MONICA PATRICHE
G. Debreu’s model of abstract economy was extended in the last years by several authors. In a paper from 2003, W. Kim proved an existence theorem of equilibrium for a generalized quasi-game with an infinite number of agents. The purpose of this paper is to obtain new existence theorems of equilibrium in a generalized quasi-game.
AMS 2000 Subject Classification: 91B52, 91B50, 91A10.
Key words: quasi fixed-point theorem, generalized quasi-game, equilibrium, con- tinuous selection.
1. INTRODUCTION
The pioneering works of G. Debreu on the existence of equilibrium in a generalized N-person game (see [4]) or on abstract economy (see [1]) were ex- tended by several authors. Shafer and Sonnenschein [16] proved the existence of equilibrium of an economy with finite dimensional commodity space and irreflexive preferences reprezented as correspondences with open graph.
Within different framework (countable infinite number of agents, infi- nite dimensional strategy spaces), Yannelis and Prahbakar [19] developed new techniques based on selection theorems and fixed-point theorems. Their main result concerns the existence of equilibrium when the constraint and preference correspondences have open lower sections.
A question raised in their paper was whether the assumptions that the correspondences have open sections can be weakened. The answer is affirma- tive: for Banach spaces we can use selection theorems for lower semicontinuous correspondences while for locally convex spaces we can use the approximation method developed by Yuan [20].
Following the line in [19], Zhou [22] and Zheng [23] obtained new exis- tence theorems based on continuous selections.
In most results on the existence of equilibria for abstract economies, the choice sets are compact and convex. However, several authors (for example, Ding [5], Yuan [21], Ding and Tan [6]) have considered the underlying spaces to be noncompact.
MATH. REPORTS10(60),2 (2008), 185–195
In 1976, Borglin and Keiding [3] used for their existence results the new concepts of K.F.-correspondence and KF-majorized correspondence. The sec- ond notion was extended by Yannelis and Prabhakar [19] to L-majorized cor- respondences. In 2001, Liu and Cai [13] introduced the notion ofQ-majorized correspondence and gave a new existence theorem of a maximal element. As its applications, they obtained some new existence theorems of an abstract economy.
In the last years, many authors generalized the classical model of abstract economy. For example, Vind [18] defined the social system with coordination, Yuan [20] proposed the model of the general abstract economy. Motivated by the fact that any preference of a real agent could be unstable by the fuzziness of consumers’ behaviour or market situation, Kim and Tan [10] defined the generalized abstract economies.
Also, Kim [9] obtained a generalization of the quasi fixed-point theorem due to Lefebvre [12] and, as an application, he proved an existence theorem of equilibrium for a generalized quasi-game with an infinite number of agents.
Kim’s result concerns generalized quasi-games where the strategy sets are metrizable subsets in linear topological convex spaces.
In this paper, we prove an equilibrium existence theorem for a non- compact generalized quasi-game that extends Kim’s theorem. Then we con- sider the case when the strategy sets are subsets of Banach space, the number of agents is infinite and the correspondences are lower semicontinuous. In order to prove our results we use different continuous selection theorem than Kim does.
The paper is organized in the following way: Section 2 contains prelim- inaries and notations. The model of the generalized quasi-game and Kim’s main result are presented in Section 3; a version of Kim’s fixed-point theorem appears in Section 4. The equilibrium theorems are stated in Section 5 while their proofs are collected in Section 6.
2. PRELIMINARIES AND NOTATION
Throughout this paper, we shall use the following notation and defini- tions. Let A be a subset of a topological spaceX.
1. 2A 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→2X are correspondences, then coT, clT,T∩F :A→2X are correspondences defined by (coT)(x) = coT(x), (clT)(x) = clT(x) and (T∩F)(x) =T(x)∩F(x) for eachx∈A, respectively.
Definition 1. Let X, Y be topological spaces and T : X → 2Y be 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.
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 neighborhood U of x in X such thatT(y)∩V 6=∅ for each y∈U.
3. T is said to becontinuousifT is both upper semicontinuous and lower semicontinuous.
Definition 2. A normal topological space in wich each open set is anFσ
is called perfectly normal.
An Fσ set in a paracompact set is also paracompact.
Definition 3 ([13]). Let X be a topological space, Y a nonempty subset of a vector spaceE,θ:X→E a mapping andT :X →2Y a correspondence.
(1)T is said to be of class Qθ (or Q) if (a) for each x∈X,θ(x)∈/ clT(x), and
(b) T is lower semicontinuous with open and convex values in Y.
(2) A correspondenceTx :X →2Y is said to be a Qθ-majorant of T at x if there exists an open neighborhoodN(x) ofx such that
(a) for each z∈N(x),T(z)⊂Tx(z) and θ(z)∈/ clTx(z), (b) Tx is l.s.c. with open and convex values.
