EQUILIBRIUM IN ABSTRACT ECONOMIES WITH
WEAKLY CONVEX GRAPH CORRESPONDENCES
MONICA PATRICHE
We establish new existence results for the equilibrium of the abstract economies.
The constraint correspondences have the WCG property. This notion was proposed in 1998 by Ding and He [18]. Basically, our proofs are based on a continuous
selection theorem for correspondences with WCG property. We use also Wu’s fixed
point theorem (see [21]).
AMS 2000 Subject Classification: 91B52, 91B50, 91A80.
Key words: weakly convex graph, continuous selection, abstract economy, equilibrium.
1. INTRODUCTION
In game theory, most authors studied the existence of equilibria for abstract economies with preferences represented as correspondences which have
continuity properties. Thus, the results obtained by Shafer and Sonnenschein
[15] concern economies with finite dimensional commodity space and preference correspondences having an open graph. Yannelis and Prahbakar [23]
used selection theorems and fixed-point theorems for correspondences with
open lower sections defined on infinite dimensional strategy spaces. Then,
a problem of great interest was to weaken the assumptions to lower semicontinuity. Yuan [25] developed an approximation method for economies with
lower semi-continuous correspondences defined on locally convex spaces. Some
authors developed the theory of continuous selections of correspondences and
gave numerous applications in game theory. Michael’s selection theorem [12]
is well known and basic in many applications. Browder [4, 5] first used a continuous selection theorem to prove the Fan-Browder fixed point theorem. Later,
Yannelis and Prabhakar [23], Ben-El-Mechaiekh [2], Ding, Kim and Tan [7],
Horvath [10], Wu [21], Park [14], Yu and Lin [24] and many others established
several continuous selection theorems with applications.
Other general existence results appeared in the literature dealing with
economies with upper semi-continuous correspondences representing the constraints and preferrences of each agent. For example, Tan and Wu [17], Ding
[6] obtained some results in this area.
MATH. REPORTS 10(60), 4 (2008), 359–373
360
Monica Patriche
2
In this paper, we use the concept of weakly convex graph for correspondences to show that an equilibrium for an abstract economy exists without
continuity assumptions. This concept was proposed in 1998 by Ding and He
[18] who obtained some new coincidence, best approximation and fixed point
theorems for correspondences without compact convex values and continuity
in topological vector spaces.
We introduce the notions of correspondences with the WCGS or e-WCGS
property. Then, using a tehnique based on a continuous selection, we prove
new equilibrium existence theorem for an abstract economy.
The paper is organized as follows Section 2 contains preliminaries and
notation. The selection theorem is presented in Section 3. The equilibrium
theorems are stated in Section 4 and their proofs are collected in Section 5.
2. PRELIMINARIES AND NOTATION
Throughout this paper we shall use the following notation and definitions.
Let A be a subset of a topological space X.
1. 2A denotes the family of all subsets of A.
2. cl A denotes the closure of A in X.
3. If A is a subset of a vector space, co A denotes the convex hull of A.
4. If F , T : A → 2X are correspondences, then co T , cl T , T ∩F : A → 2X
are correspondences defined by (co T )(x) = co T (x), (cl T )(x) = cl T (x) and
(T ∩ F )(x) = T (x) ∩ F (x) for each x ∈ A, respectively.
5. The graph of T : X → 2Y is the set Gr(T ) = {(x, y) ∈ X × Y | y ∈
T (x)}
6. The correspondence T is defined by T (x) = {y ∈ Y : (x, y) ∈
clX×Y Gr T } (the set clX×Y Gr(T ) is called the adherence of the graph of T ).
It is easy to see that cl T (x) ⊂ T (x) for each x ∈ X.
Lemma 1 ([25]). Let X be a topological space, Y a non-empty subset of a
topological vector space E, ß a base of the neighborhoods of 0 in E and A : X →
2Y . For each V ∈ ß, let AV : X → 2Y be defined by AV (x) T
= (A(x) + V ) ∩ Y
for each x ∈ X. If x
b ∈ X and yb ∈ Y are such that yb ∈ V ∈ß AV (b
x), then
x).
yb ∈ A(b
Definition 1. Let X, Y be topological spaces and T : X → 2Y a correspondence.
1. T is said to be upper semicontinuous if for each x ∈ X and each open
set V in Y with T (x) ⊂ V there exists an open neighborhood U of x in X
such that T (x) ⊂ V for each y ∈ U .
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Equilibrium in abstract economies with weakly convex graph correspondences
361
2. T is said to be lower semicontinuous if for each x ∈ X and each open
set V in Y with T (x) ∩ V 6= ∅ there exists an open neighborhood U of x in X
such that T (y) ∩ V 6= ∅ for each y ∈ U .
