Cores of ConYex Games1)
By Lrovo S.Snanrv 2)
gameis the setof feasibleoutcomesthat cannotbe improvedupon
Abstract:The coreof an rr-per$on
by any coalitionof players,A convexgameis definedas onethat is basedon a convexsetfunction.
In this paperit is shownthat the core of a convexgameis not emptyand that it has an especially
regularstructure.It is furthershownthat certainothercooperative
solutionconceptsare relatedin a
sirnplewayto thecore:The valueof a convexgameis thecenterof gravityof theextremepointsof the
corq andthe von Neumann-Morgenstern
stablesetsolutionof a convexgameis uniqueandcoincides
with the core.
|' Introduction
The core of an n-persongame is the set of feasibleoutcornesthat cannot be
improved upon by any coalition of players3).A convexgameis one that is based
on a convexsetfunction (seebelow);intuitively this meansthat the incentivesfor
joining a coalition increaseas the coalition growq so that one might expecta
'osnowballing"or "band-wagon" effect when
the game is played cooperatively.
In this paper we show that the core of a convcxgameis not ernpty - in fact,
it is quite large - and that it has an especiallyregularstructure.We further show
that certain other cooperativesolution conceptsare related in a simple way to
the core.Specifieally(l) the value of a convexgameis the centerof gravity of the
extreme points of the core, and (2) the voN NruunNN-MoncENsTERN
stable
set solution of a convexgameis uniqueand coincideswith the core.In a subsequent paper [MnscuLER,Pntnc, and SHepLEy]rather similar results will be
presentedfor two other cooperativesolutions:the kerneland the bargainingset.
r) Presentedat the Fifth Informal Conference
on GameTheory,held at PrincetonUniversityin
April 1965.This paper is basedon a Rand Corporationresearchmemorandum[SxlpLEy, 1965],
written under the sponsorshipof Air Force ProjectRand.The author wishesto acknowledge
the
stirnulusof a queryfrom JackEdmondsconcerningthe propertiesof convexsetfunctions.
2)The Rand Corporation,Santa
MonicA California.
3; The coreis sometimesincorrectlydescribed
as the set of outcomes"that cannotbe blockedby
anycoalition".This unfortunatelymisleadingdescriptionhasarisenasa resultof the counterintuitive
in the technicalterminologyof severalearly papers.(The presentauthor
useof the word 6{$tssl6"
must bearpart of the blarne!)In fact, the core is concernedwith what coalitionscando, not what
they can preuent.The distinctionis especiallystriking in economicgame models,where the bargainingpowerthat certaingroupsmay acquirethroughtheir ability to obstructtradeor production
is completelyignoredin the essentiallyconstuctiueconditionsthat delinethe core.
L. S. Sulprnv
T2
1.1.Notation
We shallsystematically
usethe lettersn,s|t,.. . to denotethe numberof elements
in the finite setsN,S,T .,. The letter*O)' will denotethe empty set,and "C'0,'s)"
will denote strict inclusion. "Payoff vectorsoo
are elementsof the ru-dimensional
linear spaceEil with coordinatesindexedby the elementsof N. If a e EN and
S E N we shall often write a(S) for.f,sa;, treatrng a as an additive function on
the subsetsof N. The hyperplanein EN delinrd by the equation a(S) : u(S),
0 c S C N, will be denotedby I{r.
2. ConvexGames
In this paper, a gameis a function u from a lower-casering l{ to the reals,
satisfying
u ( 0 ): P ,
It is superaddttiae
tf
0(S)+ u(?)S u(Su f),
It is carwexr\if
all S,Teff with Sn T : O.
(1)
(2)
u(S)+ u(T)* u(Su T) + u(Sn T), eil S,T erf
(3)
It is strictlyconuexif inequalityholdsin (3)whenever
neitherS E T nor T e S,
i.e.,wheneyor
S,T, S u T, andS n ? areall different.
Notethat(1)and(3)together
(2).
imply
To appreciate
the term "convexo',
definefor eachR e f a differencing
operator
A*by'
I a * u f ( s: u) ( s u R )- u ( s - R ) , a t l s e - { ,
and let Ao*, denote As(Anu).Then (3) is equivalentto the assertionthat these
"seconddifferences"
are everywherenonnegative,i. e., that
l A a * u ] ( S ) = 0 , a l l 0 , R , S e - , f.