(3)T is said to be Qθ-majorized if for each x∈X with T(x) 6=∅ there exists a Qθ-majorant Tx ofT atx.
Lemma1 ([13]). Let X be a regular paracompact topological vector space and Y a nonempty subset of a vector space E. Let θ :X → E be a mapping and T : X → 2Y \ {∅} a Qθ-majorized correspondence. Then there exists a correspondence F : X → 2Y of class Qθ such that T(x) ⊂ F(x) for each x∈X.
The following continuous selection theorems and the Himmelberg fixed point theorem are essential in proving our main results.
Lemma2 ([20]). Let X be a paracompact space, Y a Banach space and T : X → 2Y a lower semicontinuous closed convex valued correspondence.
Let F : X → 2Y be a correspondence whose graph is open in X ×Y such that T(x)∩F(x) 6=∅ for all x∈X. Then there is a continuous single-valued mapping f :X →Y such that f(x)∈co(T(x)∩F(x)) for all x∈X.
Lemma 3 ([19]). Let X be a non-empty paracompact Hausdorff topolo- gical space and Y a Hausdorff topological vector space. Let T :X →2Y be a
correspondence such that each T(x) is non-empty convex and, for each y ∈Y, T−1(y) is open in X. Then T has a continuous selection, i.e., there exists a continuous map f :X →Y such that f(x)∈T(x) for each x∈X.
Theorem1 ([8]). Let X be a nonvoid convex subset of a separated locally convex space L. Let T : X → X be a u.s.c. correspondence such that T(x) is closed and convex for all x ∈X, and T(X) is contained in some compact subset C of X. Then T has a fixed point.
3. THE MODEL
In this paper, we study the following model of a generalized quasi-game.
Definition 4. Let I be a nonempty set (the set of agents). For each i ∈ I, let Xi be a non-empty topological vector space representing the set of actions and define X := Q
i∈I
Xi; let Ai, Bi : X ×X → 2Xi be the con- straint correspondences and Pi the preference correspondence. Ageneralized quasi-game Γ = (Xi, Ai, Bi, Pi)i∈I is defined as a family of ordered quadruples (Xi, Ai, Bi, Pi). In particular, when I = {1,2, . . . , n}, Γ is called a n-person quasi-game.
Definition 5. An equilibrium for Γ is defined as a point (x, y)∈X×X such that yi ∈clBi(x, y) and Ai(x, y)∩Pi(x, y) =∅for each i∈I.
If Ai(x, y) = Bi(x, y) for each (x, y) ∈ X×X and i ∈ I, this model coincides with that introduced by Kim [9].
If, in addition, for eachi∈I,Ai, Pi are constant with respect to the first argument, this model coincides with the classical model of abstract economy and the definition of equilibrium is that given in [3]. In this work, W.K. Kim established an existence result for a generalized quasi-game with a possibly uncountable set of agents, in a locally convex Hausdorff topological vector space. Here is his result.
Theorem2. Let Γ = (Xi, Ai, Pi)i∈I be a generalized quasi-game and set X := Q
i∈I
Xi and Z := X ×X. Assume I is a (possibly uncountable) set of agents such that, for each i∈I,
(1)Xi is a non-empty compact convex subset of a Hausdorff locally convex space Ei;
(2) the correspondence Ai :X×X →2Xi is upper semicontinuous such that Ai(x, y) is a non-empty convex subset of Xi for each (x, y)∈Z;
(3)A−1i (xi) is (possibly empty) open for each xi ∈Xi;
(4) the correspondence Pi : Z → 2Xi is such that (Ai ∩Pi)−1(xi) is (possibly empty) open for each xi∈Xi;
(5)the set Wi : ={(x, y)∈Z |(Ai∩Pi) (x, y)6=∅} is perfectly normal;
(6)for each (x, y)∈Wi, xi ∈/coPi(x, y).
Then there exists an equilibrium point (x, y) ∈X×X for Γ, i.e., yi ∈ clAi(x, y) and Ai(x, y)∩Pi(x, y) =∅ for each i∈I.
4. A GENERALIZED FIXED-POINT THEOREM
In order to prove the existence theorems of equilibria for a generalized quasi-game, we need the following new version of Kim’s quasi fixed-point the- orem.
Theorem3. Let I and J be any (possible uncountable) index sets. For each i∈I and j∈J, let Xi and Yj be non-empty compact convex subsets of Hausdorff locally convex spaces Ei and Fj, respectively.
Let X :=Q
Xi,Y := Q
i∈I
Yj and Z :=X×Y. For each i∈I let Φi :Z → 2Xi be a correspondence such that the set Wi ={(x, y)∈Z |Φi(x, y)6=∅} is open and Φi has a continuous selection fion Wi. For each j∈J let Ψj :Z → 2Yj be an upper semicontinuous correspondence with non-empty closed convex values.
