Let N be a set of players, C the set of permissible coalitions and G an N-playerstrategic game. A profile is a coalitional-equilibrium if no coalition permissible coalition in C has a unilateral deviation that profits to all its members. Nash-equilibria consider only single player coalitions and Aumann strong-equilibria permit all coalitions to deviate. A new fixed point theorem allows to obtain a condition for the existence of coalitional equilibria that covers Glicksberg for the existence of Nash-equilibria and is related to Ichiishi's condition for the existence of Aumann strong-equilibria.Fixed point theorems, maximum of non-transitive preferences, Nash and strong equilibria, coalitional equilibria

Rida Laraki

English

Research Papers in Economics

 
 
COALITIONAL EQUILIBRIA OF STRATEGIC GAMES 
 
 
 
 
Rida LARAKI 
 
 
 
October 2009 
 
 
 
Cahier n° 2009-42 
 
 
 
 
                              ECOLE POLYTECHNIQUE                       
CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE 
 
 
DEPARTEMENT D'ECONOMIE 
Route de Saclay 
91128 PALAISEAU CEDEX 
(33) 1 69333033 
http://www.enseignement.polytechnique.fr/economie/  
mailto:chantal.poujouly@polytechnique.edu 
 
 
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9Coalitional Equilibria of Strategic Games ∗
Rida Laraki
Economics Department, ´ Ecole Polytechnique, C.N.R.S., October 1, 2009.
Abstract
Let N be a set of players, C ⊂ 2N a set of permissible coalitions and G an N-
player strategic game. A proﬁle is a coalitional-equilibrium if no coalition in C has
a unilateral deviation that proﬁts all its members. Nash-equilibria [10] correspond
to C = {{i},i ∈ N}, Aumann-equilibria [2] (usually called strong-equilibria) to
C= 2N. A new ﬁxed point theorem allows to obtain a condition for the existence of
coalitional equilibria that covers Glicksberg [7] for the existence of Nash-equilibria
and is related to Ichiishi’s [9] condition for the existence of Aumann-equilibria.
JEL classiﬁcation: C62, C72.
Keywords: Fixed point theorems, maximum of non-transitive preferences, Nash and
strong equilibria, coalitional equilibria.
Introduction
Nash [10] and Aumann [2] equilibria (also known as strong equilibria) are the two main
solution concepts in non-cooperative game theory. The ﬁrst asks the stability of the
strategy proﬁle against all single player unilateral deviations while the second asks the
stability against all coalitions unilateral deviations. The technics that are used to show
existence are of diﬀerent types. To establish the existence of Nash equilibria, Glicksberg
[7] needs quasi-concavity of the payoﬀ functions and uses a standard ﬁxed point theorem
(i.e. Brouwer/Kakutani). For the existence of Aumann equilibria, Ichiishi [9] assumes the
game to be balanced and uses the KKMS-lemma (for Knaster-Kuratowski-Mazukiewicz-
Shapley, lemma 22.4 in [4]) established by Shapley.
The paper has two main contributions. First, it provides a new ﬁxed point theorem.
Second, it deﬁnes the concept of coalitional equilibrium and provides a suﬃcient condition
for its existence. Coalitional equilibria lies between Nash and Aumann. An exogenous
coalitional structure deﬁnes which coalitions are admissible to jointly deviate. A strategy
proﬁle is a coalitional-equilibrium if it is stable against all admissible coalitions unilateral
∗I would like to thank Michel Balinski, Philippe Bich, Fran¸ coise Forges, Eilon Solan and Tristan
Tomala for their very useful comments.
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9deviations. When only single-player coalitions are permissible it is Nash-equilibrium and
when all coalitions are admissible it is Aumann-equilibrium. The motivation is straight-
forward. In many applications (voting, council of Europe, or market competition) some
coalitions are not natural and so cannot be expected to coordinate (extreme-leftists and
-rightists, western and eastern countries, or ﬁrms of diﬀerent areas).
Ray and Vohra [12] consider a solution concept that uses coalitional equilibria but only
for partitions. First, for each partition of the players, they associate the set of coalitional
equilibria. Second, given a coalitional-equilibrium for a given partition, one can deﬁne
internal stability with respect to stable coalitional equilibria of all ﬁner partitions. The set