(4)
This is analogousto the nonnegativesecondderivativesassociatedwith convex
functionsin real analysis.
Two gamesare termed equiualentif their differenceis an additive game,i.e.,
obeysthe equality in (2) (or (3)
or (a)).It is easilyseenthat any gameequivalent
'that
to a convexgameis convex;
any positivescalarmultiple of a convexgame
is convex;and that the sum of two or more convexgamesis convex.It follows that
the ensembleof convexgames(for fixed .4\ formsa convexconein a suitablelinear
space(say gr-$7, and that this cone containsthe subspaceof additive games.
r) In potential theory, functionssatisfyingthe reverseinequality to
{3) (i.e., negativesof convex
functions)are called stronglysubadditlueot, wheu certain other propertiesare adduced,capacities
lC}toquer; Mnven]. In lattice theory and combinatorialmathematics,suchfunctions,oftenrestricted
to be monotonic and integervalued,as in the rank function of a matroid, are calledsubmodulmor
lowersemi-modular,
rnaking our convexfunctionssupermodularor upper semi-modular[WrurNEy;
Eutr,toNos,
Rorn; CRnro,Ror.lr;Eouowos].
Cores of Convex Games
l3
2.1.Interpretation
Throughoutthispaper-,f will bethelower-case
ringof subsets
of afinite setN ').In
thus,sf
thestandardapplicationin gametheorytheelements
?!
of N are
"players",the elementsof ,ff are"coalitions",anda(S),calledlhe "characteristic
function",givesfor eachcoalitionthebestpayoffit canachievewithouthelpfrom
otherplayers.
(2) arisesnaturallyin this interpretation,but convexity(3) is
Superadditivity
in a votingsituationS and T, but not,Sfi T,rnight
anothermatter,For example,
bewinningcoalitions,causing(3)to fail.To seewhatconvexitydoesentail,regard
thefunctionmt
(5)
*(s,T) : u(sv T)- u(s)- 1)(T)
"incentive
as deliningthe
to merge'o
betweendisjointcoalitionsS and T. Then
it is a simple'exercise
to verifythat (3)is equivalent
to the assertionthatm(S,T)
'osnowballing"
is nondecreasing
in eachvariable- whencethe
or "bandwagen"
effectmentionedin theintroduction.
Anotherconditionthat is equivalentto (3)(provided,il is finite)is to require
that 2)
p(su {i}) - u(s)s u(r v {4) - u(r),
(6)
for all individualsi e N andall S E T e N - tt). Thisexpresses
a sortofincreasing
marginalutility for coalitionmernbership,
and is analogousto the "increasing
returnsto scale'o
associated
with convexproductionfunctionsin economics.
2.2. ConvexMeasureGames
Let u be given by
-
t ) ( S ) : / 0 r ( S ) ), a l l S . 4 ,
(7)
where p is a nonnegativeadditive function on rl, i. e., a measure,and/ is a real
function'with/(0) :03).Then it is easyto verifythat u is convexiflis convex,
and strictly convexif / is strictly convox.A gamethat is a convexfunction of a
measureis calleda conuexrneasure
gameo).
Not all convex gamesare convex measuregames.A sirnplecounterexample
is the two-personconvexgamewith u({1}): -u({2}) - l, u({1,2}): u(O)- 0.
But this game, being additive, is equivalent to the convex measure game
u(S)= Q so we might ask whether every convexgame is at least equivalentto
1) For the infinite case
seeRosrNMtlrr,ER,
'l Ct,wurrNey.
3)ActuallY,atrYgamecan be put into this
form; we needmerelychoosep so that all the numbers
p(,S)aredifferentand define/accordingly.Thus,the generalconceptof "measuregame",with/and p
not further restricted,is of no interestin the finite case.[Cf. AurrLlNwand Snepr"rv].
In economicapplications,p may representthe initial distributionof someresouice,whilef may
representa productionfunction.
a; Curiously,if/is a functionof
severalvariablesandp a vectorof measures,
then convexityof/
doesnot imply convexityof u in (7),or evensuperadditivity.
Indeed,in the caseof a homogeneous
function/(l.x) = Lf(x), it is concavityofl not convexity,that impliessuperadditivityof u.
