In pure exchange economies, a poor attention has been given to how the individual consumption possibilities of the members of a coalition should be represented. It seems economically reasonable that our knowledge and our possibility to make decisions depend on the coalition we belong to. We define a coalitional exchange economy by considering a pure exchange economy in which the individual consumption sets of consumers within a coalition depend on the membership of the coalition. Our definition includes as a particular case the classical definition of pure exchange economy. We adapt the core concept to a coalitional exchange economy, and we show the non-emptiness of the core. Finally, we discuss more general setting where individual preferences are also depending on the coalitions.Cooperative game, core, exchange economy, consumption possibility.

Elena L. Del Mercato

English

Research Papers in Economics

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http://mse.univ-paris1.fr/Publicat.htm
ISSN : 1624-0340
Centre d’Economie de la Sorbonne
UMR 8174
       
The Core of a Coalitional Exchange Economy
Elena L. del MERCATO
2006.47
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6The Core of a Coalitional Exchange Economy∗
Elena L. del Mercato†
CSEF, Universit` a degli Studi di Salerno
May 4, 2006
Abstract
In pure exchange economies, a poor attention has been given to how the individual con-
sumption possibilities of the members of a coalition should be represented. It seems eco-
nomically reasonable that our knowledge and our possibility to make decisions depend on
the coalition we belong to. We deﬁne a coalitional exchange economy by considering a pure
exchange economy in which the individual consumption sets of consumers within a coalition
depend on the membership of the coalition. Our deﬁnition includes as a particular case the
classical deﬁnition of pure exchange economy. We adapt the core concept to a coalitional ex-
change economy, and we show the non-emptiness of the core. Finally, we discuss more general
setting where individual preferences are also depending on the coalitions.
JEL classiﬁcation: C71, D50.
Keywords: cooperative game, core, exchange economy, consumption possibility.
R´ esum´ e
Dans les ´ economies d
0´ echange, peu d
0attention a ´ et´ e donn´ ee ` a la mani` ere de repr´ esenter les
possibilit´ es individuelles de consommation des membres d
0une coalition. Il semble raisonnable
que notre connaissance et notre possibilit´ e de d´ ecision d´ ependent de la coalition ` a laquelle
on appartient. On d´ eﬁnit une ´ economie d
0´ echange coalitionnelle en consid´ erant une ´ economie
d
0´ echange dans laquelle les ensembles individuels de consommation des membres d
0une coali-
tion d´ ependent de la coalition. Notre d´ eﬁnition inclut comme cas particulier la d´ eﬁnition d
0une
´ economie d
0´ echange. Nous adaptons le concept standard de cœur ` a une ´ economie d
0´ echange
coalitionnelle et nous montrons la non vacuit´ e du cœur. Finalement, nous discutons un cadre
plus g´ en´ eral o` u les pr´ ef´ erences individuelles d´ ependent aussi des coalitions.
Journal of Economic Literature Classiﬁcation Numbers: C71, D50.
Mots-cl´ es: jeu coop´ eratif, cœur, ´ economie d
0´ echange, possibilit´ e individuelle de consom-
mation.
∗This note has been done during a visit to Centre d’Economie de la Sorbonne, Universit´ e Paris 1. I wish to
thank J.-M. Bonnisseau and V. Iehl´ e for stimulating discussions.
†Elena L. del Mercato, CSEF, Dipartimento di Scienze Economiche e Statistiche, Universit` a degli Studi di
Salerno, Via Ponte don Melillo I, 84084 Fisciano (SA), Italy. Fax: –39–089–962049. Phone: –39–089–964438.
E-mail: edmercat@unisa.it.
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61 Introduction
The central concept in cooperative game theory is the core, which is the set of feasible payoﬀs
for the grand coalition which no coalition can improve upon. Scarf (1967) provided suﬃcient
conditions for the non-emptiness of the core in a general cooperative game. Then, the author used
the result to show that the core of a pure exchange economy with convex preferences is non-empty.
While Debreu (1959) considered for each consumer i an individual consumption set Xi deﬁned as
the set of all possible consumptions for consumer i, the analysis of Scarf (1967) in a pure exchange
economy simply required to specify arbitrary redistributions of the resources of any coalition of
consumers, “...It is frequently suﬃcient to summarize the detailed strategic possibilities open to a
coalition by the set of possible utility vectors that can be achieved by the coalition. For example,
in a pure exchange economy each coalition will have associated with it the collection of all utility
vectors that can be obtained by arbitrary redistributions of the resources of that coalition.”
To the best of our knowledge, in Scarf (1967) as well as in the related developments, a poor
attention has been given to how the individual consumption possibilities of consumers within
a coalition should be represented in pure exchange economies. However, it seems economically
reasonable that our knowledge and our possibility to make decisions depend on the coalition we
belong to. Importantly, one might willing to distinguish the individual consumption possibilities
of stand alone agents from the individual consumption possibilities of the members of a coalition.
On production side, coalitional behaviors have been modeled through production sets depending
on the membership of the coalition, namely coalitional production economies. See in that line
Boehm (1974), Champsaur (1974), Border (1984), and Florenzano (1987).
In the spirit of the literature concerning coalitional production economies, we deﬁne a coalitional
exchange economy by considering a pure exchange economy in which the individual consumption
sets of consumers within a coalition depend on the membership of the coalition. Then, we adapt
the standard deﬁnitions of feasibility and core to a coalitional exchange economy.
The main features of this paper are the deﬁnition of a coalitional exchange economy and the
non-emptiness result of the core in such an economy. Our deﬁnition includes as a particular case
the classical deﬁnition of pure exchange economy, and the non-emptiness result covers the one
obtained by Scarf (1967) for the case of a pure exchange economy. Our result relies on the simple
idea that our knowledge and our possibility to make decisions increase as the coalition we belong
to gets larger. This idea reﬂects, in terms of possibilities, a “positive eﬀect” for a consumer that
belongs to a larger coalition.
The paper is organized as follows. Section 2 is devoted to our basic model and deﬁnitions. The
non-emptiness result, Theorem 5 whose proof is based on Scarf’s theorem on balanced games, is
given in Section 3. A remark concludes the paper.
2 The model
A coalitional exchange economy is
E :=
 