3. T is said to have open lower sections if T −1 (y) := {x ∈ X : y ∈ T (x)}
is open in X for each y ∈ Y.
Lemma 2 ([19]). Let X be a topological space, Y a topological linear
space, and let A : X → 2Y be an upper semicontinuous correspondence with
compact values. Assume that the sets C ⊂ Y and K ⊂ Y are closed and
respectively compact. Then T : X → 2Y defined by T (x) = (A(x) + C) ∩ K for
all x ∈ X is upper semicontinuous.
Ding Xieping and He Yiran [18] proposed the concept of weakly convex
graph for a correspondence.
Definition 2 ([18]). Let X be a nonempty convex subset of a topological
vector space E and Y a nonempty subset of E. The correspondence T : X →
2Y is said to have weakly convex graph (it is a WCG correspondence for short) if
for each finite set {x1 , x2 , . . . , xn } ⊂ X there exists yi ∈ T (xi ), i = 1, 2, . . . , n,
such that
(1.1)
co({(x1 , y1 ), (x2 , y2 ), . . . , (xn , yn )}) ⊂ Gr(T ).
Let ∆n−1 =
n
o
n
P
(λ1 , λ2 , . . . , λn ) ∈ Rn :
λi = 1, λi > 0, i = 1, 2, . . . , n
i=1
be the standard (n − 1)-dimensional simplex. Then relation (1.1) is equivalent to
(1.2)
n
X
i=1
λ i yi ∈ T
X
n
λi xi ,
∀(λ1 , λ2 , . . . , λn ) ∈ σ.
i=1
It is clear that if either the graph Gr(T ) is convex or
∅, then T has a weakly convex graph.
T
{T (x) : x ∈ X} =
6
Remark 1. A WCG correspondence has non-empty values.
Remark 2. A WCG correspondence T : X → 2Y may not be convexvalued.
(
[0, 21 ] ∪ [ 32 , 2] if x = 1,
Example 1. T1 : [0, 2] → [0, 2], T1 (x) =
if x ∈ [0, 2] \ {1}
[0, 2 − 12 x]
T
is a WCG correspondence (since {T1 (x) : x ∈ [0, 2]} = {0} =
6 ∅), but T1 (1)
is not convex and Gr(T1 ) is not convex either.
Remark 3. A WCG correspondence may not enjoy topological properties.
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(
Example 2. T2 : [0, 2] → [0, 2], T2 (x) =
4
[0, 2] if x ∈ [0, 1],
is a WCG
{1} if x ∈ (1, 2]
correspondence, but it is not lower semicontinuous and it has not open lower
sections.
(
{1} if x ∈ [0, 1],
Example 3. T3 : [0, 2] → [0, 2], T3 (x) =
is a WCG
[0, 2] if x ∈ (1, 2]
correspondence, but it is not upper semicontinuous.
Example 4. T1 is not upper semicontinuous at x = 1.
Following Ding and He [18], we can formulate the selection theorem which
we shall use to prove our equilibrium theorem.
Theorem 1. Let Y be a non-empty subset of a topological vector space
E and K an (n − 1)-dimensional simplex in a topological vector space F . Let
T : K → 2Y be a WCG correspondence. Then T has a continuous selection
on K.
Proof. Let a1 , a2 , . . . , an be the vertices af K. Since Gr(T ) is weakly
convex, there exist bi ∈ T (ai ), i = 1, 2, . . . , n, such that
n
X
(3.1)
λ i bi ∈ T
X
n
i=1
λi ai
i=1
for all (λ1 , λ2 , . . . , λn ) ∈ ∆n−1 .
Define a mapping f : K → Y by
(3.2)
f
X
n
λi ai
i=1
=
n
X
i=1
λ i bi ∈ T
X
n
λi ai
∀(λ1 , λ2 , . . . , λn ) ∈ ∆n−1 .
i=1
We show that f is continuous. Since K is a (n − 1)-dimensional simplex with
vertices a1 , a2 , . . . , an , there exist unique continuous functions λi : K → R,
i = 1, 2, . . . , n, such that for each x ∈ K we have (λ1 (x), λ2 (x), . . . , λn (x)) ∈
n
P
∆n−1 and x =
λi (x)ai . Hence f (x) ∈ T (x) for each x ∈ K. Let (xm )m∈N
i=1
be a sequence which converges to x0 ∈ K, where xm =
x0 =
n
P
n
P
λi (xm )ai and
i=1
λi (x0 )ai . It follows from the continuity of λi that λi (xm ) → λi (x0 )
i=1
for each i = 1, 2, . . . , n as m → ∞. Hence we must have f (xm ) → f (x0 ) as
m → ∞, i.e., f is continuous. 5
Equilibrium in abstract economies with weakly convex graph correspondences
363
Remark 4. Let T : X → 2Y be a WCG correspondence and X0 be a
non-empty convex subset of X. Then the restriction T|X0 : X0 → 2Y of T to
X0 is a WCG correspondence, too.
