EQUILIBRIUM OF ABSTRACT ECONOMIES IN L.C.-METRIC SPACES

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IN L.C.-METRIC SPACES

MONICA PATRICHE

We establish new existence results for the equilibrium of abstract economies in l.c. (locally convex)-metric spaces. The constraint and preference correspondences are locally-uniformly-weakly lower semicontinuous (w.l.s.c.) or have the local intersection property. Basically, our proofs are based on J.C. Hou’s continuous selection theorem in [12] for w.l.s.c. correspondences and H.L. Zhang’s fixed point theorem in [24]. We use also C.D. Horvath’s fixed point theorem in [8] for Kakutani correspondence in l.c.-metric spaces.

AMS 2000 Subject Classification: 91B52, 91B50, 91A80.

Key words: l.c.-metric space, locally-uniformly-weakly lower semicontinuous correspondence, continuous selection, abstract economy, equilibrium.

1. INTRODUCTION

In game theory, most authors studied the existence of equilibrium for abstract economies with preferences represented as correspondences which have continuity properties. Thus, the results obtained by Shafer and Sonnenschein [17] concern economies with finite dimensional commodity space and preference correspondences having an open graph. Yannelis and Prahbakar [22] 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.

Some authors developed the theory of continuous selections of correspondences and gave numerous applications in game theory. Michael’s selection theorem in [14] is well known and basic in many applications. Browder [5,6] first used a continuous selection theorem to prove the Fan-Browder fixed point theorem.

Later, Yannelis and Prabhakar [22], Ben-El-Mechaiekh [3], Ding, Kim and Tan [7], Horvath [8], Wu [21], Park [14], Yu and Lin [23], and many others established several continuous selection theorems with applications.

MATH. REPORTS11(61),1 (2009), 47–57

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Motivated by Horvath’s papers [9], and [10], Bardaro and Ceppitelli [1, 2]

introduced the notion of H-space, which includes topological vector spaces, convex subsets of normed linear spaces. Many authors have investigated continuous selection theorems, fixed point theorems and existence theorems of equilibrium for abstract economies in H-spaces (see [1–11]).

Hou [12] gave a new continuous selection theorem for locally-uniformly- weakly lower semicontinuous (w.l.s.c.) correspondences in l.c.-metric spaces, that unifies and generalizes known results by Michael [14], Horvath [8], Przes- lawski and Rybinski [16], and Zheng [25]. Zhang [24] used Hou’s selection theorem to prove a fixed point theorem for w.l.s.c. correspondences with closed H-convex values in l.c.-metric spaces. This theorem contains the Brouwer- Schauder fixed point theorem as a special case.

In this paper we prove new equilibrium existence theorems for an abstract economy in l.c.-metric spaces. Using a technique based on a continuous selection, we prove the existence of equilibrium for w.l.s.c. correspondences or correspondences which have the local intersection property. We use also C.D. Horvath’s fixed point theorem in [8] for Kakutani correspondence in l.c.- metric spaces.

The paper is organized in the following way. Section 2 contains preliminaries and notation. The equilibrium theorems are stated and proved in Section 3.

2. PRELIMINARIES AND NOTATION

Throughout this paper, we shall use the following notation and defini- tions:

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)}.

Definition 1. LetX,Y be topological spaces and T :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(y)⊂V for each y∈U.

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2. T is said to be almost upper semicontinuous if for each x ∈ X and each open set V inY with T(x)⊂V there exists an open neighborhood U of x inX such thatT(y)⊂clV for each y∈U.

3. 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.

4. T is said to haveopen lower sections ifT⁻¹(y) :={x∈X:y∈T(x)}

is open in X for each y∈Y.

5. T is said to have the local intersection property if for each x ∈ X with T(x) 6= ∅ there exists an open neighborhood N(x) of x such that T

z∈N(x)T(z)6=∅.

IfT has open lower sections, then T has the local intersection property.

Definition 2. Let X be a topological space and F(X) the family of all nonempty finite subset of X. Let {Γ_A} be a family of some nonempty con- tractible subsets of X indexed by A ∈ F(X) such that Γ_A ⊂ Γ_A⁰ whenever A⊂A⁰.The pair (X,{Γ_A}) is called an H-space.

