satisfiesNE, RISE, and ETP, thena is standardfor two-person games. Proof. Let thesolutiona satisfy NE, RISE, and ETP and let(N, v) be a two-person game,whereN = { i, j}, i j. Definethetwo-persongame (N, w) by w(S) := v(S) k-keS v( { k}) for all Sc N. Because the playersi and j are substitutesin the game (N, w), we obtain that xi =X = Iw({i,j}) for any xEE(N, w) = a(N, v)As such, (v({i}), v({j})). -
o1(N,v)=v({i})?+o(N,w)=v({i})?+
[v({i,j})-v({i})-v({j})].
DG
Since SYM is a stronger fromLemma 2.1 requirement thanETP, itfollowsimmediately thata valueis standardfortwo-person RISE and SYM. gameswhenever thevaluesatisfies We concludethissectionwithsome notationand definitions. Let (N, v) and (N, w) be two gameswiththesame playerset.Then we write(N, v) = (N, w), wheneverv(S) = w(S) forall S c N. Further,forany S c N and any i E N, we usuallywriteSC,S U i, and S- i, respectively, insteadof N- S, S U { i }, and S - i}. Finally,forany i, j E N, i #j, we denote the set of coalitionscontainingplayeri but not playerj, by FIJ. That is, F1J:=
{Sl
ScN,iA S,jT
S}.
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CONSISTENCY
IN COOPERATIVE
PROPERTIES
GAME THEORY
47
3. On thefunctional equationsdetermining theShapleyvalue. Sincetheintroduction of the Shapleyvalue in Shapley[24], severalaxiomaticcharacterizations of thisvalue havebeengiven,e.g.,in [3 ], [7 ], [10], [24], and [31]. The axiomatization oftheShapley value 0 on the class GN of games witha fixedplayerset N (e.g., in [3]) makes use of fourproperties: thePareto-optimality, theso-calleddummyplayerproperty, thesymmetry (the equal treatment property, respectively), and theadditivity property, i.e., + 4(N, v + w) = 4(N, v)?(N,
w) forall games(N, v) and (N, w).
Here thegame (N, v + w) is givenby (v + w)(S) := v(S) + w(S) forall S c N. We recallthatthe additivityaxiom is the mostessentialone in the above characterizationof the Shapleyvalue on GN since the firstthreeaxioms are requiredto be satisfiedin orderto determinethe value on the so-calledunanimitygameswhichform an additivebasis forthelinearspace GN (see [ 24 ] ). However,theadditivity axiom is no longerusefulforcharacterizing the Shapleyvalue on the class G of all games. First,we pay attentionto the axiomatizationof the Shapleyvalue on G withthe in termsofthereducedgamesas definedin Hartand Masaid ofa consistency property Colell [ 13]. They considera typeof reducedgamesdealingwiththeso-calledsubgames of a fixedgame. Given any game (N, v) and any coalitionT c N, the subgame(T, v) of(N, v) on T is obtainedbyrestricting thecharacteristic functionv to subsetsof T only, i.e., to 2T
Given any value a on G, any game (N, v) with INI ' 2 and any playeri E N, the v withrespectto the value a is as follows:for reducedgame (N - { i }, D) corresponding allSc:N- {i} (3.1)
v-(S):=
v(SU
{ i})-1i(SU
{i}
, V).
Note thattheplayersetofthereducedgame is obtainedherebyremovingonlyone the worthof any coalitionin the above playerof the originalplayerset. Furthermore, withthesingleplayer reducedgameis equal to theoriginalworthofthecoalitiontogether minusthe payoffto the singleplayer,accordingto the value in the subgamewiththe withthesingleplayeras playerset.The reducedgameproperty coalitiontogether requires to theplayersin thereducedgame in comparison thatthevalue yieldsthesame payoffs withtheoriginalgame,i.e., oj(N-{i},v')=oj(N,v)
foralljEN-{i}.
THEOREM 3.1 (Hart and Mas-Colell[ 13]). The Shapleyvalue0 on G is theunique value on G whichpossessesRISE, ETP, and RGP withrespectto the reducedgame of(3.1). Second,we mentionanothertypeof reducedgamesconsideredby Sobolev [ 27 ] in orderto axiomatizetheShapleyvalue on G. Given any game (N, v) with INI _ 2, any playeri E N, and any payoffvectorx E reducedgame (N - { i }, vX)withrespectto x is as follows:for RN, the corresponding allSc(N- {i}
(3.2) vx(S):= ( INJ-1)I
I SI [v(SU { i} )-x] + (INI -1 )( INI
-I
SI -1 )v(S).
Note thatthe worthof any coalitionin the above reducedgame is obtainedas a convexcombinationof the worthof the coalitionin the originalgame and the original worthof the coalitiontogetherwiththe singleplayerminusthe payoffxl to the single playeri forhis participation. THEOREM 3.2 (Sobolev [27]). The Shapleyvalue 0 on G is theuniquevalueon G whichpossessesRISE, SYM, and RGP withrespectto thereducedgame of(3.2).
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THEO S. H. DRIESSEN
48
Next,we lista thirdtypeofreducedgamesin orderto statea thirdand lastconsistency fortheShapleyvalue.Thistypeofreducedgamesis derivedfromthealternative property formula(2.3) forthe Shapleyvalue. Given any game (N, v) with INI _ 2, any playeri E N, and any payoffvectorx E reducedgame (N - { i }, vx) withrespectto x is as follows: [RN, the corresponding vX(0):= 0, (3.3)
forallotherSScN-{i},
vx(N-{i}):=v(N)-xi, S-1
vx(S):=(INI-l)'(INI-I
)[v(S)-v(N-S)]-(INI-1)-'ISlxl.
THEOREM 3.3 (Driessen). The Shapleyvalue0 on G possessestheRGP withrespect to thereducedgame of(3.3). We concludethissectionby provingthe existencepartof Theorem3.2 as well as theuniquenesspartof Theorems3.1, 3.2. For a proofof theexistencepartof Theorem 3.1 (i.e., theproofoftheRGP ofthe Shapleyvalue withrespectto the reducedgame of (3. 1)), we referto Hart and Mas-Colell [ 13]. Our purposeis to provethe uniqueness Theorem4.2) by means of one partsof Theorems3.1, 3.2 (as well as the forthcoming byplacing In otherwords,theuniquenessproofsarehighlighted specifiedargumentation. theproofsin thecontextof a singleline of argument. Proofoftheexistencepartof Theorem3.2. It is well knownthattheShapleyvalue 0 on G possessesSYM and RISE. Thus, it remainsto proveRGP for0 withrespectto thereducedgame of (3.2). Let (N, v) be a game, i E N, j E N- i, x = 4(N, v), and let (N- i, vx) be the corresponding reduced game of (3.2). We must show that 4j(N - i, vx) = 41(N, v). Put n := INI. It followsfrom(2.3) that
41(N,v)-
(3.4)
= 4j(N,v)-
)-v(S)]
'YN(S)[V(S
, SErj,
'YN(S)[V(S) )v(S)].
E SErY
By (2.3) and (3.2), we also have (3.5)
,
4j(N-i,vX)=
(3.6)
Sc {ji,j}C
'YN-i(S)[vx((N-i)-S)-vx(S)]
ISI
(n - 1)-'
=
SC
(3.7)
+(n-1)-1
(3.8)
+(n-l)-