Now, we introduce the definitions below.
Let I be an index set. For each i ∈ I, let Xi be Q
a non-empty convex
subset of a topological linear space Ei and denote X = Xi .
i∈I
Definition 3. Let Ki be a subset of X. The correspondence Ai : X → 2Xi
is said to have the WCGS-property on Ki if there is a WCG correspondence
Ti : Ki → 2Xi such that xi ∈
/ Ti (x) and Ti (x) ⊂ Ai (x) for all x ∈ Ki .
Definition 4. Let Ki be a subset of X. The correspondence Ai : X → 2Xi
is said to have the e-WCGS-property on Ki if for each convex neighbourhood V
of 0 in Xi there is a WCG correspondence TiV : Ki → 2Xi such that xi ∈
/ TiV (x)
and TiV (x) ⊂ Ai (x) + V for all x ∈ Ki .
To prove our theorems, we need
Theorem 2 ([21]). Let I be an index set. For each i ∈ I, let Xi be
a nonempty convex subset of a Hausdorff locally convex topological vector
space
Ei , Di a non-empty compact metrizable subset of Xi , and Si , Ti : X :=
Q
Xi → 2Di two correspondences such that
i∈I
(i) co Si (x) ⊂ Ti (x) and Si (x) 6= ∅ for each x ∈ X;
(ii) Si is lower semicontinuous. Q
Q
Then there exists a point x =
xi ∈ D =
Di such that xi ∈ Ti (x)
i∈I
i∈I
for each i ∈ I.
3. EQUILIBRIUM THEOREMS
First, we present the model of an abstract economy and the definition of
an equilibrium.
Let I be a non-empty set (the set of agents). For each i ∈ I, let Xi be a
non-empty
Q topological vector space representing the set of actions and define
X := Xi . Let Ai , Bi : X → 2Xi be the constraint correspondences and Pi
i∈I
the preference correspondence.
Definition 5. The family Γ = (Xi , Ai , Pi , Bi )i∈I is said to be an abstract
economy.
Definition 6. An equilibrium for Γ is defined as a point x ∈ X such that
xi ∈ B i (x) and Ai (x, ) ∩ Pi (x) = ∅ for each i ∈ I.
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Remark 5. When Ai (x) = Bi (x) for all x ∈ X and each i ∈ I, this
abstract economy model coincides with the classical one introduced by Borglin
and Keiding [3]. If in addition, B i (x) = clXi Bi (x) for each x ∈ X, which is
the case if Bi has a closed graph in X × Xi , the definition of an equilibrium
coincides with that used by Yannelis and Prabhakar [23].
Yannelis and Prabhakar [23] proved the existence of equilibrium for abstract economies with the correspondences Ai and Pi having open lower sections. They used a selection theorem for the correspondences Ai ∩ Pi , which
also have open lower sections. Zhou [20] gave existence equilibrium theorems
and the approach he adopted to prove these theorems is an extension of Yannelis and Prabhakar’s argument in [23]. When the strategy spaces are metrizable, Theorems 5 and 6 in [20] are strict generalizations of the results of
Shafer and Sonnenschein, Yannelis and Prabhakar.
The theorems we will state use the selection theorem mentioned in Section 3 and the same tehnique based on a continuous selection, as in [20] and
[23]. We show the existence of equilibrium for an abstract economy without
assuming the continuity of the constraint and preference correspondences Ai
and Pi .
Since a WCG correspondence may fail to have continuity properties or
convex values and has a continuous selection on a simplex, our results are
related to the above mentioned theorems only through the tehnique of proof.
First, we prove a new equilibrium existence theorem for a noncompact
abstract economy with constraint and preference correspondences Ai and Pi ,
which have the property that their intersection Ai ∩ Pi contains an WCG
selector on the domain Wi of Ai ∩ Pi and Wi must be a simplex. To find the
equilibrium point, we use Wu’s fixed point theorem [21].