Definition 3. Given an H-space (X, {Γ_A}), a nonempty subset Dof X is called H-convex if Γ_A⊂Dfor all A∈ F(D).

Definition 4. For a nonempty subset K of X, we define the H-convex hull of K, denoted by H−coK, as H−coK =T{D⊂X :D is H-convex and K ⊂D}.If K=∅,we always considerH−coK =∅.

If {(X_i,{Γ_A_i}_A_i_∈F(X_i₎}_i∈I (I = {1,2. . . , N}) is a family of H-spaces, then the product X=Q

i∈I

X_i can be regarded as an H-space by defining Γ_A= Q

i∈I

Γ_A_i for each A∈ F(X), where A_i denotes the projection of A onto X_i.

Lemma 1 ([19]). The product of any number (finite or infinite) of H- spaces is an H-space and the product ofH-convex subsets is H-convex.

Remark 1. The concept of an l.c.-space is different from that of a locally convex H-space. But an l.c.-space (X,{Γ_A}) with Γ_{x} ={x} for all x ∈X must be a locally convex H-space. Otherwise, a nonempty convex subset X of a locally convex topological space must be an l.c.-space with ΓA= coA for all A∈ F(X), hence (X,{coA}) must be a locally convex H-space.

Definition 5. An l.c.-metric space is a H-space with a metric d such that for anyδ >0,the setS_δ(E) isH-convex wheneverE isH-convex, where S_δ(E) ={x∈X :d(x, E)< δ}, and open balls are H-convex.

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Definition 6 ([12]). LetX be a topological space,Y a metric space with metric d, and F :X →2^Y a correspondence. For r∈R⁺ and x∈X define

F_r(x) ={y∈Y :∃U ∈ϑ(x), ∀x⁰∈U, d(y, F(x⁰))< r}

and F₀(x) = T{F_r(x) :r > 0}, where ϑ(x) denotes a local base of X at the point x; F is called almost lower semicontinuous (l.s.c.) if Fr(x) 6= ∅ for all r ∈R⁺ and x∈X.

F is said to be locally-uniformly-weakly lower semicontinuous (w.l.s.c.) if for any x ∈X there exists an open neighbourhood U(x) of x such that for any >0 and any y∈Y there existsδ >0 such that for any z∈U(x) there exists r >0 such that∅ 6=Sr(y)∩F_δ(z)⊆S(Fµ(z)) for all µ >0.

Proposition 1 ([12]). Let X be a topological space, (Y, d) a metric space, and F :X →2^Y a correspondence. If F is lower semicontinuous, then F is locally-uniformly-weakly lower semicontinuous.

Theorem 1([12]). Let X be a paracompact topological space,(Y,{Γ_A}) an l.c.-metric space, and F :X → 2^Y a locally-weakly lower semicontinuous correspondence with F(x) closed H-convex complete for all x ∈ X. Then F admits a continuous selection.

Theorem 2([24]). Let (E,{Γ_A})be a l.c.-metric space and Xa compact H-convex subset of E. Suppose that F : X → 2^X is a locally uniformly weak lower semicontinuous correspondence with closed H-convex values.

Then, there exists a point x∈X such that x∈F(x).

Corollary 1 ([24]). Let (E,{Γ_A}) be a l.c.-metric space and X a compact H-convex subset of E. Suppose that F : X → 2^X is a lower semicontinuous correspondence with closed H-convex values. Then there exists a point x∈X such that x∈F(x).

Definition 7 ([8]). Given a metric l.c-space (X, d; Γ), a correspondence T :X→X is aKakutani correspondence if

(i)T(x) is a nonempty compact H-convex for eachx∈X;

(ii)T is upper semicontinuous.

Theorem 3([8]). Let (X, d,Γ)be a compact metric l.c.-space such that for any nonempty finite subset A⊆X the set ΓA is closed. Then a Kakutani correspondence T :X →2^X has a fixed point.

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 eachi∈I,

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let Xi be a non-empty topological space representing the set of actions and define X := Q

i∈I

Xi. Let Ai, Bi : X → 2^Xⁱ be the constraint correspondences and Pi the preference correspondences.

Definition 8. A family Γ = (Xi, Ai, Pi, Bi)i∈I is said to be an abstract economy.

Definition 10. Anequilibriumfor Γ is defined as a pointx∈Xsuch that for each i∈I,x_i∈clB_i(x) andA_i(x,)∩P_i(x) =∅.