of stable equilibria for a given partition is constructed recursively. Starting from the ﬁnest
partition structure (that contains only singleton), one can construct stable coalitional
equilibria for larger partition structures, and so on. Clearly, a similar construction could
be done if the partition constraint is relaxed. This naturally leads to what may be called
stable coalitional equilibria.
In the next section, a new ﬁxed point theorem is established. It is closely related to
Sonnenschein’s [13] theorem. It provides conditions for a non-transitive preference to have
a maximal element. Its proof is based on KKM-lemma (lemma 17.43 in [1]). It is then
applied to establish a new version of the well known Gale and Mas-Colell [5] theorem. In
the last section, the ﬁxed point theorem is applied to obtain a quasi-concavity condition
for the existence of coalitional equilibria as well as for the existence of Berge equilibria
[3] (deﬁned bellow). This induces a uniﬁed existence result for Nash, Aumann and Berge
equilibria. The theorem may be useful in other contexts such as competitive equilibria
as in Sonnenschein [13] or social coalitional equilibria as in Ichiishi [8]. The latter is an
equilibrium concept in a general game model that mixes cooperation and conﬂict and
links Aumann equilibria to the Core.
A new ﬁxed point theorem
Let S denote a compact convex subset of a Hausdorﬀ and locally convex topological vector
space (TVS). The interior of X ⊂ S relatively to S is denoted intX, its convex envelope
coX and the closed convex hull coX. Recall Ky Fan’s lemma, which is nothing but a
generalization of the well known KKM-lemma to correspondences (see [1] theorem 17.46).
Lemma 1 (Ky Fan) If the correspondence F from S to S has closed values and if for
any ﬁnite family {x1,...,xk}, co{x1,...,xk} ⊂ ∪i=1,...,kF(xi) then ∩x∈SF(x)  = ∅.
Let A be a correspondence on S (i.e. from S to S), best viewed as a (not necessarily)
transitive preference where A(x) ⊂ S is the set of points in S better than x. The set of
maximal elements of a A is E = {x ∈ S such that A(x) = ∅}.
A famous implication of the Ky Fan lemma is the following important result of Son-
nenschein [13]. For its proof and the numerous applications, see [1] and [4].
Theorem 2 (Sonnenschein) Let A be a correspondence on S. If (i) for all x ∈ S,
x / ∈ coA(x) and (ii) for any y ∈ A−1(x) there exists x′ ∈ S (possibly x′ = x) such that
y ∈ intA−1(x′), then the set of maximal elements of A is compact and non-empty.
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9The second assumption of Sonnenschein is very strong (in applications, it asks the
lower-sections A−1(x) to be open for every x). The ﬁxed point theorem that follows
reinforces slightly assumption (i) and relaxes suﬃciently (ii) to allow A to be only lower-
semi-continuous (i.e. A−1(W) is asked to be open for any W that is open).
Theorem 3 Let A be a correspondence on S. If (i’) for all x ∈ S, x / ∈ coA(x) and (ii’)
for any y ∈ A−1(x) and any convex open neighborhood W of zero, there exists x′ ∈ S
(possibly x′ = x) such that y ∈ intA−1(x′ + W), then the set E of maximal elements of
A is compact and non-empty.
We ﬁrst establish a useful lemma (to be compared with lemma 17.47 in [1]).
Lemma 4 If there is a convex open neighborhood W of zero such that for each x ∈ S,
x / ∈ coA(x) − W, then ∩x∈SF(x)  = ∅ where F(x) is the closure of the complement of
A−1(x + W): F(x) = S/A−1(x + W).
Proof. From Ky Fan’s lemma, it is suﬃcient to show that for any ﬁnite family
{x1,...,xk} and any y ∈ co{x1,...,xk}, y ∈ ∪i=1,...,kF(xi). Suppose not. This implies that
for any i, y / ∈ F(xi) implying that xi ∈ A(y) − W. Since W is convex, conclude that
y ∈ coA(y) − W, a contradiction.
We now prove the ﬁxed point theorem, inspired by the proof of Sonnenshein’s theorem
17.48 in [1].
Proof. If for any convex open neighborhood W of zero, there is an x in S such that
x + W ∩ coA(x)  = ∅ then, by compactness, Hausdorﬀ and local convexity assumptions,
there will exist x ∈ S such that x ∈ coA(x), contradicting (i’). Thus, there is an open
and convex neighborhood W of zero such that, for all x ∈ S, x + W ∩ coA(x) = ∅.
Note that E = ∩x∈SS/A−1(x + W). By (ii’), E = ∩x′∈SS/intA−1(x′ + W), so that it
is compact (as the intersection of a family of compact sets). Thus S/A−1(x + W) ⊂
S/intA−1(x + W). Hence, ∩x∈SS/A−1(x + W) ⊂ ∩x∈SS/intA−1(x + W). By (i’) and the