Theorem 3. Let Γ = (Xi , Ai , Pi , Bi )i∈I be an abstract economy, where
I is a (possibly uncountable) set of agents such that for each i ∈ I:
(1) Xi is a non-empty convex set in a locally convex space Ei and there
exists a compact subset Di of Xi containing
all the values of the corresponQ
dences Ai , Pi and Bi such that D =
Di is metrizable;
i∈I
(2) cl Bi is lower semicontinuous, has non-empty convex values and Ai (x)
⊂ Bi (x) for each x ∈ X;
(3) Wi = {x ∈ X | (Ai ∩ Pi ) (x) 6= ∅} is an (ni − 1)-dimensional simplex
in X such that Wi ⊂co D;
(4) there exists a WCG correspondence Si : Wi → 2Di such that Si (x) ⊂
(Ai ∩ Pi ) (x) for each x ∈ Wi ;
(5) xi ∈
/ (Ai ∩ Pi )(x) for each x ∈ Wi .
Then there exists an equilibrium point x ∈ D for Γ, i.e., xi ∈ cl Bi (x)
and Ai (x) ∩ Pi (x) = ∅ for each i ∈ I.
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Equilibrium in abstract economies with weakly convex graph correspondences
365
Since a correspondence
T : X → 2Y with a convex graph or having the
T
property that {T (x) : x ∈ X} 6= ∅ is a WCG correspondence, we obtain the
following corollaries.
Corollary 1. Let Γ = (Xi , Ai , Pi , Bi )i∈I be an abstract economy, where
I is a (possibly uncountable) set of agents such that for each i ∈ I:
(1) Xi is a non-empty convex set in a locally convex space Ei and there
exists a compact subset Di of Xi containing
all the values of the corresponQ
dences Ai , Pi and Bi such that D =
Di is metrizable;
i∈I
(2) cl Bi is lower semicontinuous, has non-empty convex values and Ai (x)
⊂ Bi (x) for each x ∈ X;
(3) Wi = {x ∈ X | (Ai ∩ Pi ) (x) 6= ∅} is an (ni −1)-dimensional simplex
in X such that Wi ⊂ co D;
(4) there exists a correspondence Si : Wi → 2Di such that Si has a convex
graph and Si (x) ⊂ (Ai ∩ Pi ) (x) for each x ∈ Wi ;
(5) xi ∈
/ (Ai ∩ Pi )(x) for each x ∈ Wi .
Then there exists an equilibrium point x ∈ D for Γ, i.e., xi ∈ cl Bi (x)
and Ai (x) ∩ Pi (x) = ∅ for each i ∈ I.
Corollary 2. Let Γ = (Xi , Ai , Pi , Bi )i∈I be an abstract economy, where
I is a (possibly uncountable) set of agents such that for each i ∈ I:
(1) Xi is a non-empty compact convex metrizable set in a locally convex
space Ei ;
(2) cl Bi is lower semicontinuous, has non-empty convex values and Ai (x)
⊂ Bi (x)for each x ∈ X;
(3) Wi = {x ∈ X | (Ai ∩ Pi ) (x) 6= ∅} is an (ni − 1)-dimensional simplex
in X;
(4) there exists a correspondence Si : Wi → 2Xi with closed values
such
Tn that Si has the property that for any finite set {x1 , x2 , . . . , xn } ⊂ X,
i=1 S(xi ) 6= ∅ and Si (x) ⊂ (Ai ∩ Pi ) (x) for each x ∈ Wi ;
(5) xi ∈
/ (Ai ∩ Pi )(x) for each x ∈ Wi .
Then there exists an equilibrium point x ∈ X for Γ, i.e., xi ∈ cl Bi (x)
and Ai (x) ∩ Pi (x) = ∅ for each i ∈ I.
To prove Theorem 4 we use an approximation method, to mean that
we obtain a continuous selection fiVi of (Ai + Vi ) ∩ Pi for each
Q i ∈ I, where
Vi is a convex neighborhood of 0 in Xi . For every V =
Vi , we obtain
i∈I
an equilibrium point for the associated approximate abstract economy ΓV =
(Xi , Ai , Pi , BVi )i∈I , i.e., a point x ∈ X such that Ai (x) ∩ Pi (x) = ∅ and
xi ∈ BVi (x), where the correspondence BVi : X → 2Xi is defined by BVi (x) =
cl(Bi (x) + Vi ) ∩ Xi for each x ∈ X and each i ∈ I. Finally, we use Lemma 1
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to get an equilibrium point for Γ in X. The compactness assumption for Xi
is essential in the proof.
Theorem 4. Let Γ = (Xi , Ai , Pi , Bi )i∈I be an abstract economy, where
I is a (possibly uncountable) set of agents such that for each i ∈ I:
(1) Xi is a non-empty compact convex set in a locally convex space Ei ;
(2) cl Bi is upper semicontinuous, has non-empty convex values and Ai (x)
⊂ Bi (x) for each x ∈ X;
(3) the set Wi : = {x ∈ X | (Ai ∩ Pi ) (x) 6= ∅} is non-empty, open and
Ki = cl Wi is an (ni − 1)-dimensional simplex in X ;
(4) for each convex neighbourhood V of 0 in Xi , (Ai +V )∩Pi : Ki → 2Xi
is a WCG correspondence;
(5) xi ∈
/ Pi (x) for each x ∈ Ki .