RC,N,{Xi(S)}i∈N,S∈Ni,{ui,ωi}i∈N

C is the ﬁnite number of physical commodities and RC is the commodity space. N is the ﬁnite
set of consumers labeled by subscript i. N is the family of non-empty subsets of N, and each
S ∈ N is said coalition. For each i ∈ N, Ni := {S ∈ N : i ∈ S} is the family of coalitions in N
which contain consumer i. For each i ∈ N and for each S ∈ Ni, the individual consumption set
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6Xi(S) ⊆ RC is deﬁned as the set of all possible consumptions for consumer i when S forms, where
Xi(S) ⊆ Xi(N) (1)
For each i ∈ N, consumer i’s preferences are represented by a utility function ui from Xi(N) to R,
and ui(xi) ∈ R is the consumer i’s utility associated with the consumption xi ∈ Xi(N). For each
i ∈ N, ωi ∈ RC is the initial endowment of commodities owned by consumer i.
Remark 1 In a pure exchange economy, for each i ∈ N, Xi(S) = Xi for every S ∈ Ni.
A consumer i is said to be alone when coalition {i} forms. Observe that in the above deﬁnition,
the individual consumption set Xi ({i}) of a stand alone consumer i possibly diﬀers from the
individual consumption set Xi(S) of the same consumer i within a coalition S ⊃ {i}.
As stressed in Introduction, it seems economically reasonable that for each i ∈ N and for each
S and T in Ni, S ⊆ T implies Xi(S) ⊆ Xi(T). This condition reﬂects, in terms of possibilities,
a “positive eﬀect” for a consumer that belongs to a larger coalition. Furthermore, note that the
latter condition is stronger than condition in (1).
Deﬁnition 2 Let S ∈ N be a coalition.
1. xS := (xi)i∈S is a S-feasible bundle of E if xi ∈ Xi(S) for each i ∈ S, and
X
i∈S
xi ≤
X
i∈S
ωi.
Let FS(E) denote the set of S-feasible bundles of E, i.e.
FS(E) :=
(
xS ∈
Y
i∈S
Xi(S) :
X
i∈S
xi ≤
X
i∈S
ωi
)
(2)
2. The coalition S blocks a N-feasible bundle x ∈ FN(E) if there exists xS = (xi)i∈S ∈ FS(E),
such that ui(xi) > ui(xi) for each i ∈ S.
Deﬁnition 3 The core C (E) of the coalitional exchange economy E is deﬁned as the set of all
N-feasible bundles of E that no coalition can block.
3 Main result
Theorem 5 provides a core non-emptiness result for the coalitional exchange economy deﬁned in
Section 2. The proof of Theorem 5 is based on a weak version of the celebrated Theorem 1 of Scarf
(1967) for balanced games. Following is the statement of that weak version.
Theorem 4 (Scarf’s Theorem) Let Γ = (N,{VS}S∈N) be a balanced game. Γ has non-empty core
if the following assumptions are satisﬁed.1
1. VN 6= ∅, and V{i} 6= ∅ for each i ∈ N;
2. for each S ∈ N, VS 6= ES and VS is a closed set;
1Note that we do not require VS to be non-empty for S 6= N and S 6= {i}, nor {v = (vi)i∈S ∈ VS : vi ≥
maxV{i},∀i ∈ S} to be non-empty for each S ∈ N. Indeed, observe that the core would be unchanged if we replace
the set VS by VS := {v = (vi)i∈S ∈ ES : vi ≤ maxV{i},∀i ∈ S}.
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63. for each S ∈ N, if v ∈ VS and w ∈ ES with w ≤ v, then w ∈ VS;