L. S. Snapmy
T4
Table1. Extremalconvexgameson four players
a
1
2
3
4
L2 13 L4 23 24 34
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
t
1
0
1
1
1 1 0 |
1
1
1
0
2
2
2
2
2
2
2
2
7
2
2
2
2
2
2
2
2
2
2 2 2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
1
1
1
1
l
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
0
1
1
0
0
1
0
1
1
0
1
0
1
0
1
1
0
0
1
2
2
2
2
1
2
7
2 2
2 2
t 2
2 1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0
0
0
0
0
0
0
0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
r23 r24 r34 234
1234
1
1
1
1
t
1
0
0
0
0
0
0
0
1
1
1
0 0 0 0 0 0
1 0 1 1
0 0 0 0 0 0
1 1 0 t
0 0 0 0 0 0
1 1 1 0
0
0
1
0
0
0
1
0
0
0
0
0 0 0 0
0 0 0 0
1 1 1 1
0
0
0
0
1
1
2
2
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1.
2
1
l
1
1
2
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
0
1
0
0
1
1
0
1
0
0
1
0
1
0
1
1
0
0
0
0
0 0 0 0
0
1
0
0
0
0
0
0
1
0
0
0
0' 0 0 1 0 0
0
0
0
0
1
0
0
0
0
0
0
1
0
1
2
0
0
1
0
0
1
2
0
0
0
1
1
1
1 1
t
2
1
1
1
2
1 1 0 0
1
0
1
0
0
1
1
0
I
0 0 1
0
1
0
1
0
0
1
1
a convexmeasuregame.Again the answeris no. lndeed,if this were true, then
everyconvexgamewould be the sum of threeconvexmeasuregames,sinceevery
additivegameis the sum of two r;. But gonvexgamesexistthat are not the sum of
let/, (x) = -fz(x) = x.
) Siutpty expressthe game as the difference of two measures,h - !r2, arrLd
Coresof ConvexGames
l5
any number of convsx rneasuregames;indeed,the first twelve gamesof Table 1
have this property.
The situation can be viewed geometrically.As already remarked,the convex
gamesform a cone in E{ which is invariant under equivalence.The subconeof
convex measuregames,however,is neither convex nor invariant under equivalence(unlessn is trivially small). The convex hull oI that subcone,which is
what we obtain when we take all sums of convexmeasuregames,is invariant
under equivalencebut still doesnot include all convexgamesif n > 3.
The thirty-seven extremals of the cone of four-person convex games were
determinedin 1965by S.A. Coor t) (Table1).Thegamesin the first sevensymnretry
(seebelow);the remainderare obtainedfrom
classeslisted are indecomposable
smallergamesby addingdummy players.For largern,little is known about the
set of all extrernals2).
2.3. DecomposableGames3)
Let P : {Nt,...,No} be a partition of If into p Z 2 nonemptysubsets.
The
gameu on .,{ is said to be decomposable
(with respectto P) if u is additive across
the partition, i.e.,for eachS e tf ,
u ( s ) : a ( S n N r ) * u ( S n N z ) + ' . , * a ( S n N o .)
(8)
Note that u is completelydeterminedby its valueson the subsetsof the N,. The p
smallergames,obtainedby restrictingu to the subsetsof eachN; in turn, are the
componentsof the decomposition.A decomposablegame has a unique finest
decomposition,that is, a decompositionin which none of the componentsare
u).
themselves
decomposable
ThearemI.
(a) A decomposable
gameis convexif and only if eachcomponentis convex.
(b)A convex gameu is decomposable
if and only if
u(N) : u(l[r) * u{}fr) + ... * u(Nr)
(9)
holdsfor somepartition{Nr,...,1{o} of lf into p ZZ nonemptysubsets.
Proof s).
The first statementis immediate,For the second,suppose(9) holds and let
S e tf . By convexity,
t) Private communication.
2; For the inlinite case,see
RosrNrvrUu*n.
3; This section is based on Mescrrr-eR,PnLEc
and Snepuev 1L9671.
4; The notion of decomposition
was introduced by vox ttliuuariN and MonGENSTERN;
see also
Cnrms [1953,1959] and pruc [1965a,bJ.
") Severalproofs can
be given; this one is due to Y. Kl.t.rx*r.
t6
L. S. SH.rplrv
u ( S )* u ( N , ) S u ( S^ N r ) + u ( S u N r )
u(Su /V,) + u(l{z){ u(Sn Nz) + u(Su Nr u Nz)
u ( Su l v r u . . . u N p - r ) + u ( N r ) { u ( Sn N o )+ u ( Su N 1 L r . . ' u N p ) .
