Abstract. We give general conditions, based on the largeness of the core,under which cores of exact TU games are their unique von Neumann-Morgenstern stable sets. We show that this condition is satisfied by convexgames and by nonatomic exact market games. In this way, we extend andunify earlier results existing in literature. Under some additional conditionswe also prove the equivalence between the core and the Mas-Colell bar-gaining set.

Key words: TV games, vN-M stable sets, large cores, bargaining sets.

1. Introduction

Along with the Shapley value, the core is the most popular solution conceptfor cooperative games and it has found many economic applications. A majorattraction of the core is its uniqueness and its characterization through sets ofinequalities, which makes it a relatively easy object to handle.

On the other hand, a conceptually better solution concept is the so-calledvon Neumann-Morgenstern stable set, introduced by von Neumann andMorgenstern (1944). For many years stable sets were the standard solutionconcept for cooperative games and the subject of many investigations (see,e.g., the surveys of Lucas, 1992, and Owen, 1995).

Int J Game Theory (2005) 33: 189–213DOI: 10.1007/s001820400191

We thank Jean-Francois Mertens, Enrico Minelli, William Thomson, and two anonymousreferees for helpful comments. We also thank seminar audiences at CORE, Cornell, Pescara, andRochester. We gratefully acknowledge the financial support of the Ministero dell’Istruzione,dell’Universita e della Ricerca.

Stable cores of large games

Massimo Marinacci and Luigi Montrucchio

Dipartimento di Statistica e Matematica Applicata and ICER, Universita di Torino, PiazzaArbarello 8, 10122 Torino, Italy.E-mails: massimo.marinacci@unito.it and luigi.montrucchio@unito.itURL: http://web.econ.unito.it/gma.

Revised: September 2004

Despite their conceptual appeal, stable sets have some serious shortcom-ings: they may not exist and not be unique. More important, they are difficultto handle. As a result, though it is a conceptually less appealing solutionnotion than stable sets, the core has gained a central importance in allapplications because of its tractability. When these two fundamental solutionconcepts happen to be equivalent — that is, when the core is the unique vonNeumann-Morgenstern stable set — we are in an ideal situation where theresulting solution combines the conceptual appeal of stability and the trac-tability of the core.

In this paper we focus our attention on such an ideal situation. Our maincontribution, Theorem 1, establishes a general result showing under whatconditions the core of a continuous exact game is its unique von Neumann-Morgenstern stable set. We then show in Corollary 1 that the two importantclasses of non-atomic exact market games and of convex games satisfy suchconditions. Therefore, the two solution concepts are equivalent for theseclassical classes of games.

Theorem 1 extends and unifies the existing results on this subject forinfinite games. In particular, Einy and Shitovitz (1996) and Einy et al. (1997)solved this problems for convex games, whereas Einy et al. (1996) treatedglove market games, that is, games defined as the minimum of finitely manynon-atomic probability measures.

Our approach is based on the largeness condition that Sharkey (1982)introduced to study the stability of cores in finite games. This makes it pos-sible to relate our results for infinite games with the existing ones on finitegames (see, e.g., Shapley, 1971, Kikuta and Shapley, 1986, Gellekom et al.,1998, and Biswas et al., 1999 and 2001).

We also prove some equivalence results between cores and Mas-Colellbargaining sets. Among the several concepts of bargaining sets introducedsince the seminal work of Aumann and Maschler (1964), we consider thenotion due to Mas-Colell (1989) because it can be easily adapted to infinitegames, as observed by Einy et al. (1997), who studied this issue for infiniteconvex games.

At a more technical level it is noteworthy that our main result is basedon a refinement of Lemma 3.2 of Einy et al. (1997). While they use theSion Minmax Theorem, we use the Ekeland Variational Principle (Ekeland,1974) and this permits us to go beyond the case of convex games treatedby them.

The paper is organized as follows. Section 2 contains some preliminaries.Section 3 introduces our largeness conditions and contains our main resultson stable cores and Mas-Colell bargaining sets. Section 4 studies two specialclasses of games for which our results hold, that is, games with finitedimensional non-atomic cores (a class of games studied in detail in Marinacciand Montrucchio, 2003a) and games with finitely many players. The paperends with two appendices. Appendix 1 establishes all mathematical toolsneeded for the proofs of our main results. In particular, Theorem 6, whichcharacterizes our largeness conditions through the construction of the con-cave upper envelope of a game, is of special importance. Appendix 2 collectsall the proofs of the main theorems.

190 M. Marinacci and L. Montrucchio

2. Preliminaries

2.1. Games

The set of players is X and the r-algebra R denotes the set of admissiblecoalitions. Subsets of X are in R even where not stated explicitly and they arereferred to both as sets and as coalitions.

A transferable utility (TU) game m is a function m : R! R such thatm Bð Þ ¼ 0. A game is:

positive if m Eð Þ � 0 for all E,bounded if supE2R m Eð Þj j <1,superadditive if m Eð Þ þ m Fð Þ � m E [ Fð Þ for all E \ F ¼ B,convex (or supermodular) if m E [ Fð Þ þ m E \ Fð Þ � m Eð Þ þ m Fð Þ for all sets

E and F ,continuous at E if m Enð Þ ! m Eð Þ for all the sequences En " E and En # E,continuous if m is continuous at any E,additive (or a charge) if m E [ Fð Þ ¼ m Eð Þ þ m Fð Þ for all pairwise disjoint

coalitions E and F ,countably additive (or a measure) if m

S1i¼1 Ei

� �¼P1

i¼1 m Eið Þ for allcountable collections of pairwise disjoint coalitions Eif g1i¼1.

Unless otherwise stated, charges and measures are understood to besigned. The set of all bounded charges (measures) is denoted ba Rð Þ (ca Rð Þ).Given k 2 caþ Rð Þ, the set of all bounded measures which are absolutelycontinuous with respect to k is denoted by ca R; kð Þ.

A charge m is non-atomic if for each m Eð Þ 6¼ 0 there exists F � E such thatm Fð Þ 6¼ 0 and m E n Fð Þ 6¼ 0. The set of all non-atomic measures is denoted byna Rð Þ.

The core of a game m is

coreðmÞ ¼ fm 2 ba Rð Þ : m Xð Þ ¼ m Xð Þ and mðEÞ � mðEÞ for all E 2 Rg:

The core is a weak�-compact subset of ba Rð Þ. Games having nonempty coresare called balanced.

Exact games are the main object of interest of our paper. A game m is exactif coreðmÞ 6¼ B and

m Eð Þ ¼ minm2core mð Þ

m Eð Þ

for all E 2 R. Namely, for any coalition E there exists some m 2 core mð Þ suchthat m Eð Þ ¼ m Eð Þ. Clearly, any exact game is bounded.

Given a coalition E and the relative r-algebra RE, the subgamemE : RE ! R is the restriction of the game m on RE. Games whose subgameshave all nonempty cores are called totally balanced. Exact games are totallybalanced. Moreover, bounded convex games are exact (see Lemma 1 inMarinacci and Montrucchio, 2003a).

We denote by B Rð Þ the set of all bounded R -measurable functions on X,equipped with the supnorm topology. The standard duality pairing betweenf 2 B Rð Þ and m 2 ba Rð Þ will be written as f ;mh i ¼

Rfdm. Finally, B1 Rð Þ

denotes the set of all members f of B Rð Þ such that 0 � f � 1 (its elements areoften called ideal coalitions), while Bþ Rð Þ denotes the cone of all non-negativefunctions in B Rð Þ.

Stable cores of large games 191

2.2. Stable sets

We adopt the definition of von Neumann-Morgenstern stable sets for infinitegames due to Einy et al. (1997), with only a slight modification to deal withnot necessarily positive games. Define rm : R! R [ �1f g by

rm Eð Þ ¼ infXn

i¼1m Eið Þ;

where the inf is taken over all finite partitions E1; . . . ;Enf g � R of E.1 Thevalue rm Eð Þ captures the idea of what the members of E are guaranteed toobtain without cooperation.

The set I mð Þ of all feasible and individually rational allocations, calledimputations, is defined as

I mð Þ ¼ m 2 ba Rð Þ : m � rm and m Xð Þ � m Xð Þf g. ð1ÞA coalition E is inessential if m Eð Þ ¼ rm Eð Þ and m E [ Fð Þ ¼ m Fð Þ for all F ,while it is essential if it is not inessential. The last condition in the definition ofinessential coalition plays an important role. Given a game m, a coalition N ism-null if

m N [ Fð Þ ¼ m Fð Þ ð2Þfor all F.2

The next result gives some properties of the set function rm and of essentialcoalitions.

Proposition 1. Suppose m is superadditive. Then:

(i) rm is additive and rm � m;(ii) if N is m-null, then N is rm-null;(iii) a coalition N is inessential if and only if it is m-null.

Define a dominance relation � among members of I mð Þ as follows: giventwo elements g and n of I mð Þ, say that g � n if there is an essential coalitionA such that m Að Þ � g Að Þ and g Fð Þ > n Fð Þ for all essential subcoalitionsF � A. If needed, we write g �A n when the dominance relation is viacoalition A.