Then there exists an equilibrium point x ∈ X for Γ, i.e., xi ∈ B i (x) and
Ai (x) ∩ Pi (x) = ∅ for each i ∈ I.
Theorem 5 uses the idea of Zhen’s Theorem 3.2 in [19]. These two results differ by the nature of the correspondence selectors cl Bi . To find the
equilibrium point, we use Wu’s fixed point theorem for correspondences clBi
which are lower semicontinuous and we need a non-empty compact metrizable
set Di in Xi for each i ∈ I. The spaces Xi are not compact.
Theorem 5. Let Γ = (Xi , Ai , Pi , Bi )i∈I be an abstract economy, where
I is a (possibly uncountable) set of agents such that for each i ∈ I:
(1) Xi is a non-empty convex set in a Hausdorff locally convex space Ei
and there exists a nonempty compact metrizable subset Di of Xi containing
all values of the correspondences Ai , Pi and Bi ;
(2) cl Bi is lower semicontinuous with non-empty convex values;
(3) there exists an (ni − 1)-dimensional simplex Ki in X and
Wi := {x ∈ X | (Ai ∩ Pi ) (x) 6= ∅} ⊂ intX (Ki );
(4) cl Bi has the (WCGS)-property on Ki .
Then there exists an equilibrium point x ∈ D for Γ, i.e., xi ∈ cl Bi (x)
and Ai (x) ∩ Pi (x) = ∅ for each i ∈ I.
Theorem 6 follows the idea of Zheng’s Theorem 3.1 in [19], but the correspondences cl Bi have e-WCGS property on a simplex Ki in X which contains
the domain of Ai ∩ Pi . The sets Xi are non-empty, compact, convex in locally
convex spaces Ei . As in Theorem 4, we first obtain an equilibrium for ΓV , and
the proof then coincides with the proof of Theorem 4.
Theorem 6. Let Γ = (Xi , Ai , Pi , Bi )i∈I be an abstract economy, where
I is a (possibly uncountable) set of agents such that for each i ∈ I:
(1) Xi is a non-empty compact convex set in a locally convex space Ei ;
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Equilibrium in abstract economies with weakly convex graph correspondences
367
(2) cl Bi is upper semicontinuous with non-empty convex values;
(3) the set Wi : = {x ∈ X | (Ai ∩ Pi ) (x) 6= ∅} is open and there exists
an (ni − 1)-dimensional simplex Ki in X such that Wi ⊂ intX (Ki );
(4) cl Bi has the e-WCGS-property on Ki .
Then there exists an equilibrium point x ∈ X for Γ, i.e., xi ∈ B i (x) and
Ai (x) ∩ Pi (x) = ∅ for each i ∈ I.
4. PROOFS OF THE THEOREMS
Proof of Theorem 3. Let i ∈ I. It follows from assumption (4) and the
selection theorem that there exists a continuous function fi : Wi → Di such
that fi (x) ∈ Si (x) ⊂ Ai (x) ∩ Pi (x) ⊂ Bi (x) for each x ∈ Wi .
Define the correspondence Ti : X → 2Di by
(
Ti (x) :=
{fi (x)}
if x ∈ Wi ,
cl Bi (x) if x ∈
/ Wi ;
Ti is lower semicontinuous on X. Let V be an closed subset of Xi . Then
U := {x ∈ X | Ti (x) ⊂ V } = {x ∈ Wi | Ti (x) ⊂ V } ∪ {x ∈ X \Wi | Ti (x) ⊂ V }
= {x ∈ Wi | fi (x) ∈ V } ∪ {x ∈ X | cl Bi (x) ⊂ V }
= (fi−1 (V ) ∩ Wi ) ∪ {x ∈ X | cl Bi (x) ⊂ V } .
U is a closed set because Wi is closed, fi is a continuous map on int
Q X Ki and the
Di . Then by
set {x ∈ X | cl Bi (x) ⊂ V } is closed since cl Bi is l.s.c. Let D =
i∈I
Tychonoff’s theorem, D is compact in the convex set X. By Theorem 2 (Wu’s
fixed-point theorem) applied to the correspondences Si = Ti and Ti : X → 2Di ,
there exists x ∈ D such that xi ∈ Ti (x) for each i ∈ I. If x ∈ Wi for
some i ∈ I, then xi = fi (x), which is a contradiction. Therefore, x ∈
/ Wi ,
hence (Ai ∩ Pi )(x) = ∅. Also, for each i ∈ I, we have xi ∈ Ti (x), and then
xi ∈ cl Bi (x). Remark 6. In Theorem 3, the correspondences Ai ∩ Pi don’t verify continuity assumptions and do not have convex or compact values.