Adding theseinequalitiesto eachother and to (9) yields
u ( S5
) u ( Sn N r ) + u ( Sn N z )+ . . ' * u ( S nN o ) .
(10)
But equality must hold here, by superadditivity(2). This completesthe proof.
A coalition is said to be "inessential"if its characteristic-function
valueis no
greaterthan required for superadditivity(2). Theorem L (b) expressesthe fact
that in a convexgame,decomposabilityis equivalentto inessentialityof the allplayer coalition.
Corollary.
A strictly convexgameis indecomposable.
3. CoreGeometrY
A payoffvector aeEN is said to befeasible(for u) if a(N) < u(l{)1),The core
of u is definedas the set C of all feasiblea e EN such that
a(S)Iu(S), all SEN.
(11)
The coreis obviouslya subsetof the hyperplane.t[,y(see$ 1.1),and sincethe inequalitiesai Z u({t}) areincludedin (11),the coreis bounded.Thus,C is a compact
convex polyhedron,possibly ernpty, of dimensionat most n - 1. By a "fu11dimensional"core we shall mean one of dimensionexactlyn - 1:
In order to discuss the facial structure of C, we define Cs : C n I{, for
O C S E N; note that Cw : C. It is convenientalsoto delineCo : (.If none of
the Cs are empty, we sa] that the configuration {Cr} is complete;if they all
have the highestpossibledimensionwe say that {Cr} is strictly complete.In the
latter case,the core is full-dimensionaland has exactly 7n- 2 polyhedral faces
of dimensionn - 2.
Figure 1 showsa core configurationfor n : 3 that is complete.but not strictly
complete,sinceC121
is only a point. Figure 2 depictsa strictly completecore2)
for n - 4, there being fourteen2-dimensionalfaces.(The extra lines and points,
included for perspective,indicate the intersectionsof the IJ" for s : 3 with the
1) In application, there may be also a lower bound p, such that only payoff vectors with
"feasible". Only the upper bound, however, will be of any significancein
f 5 a(N) < D(lV)are truiy
the present work.
2) Note that if we are given only the point set C, we cannot in general identify all of the sets Cr,
or even tell whether any are empty. In fact, it is not difficult to show that C uniquely determinesu,
"strictly complete
and hence{Cr}, only in the strictly completecase.Thus, it makessenseto speakof a
"complete
coreo'.
core'', but not a
Coresof Convex Games
17
.Fig. 1: Core configurationof a 3-personconvex game
simplex A in llly defined by the FIs for s : 1.)We seefrom the key that parallel
faceson opposite sides of the core correspondto complementarycoalitionst),
implies a "large" core - one which touchesall of the
Note that completeness
(n - 2)-dirnensional
facesof A.
Kcy to vertices ond fqces:
4231
Fronf:
42rS
4123
Cr+
r42l!
324L
Czsl
32t4,
3 124
l3?4
Chqrocteristic funclion:
"(,)=(ll),
Fig.2: Core of a four-personconvex game
t; In ScHffiIDLER game
a
with a complete core corrfiguration is called arLexect game.
L. S. Sseprnv
l8
3,1. RegularCore Configurations
Although the hyperplanesI{, have the same slopesin every gamesu, their
positionsin EN relative to one another are not predetermined,and even in the
caseof a strictly completecore there are many ways(if n > 3) in which the faces
"natural" arrangementstandsout from the rest; it
Cs can fit together.But one
occurs, for example, when all faces Cs, S + O , N, are equidistant from a
1).In this arrangecommoncenter - i.e.,tangentto the inscribed(ru- 2)-sphere
ment, it can be shown that two facestouch if and only if their corresponding
coalitionsare comparable,i. e.,
CsnCr*O
i f a n d o n l y i fS E T
or TES.
(12)
We shallcall property [Mrvnnf strictregularity.It obviouslyimpliescompleteness
.
:
(take S T); we shall see presently that it implies strict completsness.The
configurationof Fig. 2 is strictly regular,but not that of Fig. 1, sinceCp and C^
rneet.
We shall work primarily with a weaker form of (12).A core configuration
{Cr} will be calledregularif CN { O and
C sn C r I C s r , f t C s . , r , a l l S , T e l V .
(13)
We seeat oncethat strict regularityimpliesregularity,as the terminologywould
suggest.Condition (13) is equivalentto the statementthat for each aeEN, the
family of sets
g*: {s : c e cs}
(14)
is closedunder union and intersection.