A set V � I mð Þ of a superadditive game m is said to be a (von Neumann-Morgenstern) stable set if:

(i) V is internally stable: if n 2 V , then there is no g 2 V such that g � n;(ii) V is externally stable: if n 2 I mð Þ n V , then there is g 2 V such that g � n.

The core is internally stable. More is actually true: the core is the setof undominated members of I mð Þ. For positive games this was proved inEiny and Shitovitz (1996). Here we show that this is the case for generalgames.

1In Einy et al. (1997) the inf is taken over countable partitions of E. The value rm Eð Þmay then notbe well defined unless the game is positive, which is the case they consider. An approach similar toours can be found in Einy and Shitovitz (1996).2When the game is positive it is enough to consider coalitions F such that F \ N ¼ B. SubsectionA.1.2 contains some more details on null sets.

192 M. Marinacci and L. Montrucchio

Proposition 2. Let m be a balanced game. Assume that there exists an elementk 2 baþ Rð Þ such that k Eð Þ > 0 for all non m-null coalitions E. Then, g 2 core mð Þif and only if there is no n 2 I mð Þ such that n � g.

Proposition 8 guarantees the existence of such an element k for continuousexact games.

2.3. Market games

Market games will play an important role in this paper; they arise in modelingmonetary exchange economies (see, e.g., Hart, 1997). Their definition is basedon the na-topology of B Rð Þ, whose neighborhood base at f 2 B Rð Þ is given by

Uf e; m1; . . . ;mNð Þ ¼ g 2 B Rð Þ : g;mih i � f ;mih ij j < e, i ¼ 1; . . . ;Nf g;where m1; . . . ;mNf g � na Rð Þ and e > 0. Under this topology B Rð Þ is a locallyconvex (not Hausdorff) topological vector space. The na-topology, due toAumann and Shapley (1974), is the coarsest topology which makes contin-uous all the linear functionals f 7! f ;mh i with m 2 na Rð Þ. We will alwaysrestrict the na-topology to B1 Rð Þ.

By the Lyapunov Theorem the characteristic functions are na-dense inB1 Rð Þ. Therefore, any game m, when viewed as a function over the charac-teristic functions 1E, has at most one na-continuous extension to B1 Rð Þ.Following Aumann and Shapley (1974), we denote this extension by m�.

We say that a game m is a non-atomic market game if it is superadditive andadmits an na-continuous extension m� which is positively homogeneous (see,e.g., Mertens, 1980, for a similar definition). This name is justified by the factthat Aumann and Shapley (1974) have shown that under suitable conditionsexchange economies with a continuum of agents can be modelled as non-atomic market games. Non-atomic market games feature important proper-ties: they are totally balanced and continuous, with core mð Þ � na Rð Þ:3

3. Main results

3.1. Large cores

Sharkey (1982) showed that in games with a finite set of players the core isstable when it is ‘‘large’’ enough, a result later refined by Kikuta and Shapley(1986). In this section we introduce some notions of largeness for infinitegames that generalize the one by Sharkey.

We list our main conditions.L0ð Þ If n � m, with n 2 ba Rð Þ, then there is g 2 core mð Þ such that n � g.ðL1Þ If nþ k � m, with n 2 ca Rð Þ and k � 0 is a scalar, then there is

g 2 core mð Þ such that nþ k � g.ðL2Þ Given any E 2 R, if nþ k � mE, with n 2 ca REð Þ and k � 0, there is

g 2 core mð Þ such that nþ k � gE.For short, in what follows we will often say that a game m belongs to the

class Li when condition Lið Þ is satisfied.

3For our purposes there is no gain in using Mertens’s dna-extension in place of the na-extension.In fact, for the games we will consider the extension m� will be Lipschitz continuous over B1 Rð Þ,and in this case m� is both the dna and the na extension.

Stable cores of large games 193

Sharkey’s original notion is L0ð Þ. Clearly, L2ð Þ implies L1ð Þ. It may beproved that L1ð Þ implies L0ð Þ, provided m is continuous and exact. Moreover,though L2ð Þ is stronger than L1ð Þ, they become equivalent under a mildLipschitz condition on the game. Proofs of all these relations can be found inMarinacci and Montrucchio (2004b).

3.2. Stable cores

We begin with our main result.

Theorem 1. Let m be a continuous exact game satisfying condition L2ð Þ. Then,core mð Þ is its unique stable set.

A version of Theorem 1 can be proved only under condition L1ð Þ.

Proposition 3. Let m be a continuous exact game m satisfying condition L1ð Þ.Then,

V \ ca Rð Þ ¼ core mð Þ ð3Þfor all stable sets V of m.

Eq. (3) shows that under L1ð Þ the core is still the unique stable set, pro-vided only countably additive imputations are considered.

The next lemma, based on the Ekeland Variational Principle (see, Eke-land, 1974), is the key step in proving Theorem 1.

Lemma 1. Let m be a continuous and exact game satisfying L2ð Þ. Given acoalition E, an element n 2 ca REð Þ and e > 0, there exist A � E and g 2 core mð Þsuch that:

(i) m Að Þ � n Að Þ � a� e, wherea ¼ sup

F�Em Fð Þ � n Fð Þ½ �; ð4Þ

(ii) g Að Þ ¼ m Að Þ;(iii) g Fð Þ � n Fð Þ � e1=2k Fð Þ for all F � A.

Remarks (i) The measure k 2 ca1 Rð Þ in point (iii) is a control measure thatarises naturally in continuous exact games (see Proposition 8 and Lemma 2).(ii) Inspection of the proof shows that Lemma 1 still holds for E ¼ X when mjust belongs to L1. The special case of Lemma 1 when e ¼ 0, E ¼ X, and m is acontinuous and positive convex game, has been established by Einy et al.(1997). Notice that the condition e ¼ 0 (which requires the sup in (4) to beattained) substantially restricts the class of continuous exact games to whichLemma 1 can be applied.

Theorem 1 raises the key question of which games lie in L2. Fortunately,this is the case for two important classes of games, convex games and marketgames. We begin with convex games.

Theorem 2. Any bounded, continuous and convex game satisfies condition L2ð Þ.Turn now to exact market games. Recall that they are always continuous.

Theorem 3. Any non-atomic exact market game satisfies condition L2ð Þ.

194 M. Marinacci and L. Montrucchio

By Theorems 2 and 3, we have the following noteworthy consequence ofTheorem 1.

Corollary 1. The core of a game m is its unique stable set provided either m is anexact market game or it is a bounded, continuous and convex game.

This result for finite convex games was proved by Shapley (1971), and forinfinite positive convex games by Einy et al. (1997) (a version for games withcountably many players was proved by Einy and Shitovitz, 1996). For exactglove market games, a subclass of the exact market games, this result isknown from Einy et al. (1996). Corollary 1 extends and unifies these earlierresults by showing that they are all special cases of Theorem 1. An especiallyimportant subclass of non-atomic market games will be discussed in the nextsection.

To put in better perspective Corollary 1, as well as Theorems 2 and 3,observe that convex games and exact market games are two essentially dis-joint classes of games. In fact, the next result shows that their overlap consistsof additive games, a trivial class of games from a game-theoretic viewpoint.

Proposition 4. A bounded convex game is a non-atomic market game if andonly if it is a measure in na Rð Þ.

We close with an example showing that games satisfying L2ð Þ may beneither convex nor market games.

Example. Let X;Rð Þ be a measurable space and X1;X2f g a measurablepartition of X. Set Ri ¼ RXi , i ¼ 1; 2. Given any two games mi : Ri ! R, definetheir direct sum m1 � m2 as m1 � m2ð Þ Eð Þ ¼ m1 E \ X1ð Þ þ m2 E \ X2ð Þ. We havecore m1 � m2ð Þ ¼ core m1ð Þ � core m2ð Þ, and so m1 � m2 is a continuous exact gameon X;Rð Þ if and only if both m1 and m2 are continuous and exact. It is simple tocheck that m1 � m2 2 Li if and only if both m1 and m2 do.

An extension is easily obtained when m1 and m2 have disjoint carriers X1

and X2, with X ¼ X1 [ X2. In this case, m1 � m2 ¼ m1 þ m2. For example, letP1 and P2 be two non-atomic Borel probability measures on X ¼ 0; 1½ �, and letd0 and d1 be the Dirac measures concentrated at 0 and 1, respectively.Consider the game m ¼ min P1; P2f g þ d0d1, which is the sum of an exact glovemarket and a convex game having disjoint carriers X n 0; 1f g and 0; 1f g,respectively. The exact game m belongs to L2, without being neither convexnor a market game.

3.3. Mas-Colell bargaining sets

In this subsection we consider the Mas-Colell bargaining set MB mð Þ. Einyet al. (1997) have extended Mas-Colell (1989)’s original definition to infinitegames; for brevity, we refer the reader to them for a formal definition.

Using our techniques we can establish a version of Theorem 1 for the Mas-Colell bargaining set. Here we have to require that the sup in (4) be attained,something not needed for Theorem 1.