Remark 7. In assumption (3), Wi must be a proper subset of X. Q
In fact,
if Wi = Xi then, by applying Himmelberg’s fixed point theorem to
fi (x),
i∈I
where
Q fi is a continuous selection of Si ⊂ Ai ∩ Pi , we can get a fixed point
x ∈ (Ai ∩ Pi )(x), which contradicts assumption (5).
i∈I
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Proof of Corollary 1. Since a correspondence with a convex graph is a
WCG correspondence, Si verifies assumption (4) from Theorem 3, so that we
can apply this theorem. Proof of Corollary 2. X is a compact space and for each i T
∈ I the closed
sets Si (x), x ∈ X have the finite intersection property. Then {Si (x) : x ∈
X} =
6 ∅. It follows that Si is a WCG correspondence and the conclusion comes
from Theorem 3. Proof of Theorem 4. For each i ∈ I, let ßi denote the family
of all
Q
open convex neighborhoods of zero in Ei . Let V = (Vi )i∈I ∈
ßi . Since
i∈I
(Ai + Vi ) ∩ Pi is a WCG correspondence on Ki , from the selection theorem
there exists a continuous function fiVi : Ki → Xi such that
fiVi (x) ∈ (Ai (x) + Vi ) ∩ Pi (x) ⊂ (Ai (x) + Vi ) ∩ Xi .
for each x ∈ Ki . It follows that fiVi (x) ∈ cl(Bi (x) + Vi ) for x ∈ Ki . Since
Xi is compact, cl(Bi (x)) is compact for every x ∈ X and cl(Bi (x) + Vi ) =
cl(Bi (x)) + cl Vi for every Vi ⊂ Ei .
Define the correspondence TiVi : X → 2Xi , by
(
{fiVi (x)}
if x ∈ intX K = Wi ,
Vi
Ti (x) :=
cl(Bi (x) + Vi ) ∩ Xi if x ∈ X \ intX Ki .
The correspondence BVi : X → 2Xi , defined by BVi (x) := cl(Bi (x) + Vi ) ∩ Xi
is u.s.c. by Lemma 2. Then reasoning as in the proof of Theorem 3, we can
prove that TiVi is upper semicontinuous on X and has closed convex values.
Q Vi
Define T V : X → 2X by T V (x) :=
Ti (x) for each x ∈ X. T V is an
i∈I
upper semicontinuous correspondence and also has non-empty convex closed
values. Since X is a compact convex set, by Fan’s fixed-point theorem [8], there
exists xV ∈ X such that xV ∈ T V (xV ), i.e., for each i ∈ I, (xV )i ∈ TiVi (xV ).
S
We clain that xV ∈ X \ intX Ki . Indeed, if xV ∈ intX Ki , (xV )i ∈ TiVi (xV ) =
i∈I
fi (xV ) ∈ ((Ai (xV ) + Vi ) ∩ Pi )(xV ) ⊂ Pi (xV ), which contradicts assumption
(5). Hence (xV )i ∈ cl(Bi (xV ) + Vi ) ∩ Xi and (Ai ∩ Pi )(xV ) = ∅, i.e. xV ∈ QV
where
\
QV = {x ∈ X : xi ∈ cl(Bi (x) + Vi ) ∩ Xi and (Ai ∩ Pi )(x) = ∅}.
i∈I
Since Wi is open, QV is the intersection of non-emptynclosed sets, theno it is
Q
non-empty, closed in X. Let us prove that the family QV : V ∈
ßi has
i∈I
the finite intersection property. Let {V (1) , V (2) , . . . , V (n) } be any finite set of
11
Q
i∈I
Equilibrium in abstract economies with weakly convex graph correspondences
(k)
ßi and let V (k) = (Vi
)i∈I , k = 1, . . . , n. For each i ∈ I, let Vi =
Then Vi ∈ ßi , thus, V = (Vi )i∈I ∈
Q
ßi . Clearly, QV ⊂
i∈I
n
T
k=1
n
T
k=1
n
T
k=1
369
(k)
Vi
.