The formal sirnilarity between(13) and (3) is not accidental;in due course
we shall show that convexity of a game is equivalentto regularity of its core
configuration.First we shall developsome geometricalconsequences
of (13).
3.2.Faces
Theorem2.
In a regularcore configuration{Cr} we have
(15)
Cs,fr Cs,n ... n Cr_ * O
for any increasingsequenceSr C S, C ... C S,,,.In particular (take m: L),
a regularcore configurationis complete.
The proof proceedswith the aid of two lemmasconcerningregularcores.Let
us write S c c T to mean that S c T and t - s ) 2.
LemmaI.
If S C CT and a€Cs ACr, then thereexistsQ and b suchthat S C 0 CT
and b€Cs nCa^Cr. Moreover,we canrequirethat br: ei, a,llf eS, and that
jeQ,k+Q for any two preassigned
elementsi,kof T - S.
1)sucha gsmeis u(s)= s @
t9
Coresof Convex Games
Proof.
Fix j,ke T - S. DefinebeC by bi: a; * p, b*: ar,* P, and bt: Q for
i * i,k, giving p the largestpossiblevalue cornpatiblewith b e C; this can be
done becauseC is compact and containsa. Clearly,b(S) : a(S),bg) -- a(T);
hencebeC, n Cr- Sincea larger p would havetaken b out of the core entirely,
there must be a set R, containing"l and not containingk, such that beC*. Let
"r)"
"n", we haveb e Cq.
and
Q : (S v R) n I Since96 $ee(14)is closedunder
But Q hasall the requiredpropefiies:,S
C Q C T je Q,andkt Q; so thelernma
is proved.
Lemma2.
Let ^SC C N and a€Cs. Then,for anyJ€I{ - S thereexistsbeCrn Cs,u}
suchthat bi : ai, all f e S.
Proof.
a n d u s eL e m m al t o f i n d g , b s u c ht h a t S . r { " 1 }E Q C l { a n d
Set?:N
beC, n Cs, the latter agreeingwith a on S. If 8: S, fi) we are through;if
not, set | - Q and repeatthe argument.
Proof of Theorem2.
We may assumethat the ssquence{So} is of maximum length, i.e., that
rn - n * I. Then we have Cs, : C6, which is nonempty by definition. Take
n - 7, apply Lernma2 to find a point e1rt,+s€
a6.1€Csr,and for k: 7
CsuA Csu*,, agreeingwith a1r1on 51. The point a,n,will be in all of the Csu,
I S k 5 n, and it will alsobe in Cr,,*, : CN- C. Hence(15)follows.
3,3. Vertices
Let u representa simple orderingof the players.Specifrcally,
let o be one of
the n! functionsthat map /{ onto {1,2,..r, ru}.Define
S a , , k :{ f e N : a t ( i )S k } ,
k - 0 ,1 , . . . , n i
(16)
these arc the "initial segments"of the ordering.Thus, Sr,o- O and Sr,n: 1{.
: D(Sr,r),
k : L,2,,..,n. Linear independence
Considerthe equationsa(Sr,1,)
is
assured,and it is easy to solve the set of equationsto find the coordinatesof
the intersectionof the hyperplanesH s*,*, namely
- u(S.,r(;i-r), all
a? : 0(S@,.(;l)
i E N.
(r7)
In this w&y,each ordering al definesa payoff vector a'; thereis, of course,no
assurance
that the n! points a* areall distinct.
Theorem3L).
The verticesof the core in a regularconfigurationare preciselythe points a@.
1) Theorem 22 of Eouonos gives a similar result for "polymatroids", which are similar in
many
respectsto cores of convex games.Thus, Theorem I of EorrtoNDsmay b6 compared with our (ll),
Theorem 42 of Eor''rouoswith our (13),etc.
L. S. Sneplnv
Proof.
We have,for any @,
C r - , 1 n " ' n C s o , , ,C, H r o , , ,f r " ' A
Hs-,n.
The left-handside is not empty,by Theorem2; it thereforeconsistsof the single
point d'. Hencea* is in the core.If it werenot a vertex(i.e.,an extremepoint),
thenfor somenonzerovectord we would haveboth a* t d e C. But this is impossible,sinceat least one of the hyperplanesHr-,n that meet at and determinea*
must passstrictly betweenthe two points a' t d, excludingone of them from
possiblemembershipthe core. Hencea' is a vertex of C, for eachar.