4. for each S ∈ N, the set {v = (vi)i∈S ∈ VS : vi ≥ maxV{i},∀i ∈ S} is bounded from above.
It has been shown by Scarf (1967) that a pure exchange economy has a non-empty core. In
Theorem 5, we extend his result in the more general setting of coalitional exchange economies.
Theorem 5 Let E be a coalitional exchange economy. The core C (E) of E is non-empty if for
each i ∈ N, the following assumptions are satisﬁed.
1. Xi(N) is a convex and bounded from below subset of RC;
2. for each S ∈ Ni, Xi(S) is a closed subset of RC;
3. ωi ∈ Xi({i});
4. ui is a continuous and quasi-concave function.
Proof. Deﬁne the cooperative game Γ = (N,{VS}S∈N), where for each S ∈ N
VS :=

v = (vi)i∈S ∈ ES : ∃ xS ∈ FS(E) s. t. vi ≤ ui(xi),∀i ∈ S
	
We want to show that the game deﬁned above satisﬁes the assumptions of Theorem 4.
Claim 1. The game Γ is balanced.
Let B be a balanced family, and u ∈ RN be a vector such that uS ∈ VS for each S ∈ B. We
want to prove u ∈ VN. For each S ∈ B, there is xS := (xS
i )i∈S ∈
Y
i∈S
Xi(S) such that
X
i∈S
xS
i ≤
X
i∈S
ωi and ui ≤ ui(xS
i ) for each i ∈ S (3)
Since B is a balanced family, it is possible to ﬁnd non-negative weights λS for each coalition S ∈ B,
such that for each i ∈ N,
X
S∈Bi
λS = 1 where Bi := {S ∈ B : i ∈ S}. Now, for each i ∈ N deﬁne
xi :=
X
S∈Bi
λSxS
i . First, observe that x := (xi)i∈N ∈
Y
i∈N
Xi(N). Indeed, from condition in (1) and
the convexity of Xi(N) (see Assumption 1), for each i ∈ N, xi ∈
X
S∈Bi
λSXi(S) ⊆
X
S∈Bi
λSXi(N) ⊆
Xi(N). Then, from (3) we get
X
i∈N
xi ≤
X
i∈N
ωi and for each i ∈ N, ui(xi) = ui(
X
S∈Bi
λSxS
i ) ≥
X
S∈Bi
λSui(xS
i ) ≥
X
S∈Bi
λSui = ui, since the function ui is quasi-concave (see Assumption 4). Then,
u ∈ VN.
Claim 2. VN 6= ∅, and V{i} 6= ∅ for each i ∈ N.
From Assumption 3, we get ui(ωi) ∈ V{i} for each i ∈ N. From Assumption 3 and condition in
(1), we get ωi ∈ Xi({i}) ⊆ Xi(N) for each i ∈ N. Then, trivially (ui(ωi))i∈N ∈ VN.
Claim 3. For each S ∈ N, VS is bounded from above and closed.
Observe that the set FS(E) deﬁned in (2) is compact. Indeed, from Assumption 2 we deduce
that it is closed. From the condition in (1) and the boundedness from below of Xi(N) (see
Assumption 1), we have that there exists b ∈ RC such that for each i ∈ S, b ≤ xi for every
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6xi ∈ Xi(S). Let xS = (xi)i∈S ∈ FS(E), for each k ∈ S xk ≤
X
i∈S
ωi − (card(S) − 1)b. Then,
FS(E) is bounded. Therefore, from the continuity of function ui (see Assumption 4) follows the
boundedness from above and the closedness of VS.
The assumptions of Theorem 4 are satisﬁed. Then, Γ has non-empty core. Let u ∈ VN an
element in the core of Γ. By the deﬁnition of VN, there exists x = (xi)i∈N ∈
Y
i∈N
Xi(N) such that
X
i∈N
xi ≤
X
i∈N
ωi and ui ≤ ui(xi), for each i ∈ N. It easy to see that x ∈ C (E).
A remark concludes the paper: we analyze a more general setting where utility functions are
also depending on the coalitions.
We generalize the deﬁnition of coalitional exchange economy as follows,
E :=
 