QV (k) so that
QV (k) 6= ∅.
n
Q o
Since X is compact and the family QV : V ∈
ßi has the finite
i∈I
n
o
T
Q
T
intersection property, we have
QV : V ∈
ßi 6= ∅. Take any x ∈ {QV :
i∈I
Q
V ∈
ßi }. Then xi ∈ cl(Bi (x) + Vi ) ∩ Xi and (Ai ∩ Pi )(x) = ∅ for each i ∈ I
i∈I
and each Vi ∈ ßi , but then xi ∈ cl(Bi (x)) by Lemma 3 and (Ai ∩ Pi )(x) = ∅
for each i ∈ I, so that x is an equilibrium point of Γ in X. Proof of Theorem 5. Since cl Bi has the WCGS property on Ki , there
exists a WCG correspondence Fi : X → 2Di such that Fi (x) ⊂ cl Bi (x) and
xi ∈
/ Fi (x) for each x ∈ Ki . Note that Ki is an (ni − 1)-dimensional simplex.
By the selection theorem, there exists a continuous function fi : Ki → Di
such that fi (x) ∈ Fi (x) for each x ∈ Ki . Because xi ∈
/ Fi (x) for each x ∈ Ki ,
we have xi 6= fi (x) for each x ∈ Ki .
Define the correspondence Ti : X → 2Di , by
(
Ti (x) :=
{fi (x)}
if x ∈ Ki ,
cl Bi (x) if x ∈
/ Ki ;
Ti is lower semicontinuous on X and has closed convex values.
Let U be a closed subset of Xi . Then
U 0 := {x ∈ X | Ti (x) ⊂ U } = {x ∈ Ki | Ti (x) ⊂ U } ∪ {x ∈ X \Ki | Ti (x) ⊂ U }
= {x ∈ Ki | fi (x, y) ∈ U } ∪ {x ∈ X | cl Bi (x) ⊂ U }
= ((fi )−1 (U ) ∩ Ki ) ∪ {x ∈ X | cl Bi (x) ⊂ U } .
U 0 is a closed set because Ki is closed, fi is a continuous map on Ki and the
set {x ∈ X | cl Bi (x) ⊂ U } is closed since cl Bi (x) is l.s.c. Then Ti is lower
semicontinuous on X and has non-empty closed convex values.
By Theorem 2 (Wu’s fixed-point theorem) applied to the correspondences
Si = Ti and Ti : X → 2Di , there exists x ∈ D such that xi ∈ Ti (x) for each
i ∈ I. If x ∈ Wi for some i ∈ I, then xi = fi (x), which is a contradiction.
Therefore, x ∈
/ Wi , hence (Ai ∩ Pi )(x) = ∅. Also, for each i ∈ I we have
xi ∈ Ti (x), and then xi ∈ cl Bi (x). 370
Monica Patriche
12
Proof of Theorem 6. For each i ∈ I, let ßi denoteQthe family of all open
convex neighborhoods of zero in Ei . Let V = (Vi )i∈I ∈ ßi . Since cl Bi has the
i∈I
e-WCGS property on Ki , there exists a WCG correspondence FiVi : X → 2Xi
such that FiVi (x) ⊂ cl Bi (x) + Vi and xi ∈
/ FiVi (x) for each x ∈ Ki .
Ki is an (ni − 1)-dimensional simplex. Then, by the selection theorem,
there exists a continuous function fiVi : Ki → Xi such that fiVi (x) ∈ FiVi (x)
for each x ∈ Ki . Because xi ∈
/ FiVi (x) for each x ∈ Ki , we have xi 6= fiVi (x) for
each x ∈ Ki .
Define the correspondence TiVi : X → 2Xi by
(
{fiVi (x)}
if x ∈ intX Ki ,
TiVi (x) :=
cl(Bi (x) + Vi ) ∩ Xi if x ∈ X \ intX Ki .
Here, BVi : X → 2Xi , BVi (x) = cl(Bi (x) + Vi ) ∩ Xi = (cl Bi (x) + cl Vi ) ∩ Xi is
upper semicontinuous by Lemma 2.
Let U be an open subset of Xi . Then
U 0 := {x ∈ X | TiVi (x) ⊂ U }
= {x ∈ intX Ki | TiVi (x) ⊂ U } ∪ {x ∈ X \ intX Ki | TiVi (x) ⊂ U }
= x ∈ intX Ki | fiVi (x, y) ∈ U ∪ x ∈ X | (cl Bi (x) + Vi ) ∩ Xi ⊂ U
= ((fiVi )−1 (U ) ∩ intK Ki ) ∪ x ∈ X | (cl Bi (x) + Vi ) ∩ Xi ⊂ U .
U 0 is an open set because intX Ki is open, fiVi is a continuous map on Ki and
the set {x ∈ X | (cl Bi (x) + cl Vi ) ∩ Xi ⊂ U } is open since (cl Bi (x) + cl Vi ) ∩ Xi
is u.s.c. Then TiVi is upper semicontinuous on X and has closed convex values.