It remainsto show that C hasno other vertices.Let a be any vertexof C and let
"longest"increasingsequgnce
of membersof
O - Sr C Sz C ... C S, : N be a
9o(see (14)),i.e.,one that maximizesm.If thereare no gapsin this $equence,
i.e.,
-tf m n * L,thenais of the form d', and we arethrough.Supposethereforethat a
gapexists,
so that S* C C So*, for somek, andlet i,Je Sr*, - So,i *;.Since a
is a vertex,it is not only a solution of all the equationsa(S) : u(S),S e .9/o,it is
their uniquesolution.But in order to determinethe coordinatesa; and a;, ?rrdnot
just their sum at * o;, there must be an equation that separatesthem, i. e., an
R e 9o that contain$preciselyone of f;. Since9ois closedunder '(rJ" and "n",
w e h a v eQ e ? o , w h e r e Q : ( S ru R ) n S r * r . B u t S r C Q C S ' * * r , c o n t r a d i c t i n g
the maximality of m. This completesthe proof.
3,4. Strictly Regular Cores
This sectionis mainly descriptive,andforrnalproofsareomitted.Strictregularity
was definedby (12)above,but the one-sidedcondition
C s n C r * O i m p l i e sS E T o r T e S ,
(18)
togetherwith C + O, would havesufficed.To seethis, note that (18)with C + g
impliesregularity,which in turn impliesthe converseof (18)by way of Theorem2
w i t hm : 2 .
Strict regularity implies strict completeness,
and hence is a property of the
point set C. A strictly regular core has n I distinct extremepoints a'. Tbe core
in a regular configurationthat is not strictly regular,on the other hand, always
has fewerthan n ! extremepoints, although it may still be strictly complete,i. e.,
havethe maximum number of (n - 2Fdimensionalfaces.
The facial,arrangementof a strictly regular core is completelydeterminedin
all dirnensions.For example, &rL(n - 2)-dimensionalface Cs is bounded by
exactly T + 2n-' - 4 (n - 3)-dimensionalfaces of the form Cs n C, with
0 C T C S or S C T C N. The automorphisnrsof this highty symmetriccombinatorial structureare generatedby the permutationsof N, togetherwith complementation,the latter automorphismrnappingeachvertexa'into its antipodea'',
wherEcu'denotesthe reversalof ar.This is illustrated for n : 4 in Fig. 2.
Coresof ConvexGames.
2t
We can now distinguishseveralclassesof games,accordingto the kinds of
coresthey posses:
Table 2
nonemptYcores
full-dimensionalcores
completecoreconfigurations
strictlycompletecoros
regular core configurations
strictlyregularcores
Eachclassincludesthoselistedbelowit and to the right. For ru7 3, the inclusions
- tot(see
$ 2.2),then eachof the
are all strict. If a gameis regardedasa point in Etr
six classesdescribesa convexcon€;and thoseon the left [right] are the closures
regularcore is a limit
finteriorsl of thoseon the right flettJ,In particular,every
of rttirtly regularcores - indeedit can be approximatedby both increasingand
of strictly regularcores.
nestedsequences
decreasing
4. Solutionsof ConvexGames
"solution concepts"have beenproposedfor n-persongames
Many different
form. We shall be concernedhere with three of them:
in characteristic-function
rhe core,alreadydefined;the (SHarr,ev)value ($4.2); and the (voN NeuunNNMoncnNsrrnu)solutions,which we call stablesets($4,3).Weshall seethat the
threeconceptsare intimatelyrelatedwhenthe garneis convex.
4.1.The Core
The core of a game?alreadydefinedat (11),may be interpretedas the set of
"sociallystable"outcornes,
in that no coalitioncanimproveupon any of them.In a
game with an empty core, at least one set of playersmust fail to realtzeits full
potential,no nratter how the winningsare divided.
First we require
Theorern4.
The core of a convexgameis not empty.
Proof.
It sufficesto show that a' is in the core,for somear (see$ 3.3).Let T C lf and
of7 are
elementof N * T, so that all the co-predecessors
let j be the ooia-earliest"
in T, but not 7 itself.Then we have at once
T u S . , , , a r ' ) f- u U ) ,
TnS*,r(r:Sr,r(j)-I:
usingthe notation of (16).Hence,by convexityof u,
t)V) * v(S*,,,r/5 u(Tv fi]l + u(S,,,orr)
L. S. Sseprnv
22
or, by (17),
a?So(rufi))*u(n.
