EQUILIBRIUM IN ABSTRACT ECONOMIES WITH WEAKLY CONVEX GRAPH CORRESPONDENCES

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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 semi- continuity. 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 con- straints and preferrences of each agent. For example, Tan and Wu [17], Ding [6] obtained some results in this area.

MATH. REPORTS10(60),4 (2008), 359–373

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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.

LetA be a subset of a topological spaceX.

1. 2^A denotes the family of all subsets ofA.

2. clA denotes the closure of Ain X.

3. IfA is a subset of a vector space, coA denotes the convex hull of A.

4. IfF,T :A→2^X are correspondences, then coT, clT,T∩F :A→2^X are correspondences defined by (coT)(x) = coT(x), (clT)(x) = clT(x) and (T∩F)(x) =T(x)∩F(x) for eachx∈A, respectively.

5. The graph of T :X → 2^Y 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) ∈ cl_X×YGrT} (the set clX×YGr(T) is called the adherence of the graph of T).

It is easy to see that clT(x)⊂T(x) for eachx∈X.

Lemma 1([25]). LetX be a topological space,Y a non-empty subset of a topological vector space E, ß a base of the neighborhoods of 0inEandA:X → 2^Y.For each V ∈ß, let A_V :X →2^Y be defined by A_V(x) = (A(x) +V)∩Y for each x ∈ X. If xb ∈ X and by ∈ Y are such that by ∈ T

V∈ßA_V(x),b then yb∈A(bx).

Definition 1. Let X, Y be topological spaces andT :X →2^Y a correspondence.

1. T is said to beupper semicontinuous if for eachx∈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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2. T is said to belower semicontinuous if for eachx∈X and each open set V inY withT(x)∩V 6=∅ there exists an open neighborhoodU ofx inX such that T(y)∩V 6=∅ for each y∈U.

3. T is said to haveopen lower sections ifT⁻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 → 2^Y 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→2^Y defined byT(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]). LetX be a nonempty convex subset of a topological vector space E andY a nonempty subset ofE. The correspondence T :X → 2^Y is said to haveweakly convex graph(it is a WCG correspondence for short) if for each finite set {x₁, x₂, . . . , x_n} ⊂X there existsy_i ∈T(x_i),i= 1,2, . . . , n, such that

(1.1) co({(x₁, y₁),(x₂, y₂), . . . ,(x_n, y_n)})⊂Gr(T).

Let ∆n−1 = n

(λ₁, λ₂, . . . , λ_n) ∈Rⁿ : Pn i=1

λ_i = 1, λ_i >0, i= 1,2, . . . , no be the standard (n−1)-dimensional simplex. Then relation (1.1) is equiva- lent to

(1.2)

n

X

i=1

λ_iy_i ∈T n

X

i=1

λ_ix_i

, ∀(λ₁, λ₂, . . . , λ_n)∈σ.

It is clear that if either the graph Gr(T) is convex or T{T(x) :x∈X} 6=

∅, thenT has a weakly convex graph.

Remark 1. A WCG correspondence has non-empty values.

Remark 2. A WCG correspondence T : X → 2^Y may not be convex- valued.

Example1. T1 : [0,2]→[0,2], T1(x) =

( [0,¹₂]∪[³₂,2] ifx= 1,

[0,2−¹₂x] ifx∈[0,2]\ {1}

is a WCG correspondence (since T

{T₁(x) :x ∈[0,2]} ={0} 6=∅), but T1(1) is not convex and Gr(T₁) is not convex either.

Remark3. A WCG correspondence may not enjoy topological properties.

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Example 2. T₂ : [0,2]→[0,2], T₂(x) =

( [0,2] ifx∈[0,1],

{1} ifx∈(1,2] is a WCG correspondence, but it is not lower semicontinuous and it has not open lower sections.

Example 3. T₃ : [0,2]→[0,2], T₃(x) =

( {1} ifx∈[0,1],

[0,2] ifx∈(1,2] is a WCG 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 → 2^Y 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 b_i ∈T(a_i), i= 1,2, . . . , n, such that

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n

X

i=1

λ_ib_i ∈T n

X

i=1

λ_ia_i

for all (λ1, λ2, . . . , λn)∈∆n−1.

Define a mapping f :K →Y by (3.2) f

n

X

i=1

λiai

=

n

X

i=1

λibi∈T n

X

i=1

λiai

∀(λ₁, λ2, . . . , λn)∈∆n−1.