Q Vi
Ti (x) for each x ∈ X; T V is an
Define T V : X → 2X by T V (x) :=
i∈I
upper semicontinuous correspondence and has also non-empty convex closed
values. Since X is a compact convex set, by Fan’s fixed-point theorem [8],
there exists xV ∈ X such that xV ∈ T V (xV ), i.e., (xV )i ∈ TiVi (xV ) for each
i ∈ I. If xV ∈ intX Ki , (xV )i = fiVi (xV ), which is a contradiction. Hence
(xV )i ∈ cl(Bi (xV ) + Vi ) ∩ Xi and (Ai ∩ Pi )(xV ) = ∅, i.e., xV ∈ QV where
\
QV =
{x ∈ X : xi ∈ cl(Bi (x) + Vi ) ∩ Xi and (Ai ∩ Pi )(x) = ∅}.
i∈I
Since Wi is open, QV is the intersection of non-empty closed sets, so it is
non-empty, closed in X.
n
Q o
We now prove that the family QV : V ∈
ßi has the finite interi∈I
Q
section property. Let {V (1) , V (2) , . . . , V (n) } be any finite set of
ßi and let
i∈I
13
Equilibrium in abstract economies with weakly convex graph correspondences
(k)
V (k) = (Vi
)i∈I , k = 1, . . . , n. For each i ∈ I, let Vi =
n
T
k=1
(k)
Vi
371
. Then Vi ∈ ßi ,
n
n
T
T
QV (k) 6= ∅. Since
QV (k) so that
ßi . Clearly, QV ⊂
i∈I
k=1
k=1 o
n
Q
X is compact and the family QV : V ∈
ßi has the finite intersection propi∈I
Tn
Q o
Tn
Q o
erty, we have
QV : V ∈
ßi 6= ∅. Take any x ∈
QV : V ∈
ßi .
thus V = (Vi )i∈I ∈
Q
i∈I
i∈I
Then xi ∈ cl(Bi (x) + Vi ) ∩ Xi and (Ai ∩ Pi )(x) = ∅ for each i ∈ I and each
Vi ∈ ßi , but then xi ∈ cl(Bi (x)) by Lemma 3 and (Ai ∩ Pi )(x) = ∅ for each
i ∈ I, so that x is an equilibrium point of Γ in X. Example 5. This is an example of an abstract economy with finite number
of agents where Theorem 3 is applicable. Let Γ = (Xi , Ai , Pi , Bi )i∈I be an
abstract economy, where I = {1, 2, . . . , n} is a finite set of agents
Q such that
Xi = [0, 4] for each i ∈ I is a compact convex choice set, X := Xi and the
i∈I
correspondences Ai , Pi , Bi : X → 2Xi are defined as
(
[2, 3 − 21i xi ] if x ∈ M,
Ai (x) =
[0, 12 ] ∪ [2, 4] if x ∈ X − M,


[2, 3]
if x ∈ M,



 [0, 1 ] ∪ [2, 4] if x = (1, 1, . . . , 1),
2
Pi (x) =

{2}
if x ∈ W \ (M ∪ {(1, 1 . . . , 1)}),




∅
otherwise,
Bi (x) = Xi
for each x ∈ X.
Here, W is the simplex with vertices
Q (0, 0, . . . , 0), (3, 0, . . . , 0), (0, 3, 0,
. . . , 0), and (0, 0, . . . , 0, 3) in X and M = [0, 1). Then we have
i∈I


[2, 3 − 21i xi ]



 [0, 1 ] ∪ [2, 4]
2
(Ai ∩ Pi )(x) =
 {2}




∅
if x ∈ M,
if x = (1, 1, . . . , 1),
if x ∈ W \ (M ∪ {(1, 1, . . . , 1)}),
otherwise.
It is easy to show that Ai ∩ Pi is not lower semi-continuous
or upper
T
semi-cotinuous on X, has not convex values but, since 2 ∈
S(x), Ai ∩ Pi
x∈X
is a WCG correspondence on W . Note that xi ∈
/ (Ai ∩ Pi )(x) for each x ∈
X. Clearly, cl Bi is lower semicontinuous, has non-empty convex values and
372
Monica Patriche
14
Ai (x) ⊂ Bi (x) for each x ∈ X. Then the assumptions of Theorem 3 are
satisfied, so we can obtain an equilibrium point x = ( 72 , 72 , . . . , 72 ) ∈ X for Γ
such that, xi ∈ cl Bi (x) and Ai (x) ∩ Pi (x) = ∅ for each i ∈ I.
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Received 16 June 2008
University of Bucharest
Faculty of Mathematics and Computer Science
Str. Academiei 14
014700 Bucharest, Romania
[email protected]