Remark 2. When A_i(x) = B_i(x) for all x ∈ X and each i ∈ I, this abstract economy model coincides with the classical one introduced by Borglin and Keiding [4] while the definition of an equilibrium coincides with that used by Yannelis and Prabhakar [22].

Yannelis and Prabhakar [22], 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 A_i∩P_i, which also have open lower sections. The theorems we will state use the selection theorems mentioned in Section 2 and a technique based on a continuous selection, as in [25]. We show the existence of equilibrium for an abstract economy assuming that the constraint and the preference correspondences Ai and Pi

are locally-uniformly-weakly lower semicontinuous (w.l.s.c.) or have the local intersection property. The sets Xi are included in the l.c.-metric spaces Ei.

To find the equilibrium point, we use Zhang’s fixed point theorem in [24]

or Horvath’s fixed point theorem in [8].

Theorem 4.Let Γ = (X_i, A_i, B_i, P_i)i∈I be an abstract economy where I is a finite set of agents. For each i∈I let Xi be a compact H-convex subset of an l.c.-metric space (Ei,{Γ_A_i}).Assume that for eachi∈I:

(i)Ai(x)⊂Bi(x) and Bi(x) is nonempty for each x ∈X:= Q

i∈I

Xi; (ii)the set Wi :={x∈X :Ai(x)∩Pi(x)6=∅} is closed;

(iii)Ai∩Pi:Wi →2^Xⁱ is locally-uniformly-weakly lower semicontinuous with closed H -convex complete values;

(iv) clBi is lower semi-continuous with H -convex values;

(v) x_i ∈/ clH−co[A_i(x)∩P_i(x)] for each x ∈W_i.

Then there exists a point x ∈ X such that x_i ∈ cl(B_i(x)) and A_i(x)∩ P_i(x) =∅ for all i∈I.

Proof. For each i∈ I, Wi is closed in a compact set, so that it is paracompact. Hence, by (ii) and (iii) and Theorem 1, there exists a continuous selection for the correspondence Ai∩Pi :Wi → 2^Xⁱ, i.e., there exists a continuous function fi :Wi →Xi such that fi(x)∈H−co[Ai(x)∩Pi(x)] for all

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x∈Wi. Define the correspondence Gi :X→2^Xⁱ by

G_i(x) =

( {f_i(x)} ifx∈Wi, cl(Bi(x)) ifx∈X\Wi.

Then Gi : X → 2^Xⁱ is a correspondence with nonempty closed H-convex values by (iv).

LetV be a closed set in Xi.Let

U ={x∈X:G_i(x)⊂V}={x∈W_i:f_i(x)∈V} ∪ {x∈X\W_i : cl(B_i(x))⊂V}

={x∈W_i :f_i(x)∈V} ∪ {x∈X : cl(B_i(x))⊂V}.

The set {x ∈W_i :f_i(x) ∈V} is closed in W_i. It also is closed inX because Wi is closed inX. Then U is closed inX by (iii) and (iv) and the continuity of fi. This shows that Gi : X → 2^Xⁱ is lower semicontinuous, and then the correspondence G:X → 2^X defined by G(x) = Q

i∈I

Gi(x) for each x ∈X, is lower semicontinuous and has nonempty compact H-convex values.

By Corollary 1, there exists a point x ∈Gi(x) for all i ∈I. By (v), we have xi∈cl(Bi(x)) and Ai(x)∩Pi(x) =∅ for all i∈I.

Since a lower semicontinuous correspondenceA_i∩P_i :X→2^Y is locally- uniformly-weakly lower semicontinuous, we obtain the result below.