We show that f is continuous. SinceK is a (n−1)-dimensional simplex with vertices a₁, a₂, . . . , a_n, there exist unique continuous functions λ_i : K → R, i = 1,2, . . . , n, such that for each x ∈ K we have (λ1(x), λ2(x), . . . , λn(x))∈

∆n−1 and x =

n

P

i=1

λi(x)ai. Hence f(x) ∈ T(x) for each x ∈K. Let (xm)m∈N

be a sequence which converges to x0 ∈ K, where xm =

n

P

i=1

λi(xm)ai and x₀ =

Pn i=1

λ_i(x₀)a_i. It follows from the continuity of λ_i that λ_i(x_m) → λ_i(x₀) for each i = 1,2, . . . , n as m → ∞. Hence we must have f(x_m) → f(x₀) as m→ ∞,i.e., f is continuous.

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Remark 4. Let T : X → 2^Y be a WCG correspondence and X0 be a non-empty convex subset of X. Then the restriction T_|X₀ :X₀ →2^Y 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 X_i be a non-empty convex subset of a topological linear space Ei and denoteX = Q

i∈I

Xi.

Definition 3. Let Kibe a subset ofX. The correspondenceAi :X→2^Xⁱ is said to have the WCGS-property on K_i if there is a WCG correspondence Ti :Ki→2^Xⁱ such thatxi ∈/ Ti(x) and Ti(x)⊂Ai(x) for all x∈Ki.

Definition 4. Let K_ibe a subset ofX. The correspondenceA_i :X→2^Xⁱ is said to have the e-WCGS-property onKiif for each convex neighbourhood V of 0 inX_ithere is a WCG correspondenceT_i^V :K_i→2^Xⁱsuch thatx_i ∈/T_i^V(x) and T_i^V(x)⊂A_i(x) +V for all x∈K_i.

To prove our theorems, we need

Theorem 2 ([21]). Let I be an index set. For each i ∈ I, let X_i be a nonempty convex subset of a Hausdorff locally convex topological vector space Ei, Dia non-empty compact metrizable subset of Xi,and Si, Ti:X :=

Q

i∈I

X_i →2^Dⁱ two correspondences such that

(i) coS_i(x)⊂T_i(x) and S_i(x)6=∅for each x∈X;

(ii)Si is lower semicontinuous.

Then there exists a point x = Q

i∈I

x_i ∈D = Q

i∈I

D_i such that x_i ∈ T_i(x) for each i∈I.

3. EQUILIBRIUM THEOREMS

First, we present the model of an abstract economy and the definition of an equilibrium.

LetI be a non-empty set (the set of agents). For eachi∈I, letXi be a non-empty topological vector space representing the set of actions and define X := Q

i∈I

X_i. Let A_i, B_i :X → 2^Xⁱ be the constraint correspondences and P_i the preference correspondence.

Definition 5. The family Γ = (X_i, A_i, P_i, B_i)i∈I is said to be anabstract economy.

Definition 6. Anequilibrium for Γ is defined as a pointx∈X such that x_i∈B_i(x) and A_i(x,)∩P_i(x) =∅ for eachi∈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, Bi(x) = clXiBi(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 A_i and P_i 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 Yan- nelis 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 Sec- tion 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 A_i 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 A_i and P_i, which have the property that their intersection A_i ∩P_i 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 D_i of X_i containing all the values of the correspondences A_i, P_i and B_i such that D= Q

i∈I

D_i is metrizable;

(2) clBiis lower semicontinuous, has non-empty convex values andAi(x)

⊂B_i(x) for each x∈X;

(3)W_i ={x∈X|(A_i∩P_i) (x)6=∅}is an (n_i−1)-dimensional simplex in X such that Wi ⊂coD;

(4)there exists a WCG correspondence S_i :W_i →2^Dⁱ such that S_i(x)⊂ (Ai∩Pi) (x) for eachx∈Wi;

(5)x_i ∈/ (A_i∩P_i)(x) for each x∈W_i.

Then there exists an equilibrium point x ∈D for Γ, i.e., xi ∈clBi(x) and A_i(x)∩P_i(x) =∅ for each i∈I.

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Since a correspondence T :X →2^Y with a convex graph or having the property that T{T(x) :x∈X} 6=∅ is a WCG correspondence, we obtain the following corollaries.