Theorem 4. Let m be a continuous exact game satisfying L1ð Þ. IfsupF2R

m Fð Þ � n Fð Þ½ � ð5Þ

is attained for all n 2 ca R; kð Þ, then core mð Þ ¼ MB mð Þ \ ca Rð Þ.

Stable cores of large games 195

The sup in (5) is trivially achieved for finite games, and the same is true forcountably infinite games (see the proof of Corollary 2). Besides them, this isthe case for bounded and continuous convex games (see Einy et al., 1997).Another important class of market games for which Theorem 4 holds are theexact games whose cores are finite dimensional subsets of na Rð Þ. Because oftheir importance, we devote Subsection 4.1 to them.

By Theorems 2 and 3, we have the following consequence of Theorem 4.

Corollary 2. For a game m, core mð Þ ¼ MB mð Þ \ ca Rð Þ provided at least one ofthe following conditions holds:

(i) X is at most countable, m is continuous, exact, and satisfies L1ð Þ;(ii) m is bounded, continuous, and convex;(iii) m is exact and core mð Þ is contained in a finite dimension subspace of

na Rð Þ:4

Since for finite games we trivially have ca Rð Þ ¼ ba Rð Þ, point (i) says thatcore mð Þ ¼ MB mð Þ for finite exact games satisfying L1. Dutta et al. (1989, Prop.3.3) proved this result for finite convex games, while Einy et al. (1997, Thm.B) extended it to infinite positive and continuous convex games (see also Einyet al., 1998, for games with countable many players). In Section 4 we willdiscuss in detail the scope of our extension by showing that there are finiteexact games in L1 that are not convex.

Point (i) also applies to the consistent bargaining set CB mð Þ introduced byDutta et al. (1989); that is, core mð Þ ¼ CB mð Þ holds for finite exact games in L1.In fact, they observe that core mð Þ � CB mð Þ � MB mð Þ.

4. Applications

So far we have given results for rather general classes of exact games. In thissection we study two special types of games for which the core is stable andagrees with the Mas-Colell bargaining set.

4.1. Finite dimensional games

Our objective here is to identify exact games having finite dimensional coresin na Rð Þ. To do this, let us define a class of measure games introduced byMarinacci and Montrucchio (2003a), which includes important games origi-nating from economics, such as production games and games arising fromexchange economies (see, e.g., Owen, 1975 and 1995, Billera and Raanan,1981, Hart and Neyman, 1988).

Given a compact (not necessarily convex) set C � Rn, define the positivelyhomogeneous and concave function gC : Rn ! R as follows:

gC xð Þ ¼ minc2C

c x:

The function gC xð Þ satisfies many well-known properties. For instance,gC ¼ gC1

, where C1 is the convex hull of C.

4By Theorem 5, this is a subclass of non-atomic market games.

196 M. Marinacci and L. Montrucchio

Given the function gC : Rn ! R and an n-vector measure m ¼ m1; . . . ;ðmnÞ, with mi 2 na Rð Þ for i ¼ 1; 2; . . . ; n, define the measure game

m Eð Þ ¼ gC m Eð Þð Þ ¼ minc2C

c m Eð Þ: ð6Þ

We call them generalized linear production games,5 as they extend the linearproduction games of Owen (1975) and Billera and Raanan (1981).

Setting C Eð Þ ¼ c 2 C : c m Eð Þ ¼ m Eð Þf g for any coalition E, the core ofthe measure game (6) is given by

core mð Þ ¼ c m : c 2 coC Xð Þf g: ð7ÞThis formula is proved by Marinacci and Montrucchio (2003a) and it extendsearlier results for linear games of Billera and Raanan (1981). The generalizedlinear production games are non-atomic market games, though not neces-sarily exact. By (7), the core is a finite dimensional subset of na Rð Þ.

We can now characterize finite dimensional exact games.

Theorem 5. Given a game m, the following conditions are equivalent:

(i) m is exact and core mð Þ is a finite dimensional subset of na Rð Þ.(ii) m is an exact generalized linear production game.(iii) m is a generalized linear production game such that C Eð Þ \ C Xð Þ 6¼ B for all

E:

Moreover, a game satisfying any of these conditions is a non-atomic marketgame for which Theorem 4 holds.

Summing up, exact finite dimensional games are a significant class ofmarket games for which our results on stability and on the Mas-Colell bar-gaining set hold (see Corollary 2-(iii)).

The hypothesis that core mð Þ is a finite dimensional subset of na Rð Þ isneeded in Theorem 5 to conclude that m is a market game. In fact, considerm ¼ k2 , where k is the Lebesgue measure on 0; 1½ �. The game m is continuousand exact with core mð Þ � na Rð Þ. Nevertheless, m is not a market game as itsna-continuous extension, given by m� fð Þ ¼ k fð Þð Þ2 for all f 2 B1 Rð Þ, is notconcave. On the other hand, m does not satisfy condition (i) of Theorem 5because core mð Þ is not finite dimensional (this can be proved using, forexample, the results of Marinacci and Montrucchio, 2004c).

An important example of finite dimensional games is given by exact glovemarket games, which are studied by Einy et al. (1996). They are defined asminima of finitely many non-atomic measures mi, that is, m Eð Þ ¼min m1 Eð Þ; :::;mn Eð Þf g. Next we show that, among exact games, glove marketgames are characterized by having cores that are polytopes. As a result, exactglove market games are a very special example of exact finite dimensionalgames.6

Proposition 5. An exact game m : R! R is a glove market game if and only ifits core is a polytope in na Rð Þ.

5By suitably modifying the set C, we can assume mi 2 naþ or, even, mi 2 na1.6A polytope is the convex hull of a finite set. Polytopes are nothing but compact polyhedra (seeAliprantis and Border, 1999).

Stable cores of large games 197

4.2. Finite games

The techniques used in the present paper are especially suited for infinitegames. This emerges in Proposition 3, which inter alia says that a finite gamehas a stable core as long as it satisfies L1ð Þ. This is too strong a condition:Sharkey (1982) has shown that condition L0ð Þ is enough for core stability infinite games, and Kikuta and Shapley (1986) extended his result by showingthat a weaker notion of extendability suffices. We refer the reader to Gel-lekom et al. (1998) and Biswas et al. (1999, 2001) for a detailed analysis of therelationships existing among stability, largeness, and extendability in finitegames.

Our unique novel contribution in the finite setting is Corollary 2 -(i),which shows that core mð Þ ¼ MB mð Þ provided the exact game is in L1. To see thescope of our extension, in the rest of this section we show that, beyond convexgames, there is a quite rich class of finite exact games living in L1.

Here it is useful to adopt the notion of the superdifferential @m Eð Þ of agame m, introduced by Marinacci and Montrucchio (2003a).

Given a game m and a coalition E, the superdifferential @m Eð Þ is the weak�-closed and convex (possibly empty) set

@m Eð Þ ¼ m 2 ba Rð Þ : m Fð Þ � m Eð Þ þ m Fð Þ � m Eð Þ, 8F 2 Rf g:

Observe that @m Bð Þ is nothing but the set of all acceptable vectors of m (seeBiswas, 2001). Using this notion it is possible to derive simple computablecharacterizations of L1ð Þ, discussed in Appendix A.3.

Given a scalar k � 0, define the k-modification mk of m as follows

mk Eð Þ ¼ m Eð Þ � k if E 6¼ B;0 if E ¼ B:

�

We can view mkf gk�0 as a family of games generated by the ‘‘parent’’ game m viasuitable translations. Consider the following strengthening of condition L0ð Þ:

K0ð Þ For all k � 0 the games mk satisfy L0ð Þ.

Proposition 6. Let m be a finite game satisfying condition K0ð Þ. Then, there is

some k such that mk 2 L1 for all k � k. Moreover, for all k � k:

(i) mk is exact if @m Eð Þ 6¼ B for all E,(ii) mk is convex only if m is convex.

By Proposition 6, to generate exact games satisfying L1ð Þ that are notconvex, it is enough to pick a non convex ‘‘parent’’ game m satisfying K0ð Þ. Infact, for all k large enough, the associated games mk are exact games in L1 thatare not convex.

As to what games satisfy K0ð Þ, we know from the literature at least threeclasses of such games:

(i) all symmetric exact games (see Thm. 5 in Biswas et al., 1999),(ii) all exact games with n � 4 players (see Thm. 3 in Biswas et al., 1999),(iii) all subconvex games (see Sharkey, 1982, and Gellekom et al., 1998).

All such games generate many non-convex exact games in L1.

198 M. Marinacci and L. Montrucchio

5. Concluding remarks

Our stability results rest on two crucial assumptions, the exactness of thegame and its continuity. In this concluding section we discuss them.

As to exactness, we know that core stability is no longer true in general fortotally balanced games. Studies on non exact glove market games, like Einyet al. (1996), show that the stability of the core no longer holds, with theexception of very particular examples. In any case, our techniques are notsuited to study non exact games: under a suitable Lipschitz condition, totallybalanced games in L0 are actually exact.

Continuity as well enters the proofs of our results in an essential way. Tosee how things can go wrong without continuity, consider the unanimitygames uA : R! 0; 1f g, a classical class of convex games defined by

uA Eð Þ ¼ 1 A � E;0 else,

�

for a given coalition A. The next result can be proved using some results ofEiny and Shitovitz (1996).