last lemma, ∩x∈SS/A−1(x)  = ∅.
This provides a new short proof of Brouwer-Schauder-Tychonoﬀ’s theorem [1]. Actu-
ally, if f is continuous from S to S then A(x) = {f(x)} satisﬁes (ii’) and is never empty,
thus there exists x such that x ∈ coA(x) = {f(x)}.
In [1], it is shown that lemma 1 and theorem 2 hold without the local convexity
assumption. Schauder conjectured in 1935 that every convex compact set S of a Hausdorﬀ
TVS has the ﬁxed point property (i.e. any continuous function from S to S has a ﬁxed
point). His conjecture has been proved only recently by Robert Cauty [6]. Using Cauty’s
result, one could drop local convexity in theorem 3 as well (but in that case, the proof
will depend on a highly non trivial result).
To illustrate the power of the theorem, let us use it to establish a new version of
the very helpful Gale and Mas-Colell [5] theorem. Assume S =
Q
i∈N Si where each
Si is a compact-convex subset of a locally convex TVS (typically, the strategy set of
player i where N is the set of players, not necessarily ﬁnite.). By Tychonoﬀ [1], S is
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9convex, compact, Hausdorﬀ and locally convex for the product topology. As usual, let
S−i =
Q
j =i Sj be the set of proﬁles of players other than i.
Let {Ai}i∈N denotes a family of correspondences, where for each i in N, Ai is a
correspondence from S to Si. The set of maximal elements of {Ai}i∈N is {x such that
Ai(x) = ∅ for all i in N}.
Theorem 5 If for each i ∈ N and x ∈ S (1) xi / ∈ coAi(x) and (2) Ai is lower-semi-
continuous then the set of maximal elements of {Ai}i∈N is nonempty and compact.
Proof. Deﬁne the correspondence A on S as follows: A(x) :=
S
i∈N Ai(x) × {x−i}.
Clearly A is lower-semi-continuous. Also, if for some x ∈ S, x ∈ coA(x) then there is
i ∈ N such that xi ∈ coAi(x). This contradicts (1). Thus, A satisﬁes (i’) and (ii’).
Consequently, there is x ∈ S such that for all i ∈ N, Ai(x) is empty.
Gale and Mas-Colell [5] proved the result under the assumptions (1’) xi / ∈ coAi(x) for
every x and (2’) Ai has an open graph. The later assumption is very strong. Assumption
(2) is much weaker, but (1) is slightly stronger than (1’).
Application to coalitional equilibria
Let N be a set of players, not necessarily ﬁnite. Let G = (N,{Si}i∈N,{gi}i∈N) be a
strategic game. Assume that for each i in N, Si, the strategy set of player i, is a compact-
convex subset of a Hausdorﬀ and locally convex TVS and the payoﬀ function of player i,
gi : S → R, is continuous. This deﬁnes a compact-continuous strategic game. Let C ⊂ 2N
be the set of permissible coalitions. As usual, S =
Q
i∈N Si is the set of strategy proﬁles,
SC =
Q
j∈C Sj and N/C denotes the set of players outside C.
Deﬁnition 6 s is a coalitional-equilibrium of G if no admissible coalition in C has a
unilateral deviation that proﬁts all its members; That is, there is no C in C and no
tC ∈ SC such that for any i ∈ C, gi(tC,sN/C) > gi(s).
A particularly interesting class is C= ∪K
k=12Pk where {P1,...,PK} is some partition of
N. Restricting the model of Ichiishi [8], one could obtain a balanced-condition for the
existence of this type of coalitional equilibria. The two conditions are related, as it is
proven below. The interpretation is simple. Members in diﬀerent Pk’s cannot talk to
each others while members inside Pk could try to coordinate but cannot commit to never
deviate.
Deﬁnition 7 G is C-quasi-concave if for all s ∈ S, ǫ > 0 and any family of permissi-
ble coalitions (Ck)k∈K with corresponding strategies tCk ∈ SCk, if for all k and i ∈ Ck
gi(tCk,sN/Ck) ≥ gi(s) + ǫ, then s / ∈ co{(tCk,sN/Ck),k ∈ K}.
In ﬁnite dimensional strategy spaces and ﬁnitely many players, Caratheodory’s theo-
rem implies that co above could be replaced by co and that only ﬁnitely many deviating
coalitions are to be considered.
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9Theorem 8 If a compact-convex-continuous strategic game is C-quasi-concave, the set of
coalitional-equilibria is compact and non-empty.
Proof. Let AC(s) = {(tC,sN/C) such that for all i ∈ C, gi(tC,sN/C) > gi(s) + ǫ} and
let A =
S
C AC. From the continuity of the game, A is lower-semi-continuous. Suppose
s ∈ coA(s) for some s. Thus there exists a family of permissible coalitions (Ck)k∈K and
strategies (tCk ∈ SCk)k∈K such that s ∈ co{(tCk,sN/Ck),k ∈ K} and gi(tCk,sN/Ck) ≥