Hence,
a'(T) - u(T)2 a'(Tu U))- u(Tu ff)).
- t - Ltimes,weobtain
theargurnentn
Repeating
a*(T) - u(T) 2 a' (N) - u(N) - 0.
SinceT was arbitrar!, &^ is in the core.
Theorem5.
A gameis convexif and only if its core conligurationis regular.
Proof.
Supposewe haveregularity(13).Let S,T e If. SinceS u T 2 S n T, we can
Iind aeCsuy A Cr.,r, by Theorern2. Then we have
u(su ?) + u(sn T) : a(su T) + a(sn T) : 4(s)+ a(T)* u(s)+ u(T).
Henceu is convex.
Conversely,supposethat u is convex.Then C + O by Theorem4 and it rernains
only to establish(13).Supposea is in Cs n C1. Then we have
u ( Su T ) + r i ( Sn f ) = u ( S )+ u ( T ) : a ( S )+ a ( T ) : a ( S u T ) + a ( S ^ T ) .
But we also have a(S u f) = u(Sv T) and a(S n T) > u(Sn T), becausea is
in the core.Henceequalityprevailseverywhere,
and a e Csuyn Cs,.,,y,
asrequired.
.
Corollary.
A gameis strictly convexif and only if its core is strictly regular.
We omit the simpleproof.
I
Theorem6 r).
(a) The core of an indecomposable
convexgameis full-dimensional
(b) The core of a decomposableconvex gameis the cartesianproduct of the
Its dimensionis n - p, wherep
coresof the componentsof any decomposition.
is the number of componentsin the finest decomposition.
Proof,
(a) If C is lessthan full-dimensional,then (, -= Cs for someO C S C N. lWe
Let aeCy-s c C - C5.Then
haveCN-r * 0,by completeness.
u ( S )+ u ( N - S ) : o ( S )* a ( i V- S ) : a ( N ) : u ( N ) ,
and decomposabilityfollows from (b) of TheoremL.
"cattesian ptoduct" property
1) Based on Mlscnrnn, Pnlrc, and Sneplrv
[1957]. Note that the
game.
does not depend on convexity, but is valid for any decomposable
Coresof ConvexGames
(b) Let u be additive acrossthe partition tlfr), and first let a be in the cartesian
product of the componentcofes.Then for all i and 5 we have
a(Ni n S) l u(Ni n ^S) and a(lfJ : u(NJ.
Applying(S)we obraina(S) 2 u(S)and a(N) : u(lf), showingthat a is in the core.
Next let a be a payoffvector that is not in the cartesianproduct of the component
cores.Then for sornei,
either4(s) < D(s) for some s e lf,,
of
4(lfJ > u(lfi) .
In the first case,o+ C.In the secondcase,we have
4(N) - 4(N - NJ: a(l{J > u(l{J = u(N) - u(N - N;,
using (9).Hence either a(l{) > o(I{) or a(N - Nr) < u(/f - NJ, so againo+C.
Finally,to determinethe dimensionof C let {I{t,.,.,lfo} yield the finestdeand havefull-dimencomposition.Then the componentsare all indecomposable
sionalcores,by part (a),so that their cartesianproduct has dimension
1): rt'- P.
itn,1
This completesthe proof.
4,2. The Value
The ualueof a gameu is the payoff vectorE e EN definedby
9i:
\-'r
L
5 ct{
SE;
(s-1)!(n-s)l
nl
[ u ( s ) - u ( s - t t ] ),1 a l l i e N .
(1e)
games,et4u({t}) for all ie.l{r).
We haveE(N): u(N) and, for superadditive
"equity
value" associatedwith the
The quantity ei may be interpreted as the
positionof the i-th player in the game.
Theorem7.
The value of a convexgameis an elementof the core.
Proof.
It is well known (seeSuneLEy[1953b]) that
o*,
e :4fl!L orca
(2 0)
whereO is the set of all orderingsof N. Hence,by Theorems3 and 5, the value
belongsto the core indeed,it is in the relativeintErior sinceit is a centerof
gravity of the vertices.
r) See SHerr,sy
[1953b]. It is also shown in SHeprrv [1953b] that the value of a decomposable
game is the cartesian product of the values of its components.