Proposition 7. Let X;Rð Þ be a Borel space. We have:

(i) uA is continuous if and only if A is finite;(ii) core uAð Þ is stable when A is countable.(iii) uA has no stable sets when A is uncountable.

In the discontinuous case, when A is uncountable, the convex game uA has,therefore, no stable sets at all.

A Appendix: Related analysis

A.1 Games

In the proofs we will often use the following basic properties of continuousexact games, essentially due to Schmeidler (1972). For the reader interested inmore details we refer to Marinacci and Montrucchio (2004a, Thm. 4.2).Proposition 8 and Lemma 2-(iv) below ensure the existence of a ‘‘control’’probability measure k 2 ca1 Rð Þ. Throughout the paper the symbol k will al-ways denote such a measure.

Proposition 8. Given an exact game m, core mð Þ � ca Rð Þ if and only if m iscontinuous at B and X: In such a case, m is continuous and there exists somek 2 caþ Rð Þ such that core mð Þ � ca R; kð Þ.

If N is m-null, then m Eð Þ ¼ 0 for any E � N . For, by (2): 0 ¼ m Nð Þ ¼m E [ Nð Þ ¼ m Eð Þ: Hence, when m is a charge, a set N is m -null if and only ifmj j Nð Þ ¼ 0:

We write m m when each m-null coalition is m-null, and m � m when acoalition is m-null if and only if it is m-null. Finally, given m 2 ba REð Þ, wewrite mE m when each m-null subset in E is m-null as well (note that thisdiffers from m mE).

Stable cores of large games 199

Let MmE ¼ m 2 ba REð Þ : mE mf g and MrmE¼ MmE \ ca Rð Þ. Whenever

E ¼ X, we simply write Mm and Mrm . The next lemma collects the main basic

properties related to the notion of null sets.

Lemma 2. For any game m we have:

(i) if N is m-null, then E is m-null for all E � N ,(ii) core mð Þ � Mm,(iii) if m 2 Mm, then mE 2 MmE ,(iv) if m is exact and continuous, then there is a control probability measure

k 2 ca1 Rð Þ such that m � k and Mrm ¼ ca R; kð Þ.

Proof. (i) Let E � N and F be any coalition. By (2), m E [ Fð Þ ¼m E [ F [ Nð Þ ¼ m F [ Nð Þ ¼ m Fð Þ and so E is m-null.

(ii) Let m 2 core mð Þ, N be m-null and E � N : Then m Eð Þ � m Eð Þ ¼ 0, andm Ecð Þ � m Ecð Þ ¼ m Xð Þ ¼ m Xð Þ ¼ m Eð Þ þ m Ecð Þ. Hence, m Eð Þ ¼ 0 for allE � N , namely, mj j Nð Þ ¼ 0.

(iii) It is trivial.(iv) Since the core is weakly compact, the existence of a positive measure k

such that m k for all m 2 core mð Þ is guaranteed by Thm. IV.9.2 ofDunford and Schwartz (1954) (see also Schmeidler, 1972, Delbaen, 1974,Marinacci and Montrucchio, 2004a). We have

k ¼X1

n¼1

1

2n

1

kn

Xkn

i¼1mn

i

���� ð8Þ

with each mni 2 core mð Þ. Let N be m-null. By (ii), core mð Þ � Mm and N is

m-null for all m 2 core mð Þ. Hence, mj j Nð Þ ¼ 0 for all m 2 core mð Þ. By (8),k Nð Þ ¼ 0. Therefore N is k-null. Conversely, suppose k Nð Þ ¼ 0. Ascore mð Þ � ca R; kð Þ, we have mj j Nð Þ ¼ 0 for any m 2 core mð Þ. By exactness,there are m;m0 2 core mð Þ such that:m N [ Fð Þ ¼ m N [ Fð Þ ¼ m Fð Þ � m Fð Þ ¼ m0 Fð Þ ¼ m0 N [ Fð Þ � m N [ Fð Þ

and so N is m-null. We conclude that m � k, and so Mrm ¼ ca R; kð Þ. n

A.2 Largeness

It is often difficult to check directly whether conditions L1ð Þ and L2ð Þ hold.Here we study alternative characterizations that, inter alia, permit us to proveTheorem 3.

A piece of notation is in order. Given a coalition E and u 2 B REð Þ, thefunction bu 2 B Rð Þ is its natural extension defined as bu xð Þ ¼ u xð Þ if x 2 E,and bu xð Þ ¼ 0 if x=2E. We denote by LE : B REð Þ ! B Rð Þ the injective mapLE uð Þ ¼ bu. Clearly, its adjoint map L�E : ba Rð Þ ! ba REð Þ is given byL�E mð Þ ¼ mE.

Given a continuous exact game m and E 2 R, define its concave upperenvelope extension mE : B1 REð Þ ! R by7

7A functional U : B1 Rð Þ ! R [ �1f g is called an extension of the game m if U 1Eð Þ ¼ m Eð Þ for allE.

200 M. Marinacci and L. Montrucchio

mE fð Þ ¼ inf m; fh i þ k : m 2 MrmE

and m Fð Þ þ k � m Fð Þ; 8F 2 RE

n o:

If E ¼ X, we write mE � m. Next we collect the main properties of thesefunctionals.

Proposition 9. Suppose m is continuous and exact. Then:

(i) there is a constant c such that mE fð Þj j � c for all E and f 2 B1 REð Þ;(ii) each mE is concave, continuous on intB1 REð Þ, and upper semicontinuous on

B1 REð Þ;(iii) mE is an extension of the subgame mE to B1 REð Þ.

Proof. (i) We prove the statement for E ¼ X. The same argument applies forall E. As m is exact, it is bounded; hence, mE Fð Þj j � c for all F 2 RE and forsome c. As 0 2 Mm, m fð Þ � c for all f . Moreover, by the definition of m fð Þ,for all e > 0, there is k and m 2 Mm such that m; fh i þ k � m fð Þ þ e, andm Fð Þ þ k � m Fð Þ � �c for all F . Then

m fð Þ � m; fh i þ k � e ¼Z 1

0

m f � tð Þdt þ k � e

� �k � cð Þ þ k � e ¼ �c� e;

which proves our claim.

(ii) The functional mE is the upper envelope of continuous affine functions.Consequently, mE is concave and upper semicontinuous. From (i), mE islocally bounded from below at any point f 2 intB1 REð Þ. Hence mE iscontinuous over the interior of B1 REð Þ.

(iii) Let F 2 RE. As m is exact, m Fð Þ ¼ m Fð Þ for some m 2 core mð Þ � Mrm . From

(iii) of Lemma 2, mE 2 MrmE. Hence, mE Fð Þ ¼ mE Fð Þ with mE 2 Mr

mE: This

implies mE 1Fð Þ ¼ mE Fð Þ. n

The next statement is the central property that relates upper envelopeextensions and the largeness of cores. It involves the following other impor-tant concave functional that extends exact games. Denote by me : B Rð Þ ! Rthe functional

me fð Þ ¼ minm2core mð Þ

m; fh i for all f 2 B Rð Þ: ð9Þ

Clearly, me 1Eð Þ ¼ m Eð Þ if m is exact.

Theorem 6. A continuous and exact game m satisfies L1ð Þ if and only if m ¼ me onB1 Rð Þ. Moreover, m satisfies L2ð Þ if and only if mE ¼ me LE on B1 REð Þ for allE 2 R; i.e., mE uð Þ ¼ minl2core mð Þ lE;uh i for all u 2 B1 REð Þ:

Proof. It suffices to prove the second statement, since the first one is a par-ticular case. One implication is simple. Assume m 2 L2. Fix E, and let n 2 Mr

mE

be such that nþ k � mE. We then have nþ k � lE for some l 2 core mð Þ. Thisimplies n;uh i þ k � lE;uh i ¼ l; LEuh i � me LEuð Þ for all u 2 B1 REð Þ . Hence,mE � me LE over B1 REð Þ. On the other hand, for any l 2 core mð Þ we havelE � mE with lE 2 Mr

mE. Hence, lE;uh i ¼ l; LEuh i � mE uð Þ. This implies

me LE � mE . We infer that me LE ¼ mE, as desired.

Stable cores of large games 201

As to the converse, fix E 2 R and assume that me LE ¼ mE. Let nþ k � mEfor some n 2 ca REð Þ. We first add the further condition that n 2 ca R; kEð Þ,where k is the usual control probability measure of Lemma 2-(iv). Clearly,supF�E m Fð Þ � n Fð Þ ¼ a � k. We adopt now the same method used in theproof of Lemma 1 (and the same notation adopted there). By the EkelandTheorem, given a scalar e > 0, there exists a coalition A � E such thatm Að Þ � n Að Þ � a� e, and

m Að Þ � n Að Þ � m Fð Þ � n Fð Þ � e1=2kE A4 Fð Þ

for all F � E (see (19) in the proof of Lemma 1). In view of (20), some tediousalgebra shows that

m Fð Þ � k þ n Fð Þ þ e1=2 kAc

E Fð Þ � kAE Fð Þ

� �; ð10Þ

where k ¼ m Að Þ � n Að Þ þ e1=2kE Að Þ.Thus, m Fð Þ � h Fð Þ for all F � E, where h is the affine function

h ¼ k þ nþ e1=2 kAc

E � kAE

� �:

Note that m Að Þ ¼ h Að Þ. Moreover, since k 2 ca R; kð Þ ¼ Mrm (see Lemma

2-(iv)), then kE 2 MrmE. Thus, kAc

E , kAE 2 Mr

mEand, being n kE, also n 2 Mr

mE.