Corollary1. LetΓ = (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 correspondences A_i, P_i and B_i such that D= Q

i∈I

D_i is metrizable;

⊂Bi(x) for each x∈X;

(3) W_i ={x∈X|(A_i∩P_i) (x)6=∅}is an(n_i−1)-dimensional simplex in X such that Wi ⊂coD;

(4)there exists a correspondenceS_i :W_i →2^Dⁱ such thatS_i has a convex graph and Si(x)⊂(Ai∩Pi) (x) for each x∈Wi;

(5)x_i∈/ (A_i∩P_i)(x) for each x∈W_i.

Then there exists an equilibrium point x ∈D for Γ, i.e., xi ∈ clBi(x) and Ai(x)∩Pi(x) =∅ for each i∈I.

Corollary2. LetΓ = (X_i, A_i, P_i, B_i)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 E_i;

⊂B_i(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 → 2^Xⁱ with closed values such that Si has the property that for any finite set {x₁, x2, . . . , xn} ⊂ X, Tn

i=1S(x_i)6=∅ and S_i(x)⊂(A_i∩P_i) (x) for each x∈W_i; (5)xi∈/ (Ai∩Pi)(x) for each x∈Wi.

Then there exists an equilibrium point x ∈ X for Γ, i.e., x_i ∈ clB_i(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 f_i^Vⁱ of (A_i+V_i)∩P_i for each i ∈ I, where Vi is a convex neighborhood of 0 in Xi. For every V = Q

i∈I

Vi, we obtain an equilibrium point for the associated approximate abstract economy ΓV = (X_i, A_i, P_i, B_V_i)i∈I, i.e., a point x ∈ X such that A_i(x) ∩P_i(x) = ∅ and x_i ∈B_V_i(x),where the correspondenceB_V_i :X →2^Xⁱ is defined byB_V_i(x) = cl(Bi(x) +Vi)∩Xi for each x ∈X and eachi∈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 = (X_i, A_i, P_i, B_i)i∈I be an abstract economy, where I is a (possibly uncountable) set of agents such that for each i∈I:

(1)X_i is a non-empty compact convex set in a locally convex space E_i; (2) clBiis upper semicontinuous, hasnon-empty convex values and Ai(x)

⊂B_i(x) for each x∈X;

(3) the set Wi : = {x∈X|(Ai∩Pi) (x)6=∅} is non-empty, open and K_i = clW_i is an (n_i−1)-dimensional simplex in X ;

(4)for each convex neighbourhood V of 0in Xi, (Ai+V)∩P_i :Ki→2^Xⁱ 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∈Bi(x)and A_i(x)∩P_i(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 clBi. To find the equilibrium point, we use Wu’s fixed point theorem for correspondences clB_i which are lower semicontinuous and we need a non-empty compact metrizable set D_i inX_i for each i∈I.The spaces X_i are not compact.

(1)X_i is a non-empty convex set in a Hausdorff locally convex space E_i and there exists a nonempty compact metrizable subset Di of Xi containing all values of the correspondences A_i, P_i and B_i;

(2) clBi is lower semicontinuous with non-empty convex values;

(3)there exists an (n_i−1)-dimensional simplex K_i in X and Wi:={x∈X|(Ai∩Pi) (x)6=∅} ⊂intX(Ki);

(4) clB_i has the (WCGS)-property on K_i.

Then there exists an equilibrium point x ∈ D for Γ, i.e., xi ∈ clBi(x) and A_i(x)∩P_i(x) =∅ for each i∈I.

Theorem 6 follows the idea of Zheng’s Theorem 3.1 in [19], but the correspondences clB_i have e-WCGS property on a simplexK_i inX which contains the domain of A_i∩P_i.The setsX_i are non-empty, compact, convex in locally convex spacesEi.As in Theorem 4, we first obtain an equilibrium for Γ_V,and the proof then coincides with the proof of Theorem 4.

(1)Xi is a non-empty compact convex set in a locally convex space Ei;

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(2) clBi is upper semicontinuous with non-empty convex values;

(3) the set W_i : = {x∈X|(A_i∩P_i) (x)6=∅} is open and there exists an (ni−1)-dimensional simplex Kiin X such that Wi⊂intX(Ki);

(4) clBi has the e-WCGS-property on Ki.

Then there exists an equilibrium point x∈X for Γ, i.e., xi ∈Bi(x) and A_i(x)∩P_i(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 f_i :W_i → D_i such that f_i(x)∈S_i(x)⊂A_i(x)∩P_i(x)⊂B_i(x) for each x∈W_i.