gi(s)+ǫ for all k and i ∈ Ck: a contradiction. Hence, A has a maximal element sǫ whose
accumulation points are coalitional-equilibria.
Other deﬁnitions of A would lead to other concepts. For example, if AC(s) = {tC ∈
SC : there is i ∈ C such that gi(tC,s−C) > gi(s)+ǫ}, the underlying concept requires that
an admissible coalition has a deviation if at least one of its members proﬁts (even if all
the other players inside the coalition lose: this is too demanding!). Hence, at equilibrium,
players outside an admissible coalition forces all the players inside the coalition to play
according to the equilibrium. The concept was deﬁned by Claude Berge [3] but only for
admissible coalitions of the form N/{i}, i ∈ N. A coalitional Berge-equilibrium exists if
G is compact-convex-continuous and for all s ∈ S, ǫ > 0 and any family of permissible
coalitions (Ck)k∈K with corresponding strategies tCk ∈ SCk, if for all k there is i ∈ Ck
such that gi(tCk,sN/Ck) ≥ gi(s) + ǫ, then s / ∈ co{(tCk,sN/Ck),k ∈ K}.
When only single player coalitions are permissible, the theorem is an improvement
of Nash-Glicksberg [7] theorem (since it considers inﬁnitely many players1). When all
coalitions are admissible, the theorem provides a quasi-concavity condition for the ex-
istence of strong-equilibria. In the later case, quasi-concavity could be related to the
balanced-condition in Ichiishi [9] and Border [4] as shown below. Recall that a ﬁnite
family of nonempty subsets B of N is balanced if for each B ∈ B, there are nonnegative
real numbers λB (balancing weights) such that for each i in N,
P
B:i∈B λB = 1.
Deﬁnition 9 G is balanced if for all α ∈ [0,1]N and any ﬁnite balanced family of coali-
tions {Ck} with weights {λk} and corresponding strategies {tCk} ∈ S, if gi(tCk) > αi for
all k and i ∈ Ck then gi(s) > αi for all players i ∈ N, where si :=
P
k:i∈Ck λkt
Ck
i .
A slightly stronger condition is:
Deﬁnition 10 G is strictly-balanced if for all α ∈ [0,1]N and any ﬁnite balanced family of
coalitions {Ck} with weights {λk} and corresponding strategies {tCk} ∈ S, if gi(tCk) ≥ αi
for all k and i ∈ Ck and gi(tCk) > αi for some k and i ∈ Ck then gi(s) > αi for some
player i ∈ N, where si :=
P
k:i∈Ck λkt
Ck
i .
Lemma 11 With ﬁnitely many players, ﬁnite dimensional strategy spaces and all coali-
tions permissible, strictly-balanced implies C-quasi-concave.
1Observe that Nash-Glicksberg’s theorem could be proved directly from our new version of Gale
& Mas-Colell theorem. Reny [11] proved Nash-Glicksberg without the Hausdorﬀ and local convexity
assumptions. We believe that a similar improvement is possible for theorem 8.
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9Proof. Take (Ck)k∈K to be a ﬁnite family of coalitions, let tCk ∈ SCk and s = P
k αk(tCk,sN/Ck). The family {{Ck} ∪ {N/Ck}}k∈K with weights {(αk,αk)}k∈K is bal-
anced. Deﬁne tCk = (tCk,sN/Ck) and tN/Ck = s. Then, si =
P
k:i∈Ck αkt
Ck
i +
P
k:i∈N/Ck αkt
N/Ck
i .
Suppose that for each k and i ∈ Ck, gi(tCk) > gi(s). Since for all i ∈ N/Ck, gi(tN/Ck) =
gi(s), taking α = g(s) leads to a contradiction with strictly-balanced.
So, quasi-concavity almost extends Ichiishi-Border’s balanced-condition and is easier
to interpret.
References
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9

A Further Generalization of the Kakutani Fixed Point Theorem with Applications to Nash Equilibrium Points,

A Social Coalitional Equilibrium Existence Theorem.

Acceptable Points in Games of Perfect Information,

An Equilibrium Existence Theorem for General Model without Ordered Preferences,

Deman Theory without Transitive Preferences, with Applications to the Theory of Competitive Equilibrium, Preferences Utility, and Demand,

Equilibrium Points in N-Person Games,

Fixed Point Theorems with Applications to Economics and Game Theory,

Inﬁnite Dimensional Analysis.

Non-Cooperation and Cooperation. in Games, Economic Dynamics and Time Series Analysis,edited by

Solution du Probl` eme de Point Fixe de Schauder (Solution of Schauder’s Fixed Point Problem),

Th´ eorie G´ en´ erale des Jeux ` a n Personnes. Gauhtier-Villars,

Coalitional Equilibria of Strategic Games

http://hal.archives-ouvertes.fr/docs/00/42/92/93/PDF/2009-42.pdf

Coalitional Equilibria of Strategic Games

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