Corollary 2. Let Γ = (X_i, A_i, B_i, P_i)i∈I be an abstract economy, where I is a finite set of agents. For each i∈I let X_i be a compact H-convex subset of an l.c.-metric space (Ei,{Γ_A_i}).Assume that for each i∈I:

(i)Ai(x)⊂Bi(x) and Bi(x) is nonempty for each x ∈X := Q

i∈I

Xi; (ii)the set Wi :={x∈X:Ai(x)∩Pi(x)6=∅}is closed;

(iii) Ai∩Pi : Wi → 2^Xⁱ is lower semicontinuous with closed H -convex complete values;

If the correspondencesA_i, P_i :W_i →2^Xⁱ are lower semicontinuous, then we have

Corollary 3.Let Γ = (Xi, Ai, Bi, Pi)i∈I be an abstract economy,where I is a finite set of agents. For each i∈I let Xi be a compact H-convex subset of an l.c.-metric space (Ei,{Γ_A_i}).Assume that for each i∈I:

(i)A_i(x)⊂B_i(x) and B_i(x) is nonempty for each x ∈X:= Q

i∈I

X_i; (ii)the set Wi :={x∈X:Ai(x)∩Pi(x)6=∅}is closed;

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(iii) Ai ∩Pi : Wi → 2^Xⁱ has open lower sections and closed H -convex complete values;

(iv) clB_i is lower semi-continuous with convex values;

(v) xi ∈/ clH−co[Ai(x)∩Pi(x)] for each x ∈Wi.

Corollarly 4. Let Γ = (Xi, Ai, Bi, Pi)i∈I be an abstract economy, where I is a finite set of agents.For each i∈I let X_i be a compact H-convex subset of an l.c.-metric (E_i,{Γ_A_i}). Assume that for each i∈I:

i∈I

Xi; (ii)the set Wi :={x∈X:Ai(x)∩Pi(x)6=∅}is closed;

(iii) Ai ∩Pi : Wi → 2^Xⁱ has open lower sections and closed H -convex complete values;

(iv) clBi is lower semi-continuous with convex values;

Then there exists a point x ∈ X such that x_i ∈ cl(B_i(x)) and A_i(x)∩ Pi(x) =∅ for all i∈I.

We now show the existence of an equilibrium for an abstract economy assuming that the constraint and the preference correspondences A_i and P_i have the local intersection property. The setsX_iare included in the l.c.-metric spaces Ei.

Theorem 5. Let Γ = (X_i, A_i, B_i, P_i)i∈I be an abstract economy, where I is a finite set of agents. For each i∈I let X_i be a compact H-convex subset of an l.c.-metric space (Ei,{Γ_A_i}).Assume that for each i∈I:

(i)A_i(x)⊂B_i(x) and B_i(x) is nonempty for each x ∈X:= Q

i∈I

X_i; (ii)the set Wi :={x∈X:Ai(x)∩Pi(x)6=∅}is closed;

(iii) A_i ∩P_i : W_i → 2^Xⁱ has the local intersection property and closed H -convex complete values;

Then there exists a point x ∈ X such that xi ∈ cl(Bi(x)) and Ai(x)∩ Pi(x) =∅ for all i∈I.

Proof. Since A_i ∩P_i : W_i → 2^Xⁱ has the local intersection property, by (ii) and (iii) and Theorem 1 there exists a continuous selection for the correspondence Ai∩Pi, i.e., there exists a continuous functionfi :Wi → Xi

such that f_i(x)∈ H−co[A_i(x)∩P_i(x)] for allx ∈W_i and the proof is from now on similar to that of Theorem 4.

We now give a new equilibrium theorem in l.c.-metric spaces for abstract economies with correspondences A_i∩P_i being locally-uniformly-weakly lower semicontinuous and the correspondences B_i being upper semicontinuous.

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Theorem 6. Let Γ = (Xi, Ai, Bi, Pi)i∈I be an abstract economy, where I is a finite set of agents. For each i∈I let X_i be a compact H-convex subset of an l.c.-metric space (Ei,{Γ_A_i}) such that for any nonempty finite subset A⊆Y the set Γ_A is closed. Assume that for each i∈I:

(i)A_i(x)⊂B_i(x) and B_i(x) is nonempty for each x ∈X := Q

i∈I

X_i; (ii)the set Wi :={x∈X:Ai(x)∩Pi(x)6=∅}is paracompact, open;

(iii)A_i∩P_i:W_i →2^Xⁱ is locally-uniformly-weakly lower semicontinuous with closed H -convex complete values;

(iv)B_iis upper semi-continuous and clB_i(x) isH -convex for each x∈X;

Then there exists a point x ∈ X such that xi ∈ cl(Bi(x)) and Ai(x)∩ P_i(x) =∅ for all i∈I.