Define the correspondenceT_i :X →2^Dⁱ by

Ti(x) :=

( {f_i(x)} ifx∈W_i, clB_i(x) ifx /∈W_i;

T_i is lower semicontinuous onX. LetV be an closed subset of X_i. Then U :={x∈X|T_i(x)⊂V}={x∈W_i|T_i(x)⊂V} ∪ {x∈X\W_i|T_i(x)⊂V}

={x∈W_i |f_i(x)∈V} ∪ {x∈X|clB_i(x)⊂V}

= (f_i⁻¹(V)∩Wi)∪ {x∈X|clBi(x)⊂V}.

U is a closed set becauseWiis closed,fiis a continuous map on intXKiand the set{x∈X|clB_i(x)⊂V}is closed since clB_iis l.s.c. LetD= Q

i∈I

D_i.Then by Tychonoff’s theorem,Dis compact in the convex setX. By Theorem 2 (Wu’s fixed-point theorem) applied to the correspondencesS_i =T_iandT_i:X →2^Dⁱ, there exists x ∈ D such that xi ∈ Ti(x) for each i ∈ I. If x ∈ Wi for some i ∈ I, then x_i = f_i(x), which is a contradiction. Therefore, x /∈ W_i, hence (A_i∩P_i)(x) = ∅. Also, for each i ∈ I, we have x_i ∈ T_i(x), and then xi∈clBi(x).

Remark 6. In Theorem 3, the correspondencesA_i∩P_i don’t verify continuity assumptions and do not have convex or compact values.

Remark 7. In assumption (3),W_i must be a proper subset ofX. In fact, if Wi =Xi then, by applying Himmelberg’s fixed point theorem to Q

i∈I

fi(x), where fi is a continuous selection of Si ⊂ Ai ∩Pi, we can get a fixed point x∈ Q

i∈I

(A_i∩P_i)(x), which contradicts assumption (5).

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Proof of Corollary 1. Since a correspondence with a convex graph is a WCG correspondence, S_i verifies assumption (4) from Theorem 3, so that we can apply this theorem.

Proof of Corollary 2. Xis a compact space and for each i∈I the closed sets S_i(x), x ∈X have the finite intersection property. Then T{S_i(x) : x ∈ X} 6=∅. It follows thatSiis 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 open convex neighborhoods of zero in E_i. Let V = (V_i)i∈I ∈ Q

i∈I

ß_i. Since (A_i +V_i)∩P_i is a WCG correspondence on K_i, from the selection theorem there exists a continuous function f_i^Vⁱ :K_i →X_i such that

f_i^Vⁱ(x)∈(A_i(x) +V_i)∩P_i(x)⊂(A_i(x) +V_i)∩X_i.

for each x ∈ K_i. It follows that f_i^Vⁱ(x) ∈ cl(B_i(x) +V_i) for x ∈ K_i. Since Xi is compact, cl(Bi(x)) is compact for every x ∈ X and cl(Bi(x) +Vi) = cl(B_i(x)) + clV_i for everyV_i ⊂E_i.

Define the correspondenceT_i^Vⁱ :X →2^Xⁱ, by T_i^Vⁱ(x) :=

( {f_i^Vⁱ(x)} ifx∈intXK=Wi, cl(Bi(x) +Vi)∩Xi ifx∈X\intXKi.

The correspondence B_V_i :X →2^Xⁱ, defined by B_V_i(x) := cl(B_i(x) +V_i)∩X_i is u.s.c. by Lemma 2. Then reasoning as in the proof of Theorem 3, we can prove that T_i^Vⁱ is upper semicontinuous on X and has closed convex values.

Define T^V :X →2^X by T^V(x) := Q

i∈I

T_i^Vⁱ(x) for each x∈ X. T^V is an upper semicontinuous correspondence and also has non-empty convex closed values. SinceXis 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 ∈T_i^Vⁱ(xV).

We clain thatxV ∈X\S

i∈I

intXKi.Indeed, ifxV ∈intXKi,(xV)i ∈T_i^Vⁱ(xV) = fi(xV) ∈ ((Ai(xV) +Vi)∩Pi)(xV) ⊂ Pi(xV), which contradicts assumption (5). Hence (x_V)_i ∈cl(B_i(x_V) +V_i)∩X_i and (A_i∩P_i)(x_V) =∅,i.e. x_V ∈Q_V where

Q_V =\

i∈I

{x∈X:x_i∈cl(B_i(x) +V_i)∩X_i and (A_i∩P_i)(x) =∅}.