Hence, m Fð Þ � h Fð Þ implies mE uð Þ � h uð Þ for all u 2 B1 REð Þ. Sincem Að Þ ¼ h Að Þ ¼ mE 1Að Þ, it follows

mE uð Þ � h ðuÞ ¼ mE 1Að Þ þ h uð Þ � h 1Að Þ

¼ mE ð1AÞ þ u� 1A; nþ e1=2 kAc

E � kAE

� �D E:

We infer that

nþ e1=2 kAc

E � kAE

� �2 @ mEð Þ 1Að Þ;

where @ mEð Þ 1Að Þ denotes the superdifferential of the concave function mE at 1A.As mE ¼ me LE over B1 REð Þ, we can also write

nþ e1=2 kAc

E � kAE

� �2 @ me LE þ Ið Þ 1Að Þ ð11Þ

where me LE is defined over B REð Þ and I is the indicator function; i.e.,I uð Þ ¼ 0 if u 2 B1 REð Þ and I uð Þ ¼ �1 elsewhere. As the concave functionme LE is continuous, the sum rule for superdifferentials (see Thm. 3.2.6 ofBarbu and Precupanu, 1986) yields

@ me LE þ Ið Þ 1Að Þ ¼ @ me LEð Þ 1Að Þ þ @I 1Að Þ:Moreover, by the chain rule for linear maps (see, e.g., Thm. 1.5 of Pshe-nichnyi, 1971)

@ me LE þ Ið Þ 1Að Þ ¼ L�E@ með Þ 1Að Þ þ @I 1Að Þ:It is easy to calculate both @I 1Að Þ and @ með Þ 1Að Þ. In fact, we have

@I 1Að Þ ¼ m 2 ba REð Þ : mA � 0, mAc � 0f g; ð12ÞL�E@ með Þ 1Að Þ ¼ lE : l 2 core mð Þ and l Að Þ ¼ m Að Þf g:

In view of (11),

nþ e1=2 kAc

E � kAE

� �¼ gE þ m ð13Þ

202 M. Marinacci and L. Montrucchio

with m 2 @I 1Að Þ, g 2 core mð Þ, and g Að Þ ¼ m Að Þ. Hence, mA � 0 and mAc � 0.Evaluating (13) at A, we obtain

n Að Þ � e1=2k Að Þ ¼ m Að Þ þ m Að Þ: ð14ÞBy using the inequality m Að Þ � n Að Þ � k, (14) yields m Að Þ � �k � e1=2k Að Þ. Inturn, this implies m � �k � e1=2k Að Þ. By (13) it follows

nþ e1=2 kAc

E � kAE

� �¼ gE þ m � gE � k � e1=2k Að Þ

nþ k � gE � e1=2k Að Þ � e1=2 kAc

E � kAE

� �:

The last inequality holds for a fixed e. Choose a sequence en ! 0. Conse-quently, there is a sequence gn 2 core mð Þ and a sequence An � E such that

nþ k � gnE � e1=2n k Anð Þ � e1=2n kAc

nE � kAn

E

� �:

As core mð Þ is sequentially weakly compact, there is a subsequence (stilldenoted by gn) weakly convergent to some g� 2 core mð Þ. Clearly, e1=2n k Anð Þ! 0 and e1=2n kAc

nE � kAn

E

� �! 0 setwise. Hence nþ k � g�E. This proves our

claim, provided n 2 ca R; kEð Þ:Now, suppose n 2 ca REð Þ. With the usual decomposition

n ¼ na þ nsþ � ns

� and N ¼ Nþ [ N� (see proof of Lemma 1), we have

na þ e1=2 kAc

E � kAE

� �¼ gE þ m

with m such that mEnA � 0 and mA � 0. Observe that the same holds if onereplaces A by A1 ¼ A n Nþ [ N�. On the other hand, it is easy to see thatns

A1� 0 and ns

Ac1� 0. Therefore,

nþ e1=2 kAc1

E � kA1

E

� �¼ gE þ m

holds and the proof goes on as before. nUsing the upper envelope m, we can give some useful characterizations of

games enjoying condition L1ð Þ. To do this we need the following lemma,where em denotes a functional game defined over the ideal coalition and mdenotes the game m Eð Þ ¼ em 1Eð Þ.

Lemma 3. For a bounded functional game em : B1 Rð Þ ! R, the following con-ditions are equivalent:

(i) em is superlinear and

em af þ 1� að Þ1Xð Þ ¼ aem fð Þ þ 1� að Þem 1Xð Þ;for all a 2 0; 1½ � and all f 2 B1 Rð Þ,

(ii) for all f 2 B1 Rð Þ it holds:em fð Þ ¼ min

m2Cm; fh i;

where C ¼ m 2 core mð Þ : m fð Þ � em fð Þf g:In such a case, the game m is exact and em fð Þ � me fð Þ for all f 2 B1 Rð Þ.

Proof. We only prove that (i) implies (ii), the converse being obvious. Thefunctional em is clearly concave and admits a unique positively homogeneousconcave extension on Bþ Rð Þ, which we still denote by em. This functional emsatisfies the condition em f þ k1Xð Þ ¼ em fð Þ þ kem 1Xð Þ for each k � 0 and all

Stable cores of large games 203

f 2 Bþ Rð Þ. For, assume first f 2 B1 Rð Þ and k > 0. Pick a 2 0; 1ð Þ such thatk ¼ 1� að Þa�1. We have

em f þ k1Xð Þ ¼em f þ 1� að Þa�11X� �

¼ a�1em af þ 1� að Þ1Xð Þ¼em fð Þ þ kem 1Xð Þ:

The extension of this equality to functions not in B1 Rð Þ is then easily obtainedby exploiting positive homogeneity. A routine exercise shows that em has aunique extension on B Rð Þ, still denoted em, superlinear and such thatem f þ k1Xð Þ ¼ em fð Þ þ kem 1Xð Þ for each k 2 R. It suffices, for a function f � �h,to define em fð Þ ¼ em f þ hð Þ � hem 1Xð Þ.

As a result, em : B Rð Þ ! R is a support function, which is continuous sinceit is locally bounded over a neighborhood of some point in B1 Rð Þ. By Hor-mander (1954), there is a weak�-compact and convex set C � ba Rð Þ such that

em fð Þ ¼ minm2C

m; fh i for all f 2 B Rð Þ;

where C ¼ @em 0ð Þ ¼ m 2 ba Rð Þ : m fð Þ � em fð Þ for each f 2 B Rð Þf g. It remainsto show that C � core mð Þ. We have em 1Xð Þ ¼ �em �1Xð Þ because

0 ¼ em 0ð Þ ¼ em �1X þ 1Xð Þ ¼ em �1Xð Þ þ em 1Xð Þ:On the other hand, m 1Xð Þ � em 1Xð Þ and m �1Xð Þ � em �1Xð Þ for all m 2 @em 0ð Þ, sothat m 1Xð Þ ¼ em 1Xð Þ for all such m. Hence, C � core mð Þ.

The last claims are obvious. Note that C � core mð Þ implies em fð Þ � me fð Þ. n

A.3 Finite games

Here we provide some characterizations of condition L1ð Þ for finite games.They will be used in Appendix B to prove Proposition 6.

Set KEc ¼ m 2 ba Rð Þ : mE � 0and mEc � 0f g. It is easily seen that KEc isthe recession cone of the set @m Eð Þ. For finite games @m Eð Þ is therefore anunbounded polyhedron, having the representation (see, e.g., Prop. 2.1.7 ofAuslender and Teboulle, 2003):

@m Eð Þ ¼ co ext@m Eð Þ½ � þ KEc : ð15ÞOur first result characterizes balanced games satisfying L1ð Þ. Set

coreE mð Þ ¼ l 2 core mð Þ : l Eð Þ ¼ m Eð Þf g. Clearly, a game is exact if and only ifcoreE mð Þ 6¼ B for all E. As coreE mð Þ � @m Eð Þ, a game is exact if and only if@m Eð Þ \ core mð Þ 6¼ B for all E:

Proposition 10. A finite balanced game m satisfies L1ð Þ if and only if

@m Eð Þ ¼ coreE mð Þ þ KEc ð16Þfor all E such that @m Eð Þ 6¼ B.

We omit the simple, but tedious, proof of this result. It rests on the factthat the sup in (4) is attained, and so (16) holds also for infinite gamesenjoying this property, provided the superdifferentials lie in ca Rð Þ. Observethat condition (16) at B becomes @m Bð Þ ¼ core mð Þ þ KX, which is nothing butm 2 L0.