Proof. For each i ∈ I, by (ii) and (iii) and Theorem 1, there exists a continuous selection for the correspondenceA_i∩P_i:W_i →2^Xⁱ,i.e., there exists a continuous function fi : Wi → Xi such that fi(x) ∈H −co[Ai(x)∩Pi(x)]

for all x∈Wi. Define the correspondence Gi :X→2^Xⁱ by

G_i(x) =

( {f_i(x)} ifx∈W_i, cl(Bi(x)) ifx∈X\W_i.

Then G_i : X → 2^Xⁱ is a correspondence with nonempty closed H-convex values by (iv).

LetV be an open set inX_i.The set

{x∈X:Gi(x)⊂V}={x∈Wi :fi(x)∈V} ∪ {x∈X\W_i : cl(Bi(x))⊂V}

={x∈W_i:f_i(x)∈V} ∪ {x∈X: cl(B_i(x))⊂V}

is open in X by (iii) and (iv) and the continuity of fi. This shows that Gi : X →2^Xⁱ is upper semicontinuous, and then the correspondenceG:X →2^X defined by G(x) = Q

i∈I

Gi(x) for each x∈X, is upper semicontinuous and has nonempty compact H-convex values.

By Theorem 3, there exists a point x ∈ Gi(x) for all i ∈ I. By (v), we have xi∈cl(Bi(x)) and Ai(x)∩Pi(x) =∅ for all i∈I.

If the correspondencesB_iare almost upper semicontinuous, then we have Corollary 5. Let Γ = (Xi, Ai, Bi, Pi)i∈I be an abstract economy where I is a finite set of agents. For each i∈I let X_i be a compact H-convex subset of an l.c.-metric space (Ei,{Γ_A_i}) such that for any nonempty finite subset A⊆Y the set Γ_A is closed. Assume that for each i∈I:

i∈I

X_i; (ii)the set Wi :={x∈X:Ai(x)∩Pi(x)6=∅} is paracompact, open;

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(iii)Ai∩Pi:Wi →2^Xⁱ is locally-uniformly-weakly lower semicontinuous with closed H -convex complete values;

(iv) Bi is almost upper semi-continuous and clBi(x) is H -convex for each x∈X;

Proof. For each i∈I, clBi :X →2^Xⁱ is upper semicontinuous by (iv), hence the conclusion follows by Theorem 6.

Corollarly 6. Let Γ = (Xi, Ai, Bi, Pi)i∈I be an abstract economy, where I is a finite set of agents. For each i ∈ I, let X_i be a compact H- convex subset of an l.c.-metric space (Ei,{Γ_A_i}) such that for each nonempty finite subset A⊆Y the set F(A) is closed. Assume that for each i∈I:

i∈I

X_i; (ii)the set Wi :={x∈X:Ai(x)∩Pi(x)6=∅}is paracompact, open;

(iii) A_i∩P_i :W_i → 2^Xⁱ is lower semicontinuous with closed H -convex complete values;

(iv)B_i is upper semi-continuous and clB_i(x) isH -convex for each x∈X;

Proof. SinceA_i∩P_i :W_i→2^Xⁱis lower semicontinuous,A_i∩P_iis locally- uniformly w.l.s.c. by Proposition 1. Hence, by (ii) and (iii) and Theorem 1, there exists a continuous selection for the correspondence Ai∩Pi, i.e., there exists a continuous function f_i : W_i → X_i such that f_i(x) ∈ H−co[A_i(x)∩ Pi(x)] for all x ∈ Wi and the proof is from now on similar to that of Theo-

rem 6.

Corollarly 7. Let Γ = (Xi, Ai, Bi, Pi)i∈I be an abstract economy, where I is a finite set of agents. For each i ∈ I let X_i be a compact H- convex subset of an l.c.-metric space (E_i,{Γ_A_i}) such that for each for any nonempty finite subset A ⊆ Y the set ΓA is closed. Assume that for each i∈I:

i∈I

Xi; (ii)the set Wi :={x∈X:Ai(x)∩Pi(x)6=∅} is paracompact, open;

(iii) Ai, Pi : Wi → 2^Xⁱ have lower open sections and closed H -convex complete values;

(iv)B_i is upper semi-continuous and clB_i(x) isH -convex for each x∈X;

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Received 16 July 2008 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

014700 Bucharest, Romania monica.patriche@yahoo.ro