Since W_i is open, Q_V is the intersection of non-empty closed sets, then it is non-empty, closed in X. Let us prove that the family

n

QV :V ∈ Q

i∈I

ßi

o has the finite intersection property. Let {V⁽¹⁾, V⁽²⁾, . . . , V⁽ⁿ⁾} be any finite set of

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Q

i∈I

ß_i and let V^(k) = (V_i^(k))i∈I, k= 1, . . . , n. For each i∈I, let V_i = Tn k=1

V_i^(k). Then V_i ∈ ß_i, thus, V = (V_i)i∈I ∈ Q

i∈I

ß_i. Clearly, Q_V ⊂ Tⁿ

k=1

Q_V(k) so that

n

T

k=1

Q_V(k) 6=∅.

Since X is compact and the family n

QV : V ∈ Q

i∈I

ßi

o

has the finite intersection property, we haveTn

Q_V :V ∈ Q

i∈I

ß_io

6=∅.Take anyx∈T{Q_V : V ∈ Q

i∈I

ß_i}.Then x_i ∈cl(B_i(x) +V_i)∩X_i and (A_i∩P_i)(x) =∅for each i∈I and each V_i ∈ ß_i, but then x_i ∈cl(B_i(x)) by Lemma 3 and (A_i∩P_i)(x) =∅ for each i∈I, so that x is an equilibrium point of Γ inX.

Proof of Theorem 5. Since clB_i has the WCGS property on K_i, there exists a WCG correspondence Fi : X → 2^Dⁱ such that Fi(x) ⊂clBi(x) and x_i ∈/ F_i(x) for eachx ∈K_i. Note that K_i is an (n_i−1)-dimensional simplex.

By the selection theorem, there exists a continuous function fi : Ki → Di

such that f_i(x) ∈F_i(x) for eachx ∈K_i.Because x_i ∈/ F_i(x) for eachx ∈K_i, we have x_i6=f_i(x) for each x∈K_i.

Define the correspondenceT_i :X →2^Dⁱ, by T_i(x) :=

( {f_i(x)} ifx∈Ki, clB_i(x) ifx /∈K_i;

Ti is lower semicontinuous onX and has closed convex values.

LetU be a closed subset of X_i. Then

U⁰ :={x∈X|Ti(x)⊂U}={x∈Ki|Ti(x)⊂U} ∪ {x∈X\K_i |Ti(x)⊂U}

={x∈K_i|f_i(x, y)∈U} ∪ {x∈X |clB_i(x)⊂U}

= ((f_i)⁻¹(U)∩K_i)∪ {x∈X |clB_i(x)⊂U}.

U⁰ is a closed set because K_i is closed, f_i is a continuous map on K_i and the set {x∈X |clBi(x)⊂U} is closed since clBi(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 S_i = T_i and T_i :X → 2^Dⁱ, there exists x ∈ D such that x_i ∈ T_i(x) for each i∈I. Ifx∈Wi for somei∈I, then xi=fi(x), which is a contradiction.

Therefore, x /∈W_i, hence (A_i∩P_i)(x) =∅. Also, for each i∈I we have xi∈Ti(x), and then xi ∈clBi(x).

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Proof of Theorem 6. For each i∈I, let ßi denote the family of all open convex neighborhoods of zero inE_i.LetV = (V_i)i∈I ∈ Q

i∈I

ß_i.Since clB_ihas the e-WCGS property on Ki, there exists a WCG correspondence F_i^Vⁱ :X→2^Xⁱ such that F_i^Vⁱ(x)⊂clBi(x) +Vi andxi∈/ F_i^Vⁱ(x) for each x∈Ki.

K_i is an (n_i−1)-dimensional simplex. Then, by the selection theorem, there exists a continuous function f_i^Vⁱ :K_i → X_i such that f_i^Vⁱ(x) ∈F_i^Vⁱ(x) for eachx∈K_i.Becausex_i∈/ F_i^Vⁱ(x) for eachx∈K_i,we havex_i 6=f_i^Vⁱ(x) for each x∈Ki.

Define the correspondenceT_i^Vⁱ :X →2^Xⁱ by T_i^Vⁱ(x) :=

( {f_i^Vⁱ(x)} ifx∈int_XK_i, cl(B_i(x) +V_i)∩X_i ifx∈X\int_XK_i.

Here, BVi :X→2^Xⁱ, BVi(x) = cl(Bi(x) +Vi)∩Xi = (clBi(x) + clVi)∩Xi is upper semicontinuous by Lemma 2.