We close with an especially simple computable characterization of L1ð Þ.

204 M. Marinacci and L. Montrucchio

Proposition 11. A finite game with large core L0ð Þ satisfies L1ð Þ if and only if,for all E;

g 2 ext@m Eð Þ¼)g Eð Þ ¼ m Eð Þ: ð17Þ

Proof : The necessity is obvious in view of (15) and (16). Let us show that itsuffices. Let g 2 ext@m Eð Þ. As g Eð Þ ¼ m Eð Þ, we deduce that g Fð Þ � m Fð Þ for allF . For, m Fð Þ � m Eð Þ þ g Fð Þ � g Eð Þ ¼ g Fð Þ for all F . Hence, g 2 @m Bð Þ. As mhas large core, g ¼ lþ m, with l 2 core mð Þ and m � 0. Hence m Eð Þ ¼g Eð Þ ¼ l Eð Þ þ m Eð Þ � m Eð Þ. It follows mE ¼ 0 and l Eð Þ ¼ m Eð Þ¼)l 2coreE mð Þ. Consequently, g ¼ lþ m 2 coreE mð Þ þ KEc . This is true for allg 2 ext @m Eð Þ. In view of (15), @m Eð Þ � coreE mð Þ þ KEc . Hence, @m Eð Þ ¼coreE mð Þ þ KEc and m 2 L1 by Proposition 10. n

B Appendix: Proofs

Proposition 1. We omit the proof of (i) because it is a standard exercise.

(ii) If N is m-null, we know that m Eð Þ ¼ 0 for all E � N . Hence, rm Nð Þ ¼ 0and rm Eð Þ ¼ 0 for all E � N as well.

(iii). Let N be m-null. By (ii), rm Nð Þ ¼ 0 ¼ m Nð Þ. Moreover, m N [ Fð Þ ¼m Fð Þ and then N is inessential. Conversely, if N is inessential, m N [ Fð Þ ¼m Nð Þ. Hence, N is m-null. n

Proposition 2. One implication is obvious. Hence, it suffices to prove that ifg 2 I mð Þ n core mð Þ, there is an element n � g, with n 2 I mð Þ. Asg 2 I mð Þ n core mð Þ, there is a coalition U such that g Uð Þ < m Uð Þ. As rm Uð Þ �g Uð Þ < m Uð Þ, by Proposition 1-(ii) coalition U is not m-null. Let m be a fixedmember of core mð Þ. Define the measure n as

n Eð Þ ¼ g E \ Uð Þ þ a1k E \ Uð Þ þ a2k E \ Ucð Þ þ m E \ U cð Þ ð18Þ

for all E, with a1 ¼ m Uð Þ � g Uð Þ½ �=k Uð Þ and a2 ¼ m Uð Þ � m Uð Þ½ �=k Ucð Þ. Ifk Ucð Þ ¼ 0, we set a2 ¼ 0: Note that a1; a2 are well-defined since k Uð Þ > 0, andthey are non-negative. Let us check that n 2 I mð Þ. Actually, n Xð Þ ¼ m Xð Þ ifk Ucð Þ > 0. This remains true when k U cð Þ ¼ 0 as well. For, if k U cð Þ ¼ 0, U c ism-null. Thus, m Ucð Þ ¼ 0 and n Xð Þ ¼ m Uð Þ ¼ m X n Ucð Þ ¼ m Xð Þ. To conclude,by (18),

n Eð Þ � g E \ Uð Þ þ m E \ U cð Þ � rm E \ Uð Þ þ rm E \ U cð Þ ¼ rm Eð Þand n is individually rational. Consequently, n 2 I mð Þ. It is now easy to checkthat n �U g. n

Lemma 1. Step 1. The metric space R kð Þ. Given a positive measure k , werecall that R kð Þ is obtained by identifying elements F � G in R, ifk F 4 Gð Þ ¼ 0. On these equivalence classes, one defines the metricq F ;Gð Þ ¼ k F 4 Gð Þ ¼

R1F � 1Gj jdk (see, e.g., III.7 in Dunford and Schwartz,

1954). The metric space R kð Þ turns out to complete, as it can be metricallyembedded into a closed subset of L1 X; kð Þ.

By Proposition 8, we can regard any continuous exact game m as a func-tion defined on the metric space R kð Þ, where the probability measure k is as inLemma 2-(iv). The map m : R kð Þ ! R will be well defined, provided

Stable cores of large games 205

k F 4 Gð Þ ¼ 0¼)m Fð Þ ¼ m Gð Þ. As k � m, and F ¼ G [ F n Gð Þ½ � n G n Fð Þ,m Fð Þ ¼ m Gð Þ holds. Moreover, as m is the lower envelope of the additivefunctionals l 2 core mð Þ � ca R; kð Þ, which are continuous in R kð Þ, the functionm : R kð Þ ! R turns out to be upper semicontinuous function (in fact, m iscontinuous since the family l 2 core mð Þ is equicontinuous on R kð Þ). It isclearly bounded from above. Note that RE kEð Þ is a complete metric spacesince it is a closed subset of R kð Þ.

Step 2. Ekeland variational principle. Fix a coalition E and assumen 2 ca RE; kEð Þ. The programming

a ¼ supF�E

m Fð Þ � n Fð Þ½ �

can be viewed as an extremum problem for the upper semicontinuous func-tion m� n over the complete metric space RE kEð Þ � R kð Þ. By the EkelandTheorem, for all e > 0 there is some A � E such that m Að Þ � n Að Þ � a� e, and

m Að Þ � n Að Þ � m Fð Þ � n Fð Þ � e1=2kE A4 Fð Þ for all F � E: ð19ÞStep 3. Since

kE A4 Fð Þ ¼ kE Að Þ þ kE F \ Acð Þ � kE F \ Að Þ; ð20Þwe can write (19) as

m Að Þ � n Að Þ þ e1=2kE Að Þ � m Fð Þ � n Fð Þ � e1=2kE F \Acð Þ þ e1=2kE F \Að Þ:ð21Þ

Set k ¼ m Að Þ � n Að Þ þ e1=2kE Að Þ, kAE Fð Þ ¼ kE F \ Að Þ, and kAc

E Fð Þ ¼kE F \ Acð Þ. In this way, (21) can be written as nþ e1=2kAc

E � e1=2kAE þ k � mE (it

becomes an equality at A). As m 2 L2, there is g 2 core mð Þ such thatnþ e1=2kAc

E � e1=2kAE þ k � gE � mE. Clearly, g Að Þ ¼ m Að Þ. Moreover,

gE � n� e1=2kAc

E þ e1=2kAE � g Að Þ � n Að Þ þ e1=2kEðAÞ:

This means that the measure gE � n� e1=2kAc

E þ e1=2kAE attains the maximum

value at A. It is easily seen that

gE � n� e1=2kAc

E þ e1=2kAE ¼ m 2 ca REð Þ; ð22Þ

where mA � 0 and mAc � 0. Hence, g Fð Þ � n Fð Þ � e1=2k Fð Þ if F � A. Thisproves our claim when n 2 ca RE; kEð Þ.

Step 4. Assume n 2 ca REð Þ is not necessarily absolutely continuous w.r.t.kE. The first observation is that the set A found in Step 2 is an equivalenceclass. Thus, all we have established remains true for any A1 ¼ A n N1ð Þ [ N2,where Ni are kE-null sets. If n 2 ca REð Þ, by the Lebesgue DecompositionTheorem, n ¼ na þ ns holds with na ? ns and na kE. Hence, there exists acoalition N such that kE Nð Þ ¼ 0 and nsj j E n Nð Þ ¼ 0: By the JordanDecomposition Theorem we also have ns ¼ ns

þ � ns� and a decomposition

N ¼ Nþ [ N� such that ns is positive on Nþ and negative on N�.By what has been already proved, we can apply the Lemma with n � na

for the programming a1 ¼ supF�E m Fð Þ � na Fð Þ½ �. Note that we have therelation

a ¼ supF�E

m Fð Þ � n Fð Þ½ � ¼ a1 � ns� Eð Þ:

206 M. Marinacci and L. Montrucchio

We find an element g 2 core mð Þ and a coalition A satisfying (i)-(iii) of theLemma. Replacing A by A1 ¼ A n Nþ [ N�, we have

m A1ð Þ � na A1ð Þ �a1 � e, g A1ð Þ ¼ m A1ð Þg Fð Þ �na Fð Þ � e1=2k Fð Þ for all F � A1:

Hence,

m A1ð Þ � n A1ð Þ ¼m A1ð Þ � na A1ð Þ � ns A1ð Þ � a1 � eþ ns� A1ð Þ ¼ a� e

g Fð Þ �na Fð Þ � e1=2k Fð Þ � n Fð Þ � e1=2k Fð Þ:The last step is true since

na Fð Þ ¼ n Fð Þ þ ns� Fð Þ � ns

þ Fð Þ ¼ n Fð Þ þ ns� Fð Þ � n Fð Þ

if F � A1, and so the Lemma is proved when n 2 ca REð Þ. Observe that allremains unchanged when m 2 L1, provided E ¼ X. n

Theorem 1. It is well-known that if V is any von Neumann-Morgensternstable set of the game m, then core mð Þ � V . It therefore suffices to prove thatany element n 2 I mð Þ n core mð Þ is dominated by some element of the core.