RC,N,{Xi(S),ui(·,S)}i∈N,S∈Ni,{ωi}i∈N

which diﬀers from the deﬁnition given at the beginning of Section 2 since for each i ∈ N and for
each S ∈ Ni, the utility function of consumer i when S forms is ui(·,S) from Xi(S) to R. Then,
feasibility conditions do not change. A natural adaptation of Deﬁnition 2 follows.
Deﬁnition 6 Let S ∈ N be a coalition. The coalition S blocks a N-feasible bundle x ∈ FN(E) if
there exists xS = (xi)i∈S ∈ FS(E) such that ui(xi,S) > ui(xi,N) for each i ∈ S.
Then, the core non-emptiness result holds replacing Assumption 4 in Theorem 5 with the following
assumptions. For a proof, it suﬃces to use the natural adaptation of the arguments used in the
proof of Theorem 5. According to condition in (1), Assumption 7.2 seems economically reasonable.
Assumption 7 For each i ∈ N,
1. ui(·,N) is a quasi-concave function, and ui(·,S) is a continuous function for each S ∈ Ni;
2. for each S ∈ Ni, ui(xi,N) ≥ ui(xi,S) for each xi ∈ Xi(S).
Condition in (1) given in Section 2 is the crucial point to demonstrate that a coalitional exchange
economy is representable as a balanced game (see Claim 1, Theorem 5). In that more general
setting, an alternative consists of the following notion of balanced coalitional economy which is the
natural counterpart of the balancedness notion given by Boehm (1974) on the production side.
Then, one recovers the core non-emptiness result by assuming in addition that for each i ∈ N and
for each S ∈ Ni, Xi(S) is bounded from below.
Deﬁnition 8 A coalitional exchange economy E is said balanced if and only if for every balanced
family B and associated weights (λS)S∈B, for each i ∈ N
X
S∈Bi
λSXi(S) ⊆ Xi(N)
Note that the above condition of balancedness is weaker than condition in (1) if Xi(N) is convex
for each i ∈ N.
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6References
Boehm, V., 1974, The Core of an Economy with Production, Review of Economic Studies 41,
429–436.
Border, K.C., 1984, A Core Existence Theorem for Games without Ordered Preferences, Econo-
metrica 52, 1537–1542.
Champsaur, P.P., 1974, Note sur le Noyau d’une Economie avec Production, Econometrica 42,
933–946.
Debreu, G., 1959, Theory of Value (Yale University Press).
Florenzano, M., 1987, On the Non-Emptiness of the Core of a Coalitional Production Economy
without Ordered Preferences, Journal of Mathematical Analysis and Applications 141, 484–490.
Scarf, H., 1967, The Core of an n-person Game, Econometrica 35, 50–69.
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6

The Core of a coalitional exchange economy

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The Core of a coalitional exchange economy

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