LetU be an open subset of X_i. Then U⁰ :={x∈X |T_i^Vⁱ(x)⊂U}

={x∈int_XK_i |T_i^Vⁱ(x)⊂U} ∪ {x∈X\int_XK_i |T_i^Vⁱ(x)⊂U}

=

x∈intXKi|f_i^Vⁱ(x, y)∈U ∪

x∈X |(clBi(x) +Vi)∩Xi ⊂U

= ((f_i^Vⁱ)⁻¹(U)∩int_KK_i)∪

x∈X|(clB_i(x) +V_i)∩X_i⊂U .

U⁰ is an open set because intXKi is open, f_i^Vⁱ is a continuous map on Ki and the set{x∈X |(clB_i(x) + clV_i)∩X_i ⊂U}is open since (clB_i(x) + clV_i)∩X_i is u.s.c. ThenT_i^Vⁱ is upper semicontinuous onXand has closed convex values.

Define T^V :X → 2^X by T^V(x) := Q

i∈I

T_i^Vⁱ(x) for each x ∈X; T^V is an 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 x_V ∈ X such that x_V ∈ T^V(x_V), i.e., (x_V)_i ∈ T_i^Vⁱ(x_V) for each i ∈ I. If x_V ∈ int_XK_i, (x_V)_i = f_i^Vⁱ(x_V), which is a contradiction. Hence (xV)i∈cl(Bi(xV) +Vi)∩Xi and (Ai∩Pi)(xV) =∅,i.e., xV ∈QV where

Q_V =\

i∈I

{x∈X :x_i ∈cl(B_i(x) +V_i)∩X_i and (A_i∩P_i)(x) =∅}.

Since W_i is open, Q_V is the intersection of non-empty closed sets, so it is non-empty, closed in X.

We now prove that the family n

Q_V : V ∈ Q

i∈I

ß_io

has the finite intersection property. Let {V⁽¹⁾, V⁽²⁾, . . . , V⁽ⁿ⁾} be any finite set of Q

i∈I

ß_i and let

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V^(k)= (V_i^(k))i∈I,k= 1, . . . , n.For eachi∈I, let V_i = Tn k=1

V_i^(k). Then V_i ∈ß_i, thusV = (Vi)i∈I∈ Q

i∈I

ßi.Clearly,QV ⊂

n

T

k=1

Q_V(k) so that

n

T

k=1

Q_V(k) 6=∅.Since Xis compact and the familyn

Q_V :V ∈ Q

i∈I

ß_io

has the finite intersection property, we have Tn

Q_V :V ∈ Q

i∈I

ß_io

6= ∅. Take any x ∈ Tn

Q_V : V ∈ Q

i∈I

ß_io . Then x_i ∈ cl(B_i(x) +V_i)∩X_i and (A_i∩P_i)(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 thatx is an equilibrium point of Γ inX.

Example5. 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 such that Xi = [0,4] for each i∈I is a compact convex choice set, X := Q

i∈I

Xi and the correspondences Ai, Pi, Bi :X→2^Xⁱ are defined as

Ai(x) =

( [2,3−₂¹_ix_i] ifx∈M, [0,¹₂]∪[2,4] ifx∈X−M,

Pi(x) =











[2,3] ifx∈M,

[0,¹₂]∪[2,4] ifx= (1,1, . . . ,1),

{2} ifx∈W \(M ∪ {(1,1. . . ,1)}),

∅ otherwise,

Bi(x) =Xi for each x∈X.

Here, W is the simplex with vertices (0,0, . . . ,0),(3,0, . . . ,0),(0,3,0, . . . ,0),and (0,0, . . . ,0,3) inX and M = Q

i∈I

[0,1). Then we have

(A_i∩P_i)(x) =











[2,3− ¹

2ⁱx_i] ifx∈M,

[0,¹₂]∪[2,4] ifx= (1,1, . . . ,1),

{2} ifx∈W \(M∪ {(1,1, . . . ,1)}),

∅ otherwise.

It is easy to show that Ai ∩Pi is not lower semi-continuous or upper semi-cotinuous on X, has not convex values but, since 2 ∈ T

x∈X

S(x), A_i∩P_i is a WCG correspondence on W. Note that x_i ∈/ (A_i∩P_i)(x) for each x ∈ X. Clearly, clBi is lower semicontinuous, has non-empty convex values and

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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 ∈clBi(x) andAi(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 monica.patriche@yahoo.com