L. S. StHplrv
we sketcha proof of (20).Fix f. Given Se f, we ask: for how
For completeness,
= S?It is clearlyboth necessary
and sufficient
manyorderingsardo we haveSar,a{i)
t h a t a r b e s u c h t h a t C I<
r (@
/ ) ( f ) f o r a l l i e S - { , } a n d c o ( i>) c l ( , ) f o r a l l j el V - S .
Thus the number of orderingsis (s - 1)l(n - s)!. The result now follows easily
frorn (19)and (17).
4.3, StableSets
A payolf vector b is said to be dominatedby a payolTvector a if there is a nonempty S e .,f suchthat
a(S) { u(S),
and
ai 2 bi all
f e ,S.
(21)
A set V of,feasiblel)payoffvectorsis said to be stableif everyfeasiblel)payoff
vector is either a member of V or dorninatedby a memberof I/, but not both.
"standardof behavior",i.e., & set of "conA stableset rnay be interpretedas a
ventional" outcomeswhich are alwaysgivena chanceto dominateany proposal'
that might be put forward during pre-playnegotiations.
It is easilyverified that every stableset containsthe core. Sinceno stableset
can properly include another,it followsthat if the core is stablethen it is the only
stabteset2).
Theorem8,
The core of a convexgameis stable(i.e.,is the uniquevoN NnuueNN-MonceNsrERNsolution).
Proof.
Let {Cr} br regular,and take any feasibleb not in C. We shall show that b is
dominatedby an elementof C. Defineg(O): 0 and
u(s)- b(s)
,
s(s):
allOcSElV,
and let g attatnitsmaximum,g*,?t S = S*. Sinceb+C but is feasiblewe seethat
gto
a e E Nb y :
fb, o* ,
Q r : 1 '
Ir,
ieS*
ieN - S*
"feasible' in these two
t) The classical definiticin
[voN NnuueNr.r and MoncENsrERN] replaces
"feasible
places by
and individually rational", where a e EN is indioidually rational rt and only if
a, 2 u({i}), all i e N. This distinction (discussedat length in Srmplnv [1951] - see also Luce and
Rerrrl [p.2t5ff.] is not important here, sinceit can be shown that the core is stable in the one sense
if and onty if it is stable in the other. (Seee.g., Theorems 12 and 13 of Grurcs [1959].)
2; The early conjectures
[voN Nnunauw and MoncENsrERN;Sulrnnv, 1953] that stable sets exist
for all superadditive games and that the core is the intersection of all stable setshave been disproved;
seeLuces LL967,1968].
Coresof ConvexGames
and we seethat
Sincea(S*): b(S*)* s*g* : u(S*),condition(21)is satisfied,
a dominatesb. Moreover,sinceo(N) : u(S*)* c(l{ - S*) : c(N) : u(N),we
(11).
thecoreinequalities
It remainsto showthata satisfies
seethat a is feasible.
Let T E N be arbitrary,and breakup T into Q : T nS* and R f - ^S*,
Thenwe have,in easysteps,
ag):a(Q)+a(R)
: b(8) + qg* + c(R)
> b(g + qs(0)+ c(Tv s*)- c(s*)
4 u(Q)+ a(T u Sn)- u(,S*)
3 u(r).
Hencea is in the core,
4.4. Remsrks
1. An example of a stable core that does not come from a regular, or even
complete,configurationis shown in Fig. 3. The core is a perfectcube,with faces
Fig. 3: Example of a stable,nonregular core
etc.ThesetsCs,s: 3,areempty.
at (1,1,!,0),(+,+,t,8),
{Cs:s:2} andvertices
2. We have obtained a completetheory of'core stability in symrnetricgarnes,
i. e.,gamesof the form
u(S) = "f(s) ;
this will be presentedelsewhere.
L. S. Snaplnv
26
"positivefraction" of all gamesaresolvable- moro
3. Theorem8 impliesthat a
precisely,that the set of solvablegameson .,f includesa full-dimensional(i.e.,
(2^
1)-dimensional)cone rn Er-{o}. This was first pointed out by Gtnmsr),
usingthe conegeneratedby the gamesin 0,l-normalizationthat satisfyu(S)< Iln
f.or2 S s < ru.This coneincludessomebut not all convexgamcs,aswell as some
that are not convex.
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ReceivedApril, 1971