Step 1. Suppose first n 2 ca Rð Þ. As n=2core mð Þ, we must have n Uð Þ < m Uð Þfor some U 2 R. Choose e > 0 so small such that n Uð Þ þ 2e1=2k Uð Þþe < m Uð Þ:

Using Lemma 1 with E ¼ X, where n is replaced by nþ 2e1=2k and e ischosen as above, we deduce the existence of A 2 R and g 2 core mð Þ such thatconditions (i)-(iii) of Lemma 1 are true. Condition (ii) yields g Að Þ ¼ m Að Þ.Denoting

a ¼ supF2R

m Fð Þ � n Fð Þ � 2e1=2k Fð Þh i

;

condition (ii) implies

m Að Þ � n Að Þ � 2e1=2k Að Þ � a� e � m Uð Þ � n Uð Þ � 2e1=2k Uð Þ > 0:

Namely, m Að Þ > n Að Þ þ 2e1=2k Að Þ. Therefore, A is an essential coalition sincem Að Þ > n Að Þ � rm Að Þ.

Finally, condition (iii) is

g Fð Þ � n Fð Þ þ 2e1=2k Fð Þ � e1=2k Fð Þ ¼ n Fð Þ þ e1=2k Fð Þ

for all F � A. If F is an essential coalition, k Fð Þ > 0 and g Fð Þ > n Fð Þ. Weconclude that g �A n and, in turn, V \ ca Rð Þ ¼ core mð Þ.

Step 2. Assume now n 2 ba Rð Þ. The imputation n can be uniquelydecomposed into the sum nc þ np, where nc 2 ca Rð Þ and np is purely additive.Therefore, there exists an increasing sequence Fn 2 R such that npj j Fnð Þ ¼ 0for all n, and k X n Fnð Þ ! 0 as n!1 (see Yosida and Hewitt, 1952). Let Ube such that n Uð Þ < m Uð Þ as in Step 1. Consider the coalitions U \ Fn. Since mis continuous, there is some n such that n Eð Þ < m Eð Þ, with E ¼ U \ Fn. As inStep 1, we have n Eð Þ þ 2e1=2k Eð Þ < m Eð Þ for some e > 0: Apply Lemma 1where E is the same of the Lemma and n is replaced by nc þ 2e1=2k. Note that,by construction, n Eð Þ ¼ nc Eð Þ. We thus get the existence of A � E andg 2 core mð Þ such that g Að Þ ¼ m Að Þ > nc Að Þ þ 2e1=2k Að Þ ¼ n Að Þ þ 2e1=2k Að Þ,and g Fð Þ � n Fð Þ þ e1=2k Fð Þ for all F � A: The same argument of Step 1implies that g �A n and the proof is completed. n

Stable cores of large games 207

Proposition 3. By the Remark after Lemma 1, it is enough Step 1 of theprevious proof. n

Theorem 2. Let us first prove that bounded and continuous convex games liesin L1 by the characterization of Theorem 6. By Lemma 2-(ii), core mð Þ � Mr

m .Hence, m � me (see (9) for the definition of me). Consider the Choquet integralR

fdm for f 2 B1 Rð Þ:8 For any m 2 Mrm such that m Fð Þ þ k � m Fð Þ for all F , we

haveZ 1

0

m f � tð Þdt þ k �Z 1

0

m f � tð Þdt;

m; fh i þ k �Z 1

0

fdm ¼ me fð Þ;

for all f 2 B1 Rð Þ. By the definition of m, we have m � me. Hence, m ¼ me andm 2 L1 by Theorem 6. To prove that m 2 L2, it suffices to observe that anysubgame mE of m is convex. Hence, if nþ k � mE with n 2 ba REð Þ, thennþ k � l with l 2 core mEð Þ. On the other hand, we know that convex gamesare extendable (see Prop. 3.8 of Einy and Shitovitz, 1996). We conclude thatnþ k � lE for some l 2 core mð Þ. n

Theorem 3. This proof requires Theorem 6 and Lemma 3. Let m be an exactnon-atomic market game and m� be its na-extension. Let us show that m� � m.Suppose, per contra, that m fð Þ < m� fð Þ for some f 2 B1 Rð Þ. This impliesm; fh i þ k < m� fð Þ for some m 2 Mr

m � na Rð Þ and such that m Eð Þ þ k � m Eð Þfor all E (see Lemma 2): Since both m and m� are na -continuous, there existsan na-neighborhood U of f such that m;uh i þ k < m� uð Þ for all u 2 U . By thena-density of the indicator functions, there is some A such thatm Að Þ þ k < m Að Þ, which is a contradiction.

As m � me, we have

m� � m � me: ð23ÞNext we show that m� ¼ me. Let us prove that

m� af þ 1� að Þ1Xð Þ ¼ am� fð Þ þ 1� að Þm� 1Xð Þ. ð24ÞActually, for any E and a 2 0; 1½ � we can write:

m� a1E þ 1� að Þ1Xð Þ � me a1E þ 1� að Þ1Xð Þ ¼ ame 1Eð Þ þ 1� að Þme 1Xð Þ¼ am Eð Þ þ 1� að Þm Xð Þ ¼ am� 1Eð Þ þ 1� að Þm� 1Xð Þ� m� a1E þ 1� að Þ1Xð Þ;

where the last inequality is due to the concavity of m�. Hence, (24) holdswhenever f is a characteristic function.

Suppose that (24) is violated by some f 2 B1 Rð Þ. Thenm� af þ 1� að Þ1Xð Þ > am� fð Þ þ 1� að Þm� 1Xð Þ. ð25Þ

Since the na-topology is a linear topology on B Rð Þ, the map u! auþ1� að Þ1X is continuous. Hence, the function u! m� auþ 1� að Þ1Xð Þ is

na-continuous. In view of (25), there is a neighborhood U of f such that

8Note that m 2 bv Rð Þ and so the Choquet integral makes sense; see Marinacci and Montrucchio(2004a) for a detailed study of Choquet integrals in such a setting.

208 M. Marinacci and L. Montrucchio

m� auþ 1� að Þ1Xð Þ > am� uð Þ þ 1� að Þm� 1Xð Þfor u 2 U . By a density argument, this last inequality violates what has beenproved for the indicator functions, and so (24) holds.

By Lemma 3 (note that m� is necessarily bounded), we have m� � me and, by(23), m ¼ me. By Theorem 6 it follows that m 2 L1:

Observe now that m�E ¼ m� LE. For, it is easy to check that the functionalm� buð Þ is na -continuous on B1 REð Þ. We can conclude that

mE � m�E ¼ m� LE ¼ me LE:

Since necessarily mE � me LE, we get mE ¼ me LE, and the market gamem 2 L2 by Theorem 6. n

Proposition 4. Let m be a convex non-atomic market game, with na-extensionm�. We show that m is a charge. We first show that m� is ‘‘ultramodular ’’ (seeMarinacci and Montrucchio, 2003b), i.e.,

m� f þ hð Þ � m� fð Þ � m� gþ hð Þ � m� gð Þ ð26Þ

for all f ; g; h; gþ h 2 B1 Rð Þ and f � g. As the na-topology is a linear topol-ogy, the map

f ; g; hð Þ#m� gþ hð Þ � m� gð Þ þ m� fð Þ � m� f þ hð Þis na-continuous. Suppose (26) is violated for some triple f ; g; hð Þ. Then, thereexist neighborhoods U fð Þ, U gð Þ and U hð Þ such that

m� f1 þ h1ð Þ � m� f1ð Þ � m� g1 þ h1ð Þ � m� g1ð Þ þ e

for all f1 2 U fð Þ, g1 2 U gð Þ, h1 2 U hð Þ and for some e > 0. By the Dvo-retzky-Wald-Wolfowitz Theorem (see Aumann and Shapley, 1974), there aresets F , G, and H belonging to these neighborhoods and such that F � G andG \ H ¼ B. Therefore, m F [ Hð Þ � m Fð Þ > m G [ Hð Þ � m Gð Þ, which contra-dicts the convexity of m. Hence, m� satisfies (26).

Setting f ¼ 0 in (26), we obtain that m� is superadditive, i.e.,

m� fð Þ þ m� gð Þ � m� f þ gð Þ: ð27Þ

On the other hand, (26) can be written as m� f þ hð Þ � m� fð Þ �m� gð Þ � m� g� hð Þ, provided f � g� h. If we set h ¼ g� fð Þ=2, we getm� 2�1 f þ gð Þ� �

� 2�1m� fð Þ þ 2�1m� gð Þ, which is valid for all f � g. Alongwith (27), this implies

m� f þ gð Þ ¼ m� fð Þ þ m� gð Þ for all f � g: ð28ÞNow, set f ¼ 1=2ð Þ1Ec and g ¼ 1E þ 1=2ð Þ1Ec , where E is any coalition.Clearly, 0 � f � g � 1: Plugging these f and g into (28), after some tediousalgebra we get m Eð Þ þ m Ecð Þ ¼ m Xð Þ for each E. Hence, m Eð Þ ¼ m Eð Þ for all Eand all m 2 core mð Þ. Consequently, m ¼ m, as desired. n

Theorem 4. We know that core mð Þ � MB mð Þ. Hence core mð Þ � MB mð Þ \ ca Rð Þ.To prove that core mð Þ ¼ MB mð Þ \ ca Rð Þ, assume per contra that there isn 2 MB mð Þ \ ca Rð Þ such that n=2core mð Þ. Then, n Uð Þ < m Uð Þ for some U . ByLemma 1, with E ¼ X and e ¼ 0, there is a pair A; gð Þ such that g 2 core mð Þ,g Að Þ ¼ m Að Þ, g Að Þ > n Að Þ and g Cð Þ � n Cð Þ for all C � A. Hence, A; gð Þ is an

Stable cores of large games 209

objection to n in the game m . We now show that A; gð Þ is a justified objectionand this will contradict the fact that n 2 MB mð Þ.

Observe that (22) of Lemma 1 holds for the pair A; gð Þ; i.e., g� n ¼ m,with mA � 0 and mAc � 0. This is actually true when n 2 ca R; kð Þ. Ifn=2ca R; kð Þ, then g� n ¼ g� na � ns ¼ m� ns ¼ m0 by the argument used inLemma 1. It is readily seen that m0A1

� 0 and m0Ac1� 0.

If C is any coalition, we have

m Cð Þ � g Cð Þ ¼ g C \ Að Þ þ g C n Að Þ¼ g C \ Að Þ þ n C n Að Þ þ mAc C n Að Þ� g C \ Að Þ þ n C n Að Þ.

There is no counter objection f;Cð Þ to A; gð Þ. For, it must satisfy f Cð Þ � m Cð Þ.That is, f Cð Þ � g C \ Að Þ þ n C n Að Þ which violates the fact that for a counterobjection it must hold f Cð Þ > g C \ Að Þ þ n C n Að Þ.

Though in the proof we have assumed that the sup is attained for anyn 2 ca Rð Þ, nothing changes if the sup is attained only for the elementsn 2 ca R; kð Þ: Actually, if A is the coalition which maximizes the programmingwith na, then A1 ¼ A n Nþ [ N� satisfies all the conditions used in the proof. n

Corollary 2. (i) Assume m is defined over N; 2N� �

. In this case the sup in (5) isattained by the Weierstrass Theorem. For, any coalition E � N can be nat-urally identified with its indicator function 1E 2 0; 1f gN, where 0; 1f gN is aCantor compact space. Viewed as a mapping m : 0; 1f gN! R, the exact game mis upper semicontinuous since it is the envelope of measures in the core, whichare continuous (for games with a countable set of players, this framework hasbeen adopted by Einy et al., 1998).

(ii) It is known from Einy et al. (1998).(iii) In view of Theorem 5, m ¼ gC mð Þ where gC is a support function and

m ¼ m1; . . . ;mnð Þ with mi 2 naþ Rð Þ. Set m� ¼Pn

i¼1 mi. Clearly, mi m�.Moreover, if N is m�-null, then N is m-null. Consequently,ca R; kð Þ � ca R;m�ð Þ. Therefore, it suffices to consider the problem:

supF2R

m Fð Þ � n Fð Þ½ � ¼ supF2R

gC m Fð Þð Þ � n Fð Þ½ �;

with n 2 ca R;m�ð Þ. Setting U ¼ u 2 L1 m�ð Þ : 0 � u � 1, m�-a:e:f g, weenlarge the domain by considering the problem:

supu2U

gC m uð Þð Þ �Z

Xudn

; ð29Þ

which is well-defined. Both functions are r L1; L1ð Þ-continuous. Actually,

gC m uð Þð Þ ¼ gC

Z

Xu

dm1

dm�dm�; . . . ;u

dmn

dm�dm�

� �

:

As gC is continuous and each dmi=dm� 2 L1 m�ð Þ, the function u! gC m uð Þð Þis r L1; L1ð Þ-continuous. By the Alaoglu Theorem, the sup in (29) is attainedat some u 2 U . Let us show that the function achieves its maximum at some1A. Consider the vector measure m1;m2; . . . ;mn; n

þ; n�� �

. As n 2 ca R;m�ð Þ is anon-atomic measure, by the Lyapunov Theorem there is some A � X such

210 M. Marinacci and L. Montrucchio

that mi Að Þ ¼R

udmi and nþ Að Þ ¼R

udnþ, n� Að Þ ¼R

udn�: Thus,gC m uð Þð Þ �

RX udn ¼ m Að Þ � n Að Þ and this completes the proof. n

Theorem 5. (i)) (ii). Assume m is exact and core mð Þ � span m1; . . . ;mkf g withmi 2 na Rð Þ. Since some mi may be non-positive, we can considercore mð Þ � span mþ1 ; . . . ;mþk ;m

�1 ; . . . ;m�k

�. By normalizing, w.l.o.g. we can

then assume core mð Þ � span P1; . . . ; Pnf g, with each Pi 2 na1 Rð Þ and Pi linearlyindependent. Define R : span P1; . . . ; Pnf g ! Rn by R c Pð Þ ¼ c, whereP ¼ P1; . . . ; Pnð Þ and c 2 Rn. The map R is clearly a linear isomorphism. Ascore mð Þ is weak* compact and convex, its image R core mð Þð Þ ¼ C � Rn is acompact and convex subset of Rn. Hence,

m Eð Þ ¼ minl2core mð Þ

l Eð Þ ¼ minc2C

c P Eð Þ ¼ gCðP Eð ÞÞ

and m turns out to be an exact linear production game.(ii) ¼) (i). Assume that the exact game is of the type m ¼ gC mð Þ, where g is

given by g xð Þ ¼ minc2C c x for some compact and convex C subset of Rn. By(7), core mð Þ � span m1; :::;mnf g, and this is enough to show the desiredimplication.

(ii) () (iii) Observe that the exact cover me of a generalized productiongame is

me Eð Þ ¼ maxc2C Xð Þ

c m Eð Þ:

Therefore, m � me if and only if C Xð Þ \ C Eð Þ is non-empty for all E. n

Proposition 5. Let m be an exact game whose core is a polytope. Then, thereexists a finite subset C ¼ m1; . . . ;mnf g � core mð Þ consisting of the extremalpoints of core mð Þ such that co Cð Þ ¼ core mð Þ (see Lemma 5.123 of Aliprantisand Border, 1999). The glove market game em Eð Þ ¼ mini¼1;...;n mi Eð Þ is exact asmi Xð Þ ¼ m Xð Þ. By (7), its core is given by co Cð Þ ¼ core mð Þ. The two exactgames m and em share the same core and therefore they coincide. Hence m is aglove market game. Conversely, if m is an exact glove market, (7) implies thatcore mð Þ is a polytope. n

Proposition 6. Let @m Eð Þ 6¼ B. It is easy to describe the superdifferentials of mk as

@mk Eð Þ ¼ @m Eð Þ \ p : p Eð Þ � m Eð Þ � kf g ð30Þ

Hence, @mk Eð Þ 6¼ B for all k � 0 .Step 1. Given a coalition E 6¼ B, we want to prove that if

k > k ¼ maxp2ext@m Eð Þ

m Eð Þ � p Eð Þ � 0, ð31Þ

then condition (17) is fulfilled by all the elements of ext@mk Eð Þ.Fix any k > k. Clearly, any p 2 ext@m Eð Þ does not lie in @mk Eð Þ. From (31)

we actually have p Eð Þ > m Eð Þ � k, which contradicts (30). In turn, this implies

co ext@m Eð Þð Þ \ @mk Eð Þ ¼ B: ð32ÞAssume that there exists some p 2 ext@mk Eð Þ such that p Eð Þ < m Eð Þ � k.Hence, p � tm=2@mk Eð Þ for all m 2 KEc with m 6¼ 0 and t > 0 small enough.Since p Eð Þ < m Eð Þ � k, we have p Eð Þ � tm Eð Þ < m Eð Þ � k for t small enough.

Stable cores of large games 211

As p � tm=2@mk Eð Þ, some other inequality is not fulfilled. Hence, p � tm=2@m Eð Þ.This implies that p 2 co ext@m Eð Þð Þ. But this contradicts (32).

Step 2. Thanks to Step 1, by taking k0 as the maximum of all k over allcoalition A 6¼ B, we have that mk satisfies (17) for k > k0. As mk 2 L0, byProposition 11 it follows that mk 2 L1.

Step 3. If @m Eð Þ 6¼ B for all E, we have @mk Eð Þ 6¼ B for all E as well. Ifmk 2 L1, coreE mkð Þ 6¼ B, which implies that mk is exact. This proves (i). Now,suppose that m is not convex. As m is balanced, there must exist A and B withA \ B 6¼ B and m A [ Bð Þ þ m A \ Bð Þ < m Að Þ þ m Bð Þ. Clearly, we have thenmk A [ Bð Þ þ mk A \ Bð Þ < mk Að Þ þ mk Bð Þ, and this proves (ii). n

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