Then there exists a point (x, y) ∈ Z such that for each i ∈ I, either Φi(x, y) =∅ or xi ∈Φi(x, y) and yj ∈Ψj(x, y) for eachj∈J.
Proof of Theorem 3. We first endow Q
i∈I
Ei and Q
j∈J
Fj with the product topologies. Then Q
i∈I
Ei× Q
j∈J
Fj also is a locally convex Hausdorff topological vector space.
For eachi∈I, define a correspondence Φ0i :Z →2Xi by Φ0i(x, y) :=
( {fi(x, y)} if (x, y)∈Wi, Xi if (x, y)∈/ Wi.
Then for each (x, y)∈Z, Φ0i(x, y) is a non-empty closed convex subset ofXi. Also, Φ0i is an upper semicontinuous correspondence on Z. In fact, for each proper open subset V of Xi, we have
U :={(x, y)∈Z |Φ0i(x, y)⊂V}
={(x, y)∈Wi|Φ0i(x, y)⊂V ∪(x, y)∈Z\Wi|Φ0i(x, y)⊂V}
={(x, y)∈Wi|fi(x, y)∈V} ∪ {(x, y)∈Z\Wi |Xi⊂V}
={(x, y)∈Wi|fi(x, y)∈V}=fi−1(V)∩Wi.
Since Wi is open andfi is a continuous map on Wi, the set U is open, hence Φ0i is upper semicontinuous on Z.
Finally, we define a correspondence Φ :Z →2Z by Φ(x, y) :=Y
i∈I
Φ0i(x, y)×Y
j∈J
Ψj(x, y) for each (x, y)∈Z.
Then, by Lemma 3 in [7], Φ is an upper semicontinuous correspondence such that each Φ(x, y) is a non-empty closed convex. Therefore, by the Fan- Glicksebrg fixed point theorem, there exists a fixed point (x,y)∈Z such that (x, y) ∈Φ(x, y), i.e., xi ∈ Φ0i(x, y) for each i∈ I, and yj ∈ Ψj(x, y) for each j ∈ J. If (x, y) ∈ Wi for some i ∈ I, then xi = fi(x, y) ∈ Φi(x, y); and if (x, y) ∈/ Wi for some i ∈ I, then Φi(x, y) = ∅. Therefore, for each i ∈ I, either Φi(x, y) = ∅ or xi ∈ Φi(x, y). Also, for each j ∈ J, we already have yj ∈Ψj(x, y). This completes the proofs.
5. EQUILIBRIUM THEOREMS
Our first result extends the existence theorem of Kim to the non-compact generalized quasi-game.
Theorem 4. Let Γ = (Xi, Ai, Bi, Pi)i∈I be a generalized quasi-game where I is a (possibly uncountable) set of agents such that, for each i∈I,
(1)Xi is a non-empty convex subset of a Hausdorff locally convex space;
(2) clBi is upper semicontinuous;
(3) Ai is a non-empty convex valued and A−1i (xi) is (possibly empty) open in Z for each xi ∈Xi;
(4) (Ai∩Pi)−1(xi) is a (possibly empty) open for every xi∈Xi; (5)there exists a non-empty compact convex set Di ⊂Xi such that:
a) clBi(x, y)∩Di is non-empty and convex, andco (Ai∩Pi) (x, y)⊂ clBi(x, y)∩Di for every (x, y)∈Z;
b) xi ∈/coPi(x, y) for each (x, y)∈D×D (here, D =Q
i∈I
Di);
(6)the set Wi: ={(x, y)∈Z |(Ai∩Pi) (x, y)6=∅}is paracompact, per- fectly normal.
Then there exists an equilibrium point (x, y) ∈D×D for Γ, i.e., yi ∈ clBi(x, y) and Ai(x, y)∩Pi(x, y) =∅for each i∈I.
Our results on Banach spaces are given in the following two theorems.
Theorem 5. Let Γ = (Xi, Ai, Bi, Pi)i∈I be a generalized quasi-game where I is a (possibly uncountable) set of agents such that, for each i∈I,
(1)Xi is a non-empty compact convex subset in a Banach space;
(2)Ai is lower semicontinuous with convex closed values;
(3) Bi is non-empty and convex valued, clBi is upper semicontinuous and Ai(x, y)⊆Bi(x, y) for every (x, y)∈Z;
(4)Pi has open graph;
(5)the set Wi: ={(x, y)∈Z / (Ai∩Pi) (x, y)6=∅} is open;
(6)xi∈/ coPi(x, y) for every (x, y)∈Wi.
Then there exists an equilibrium point (x, y) ∈ Z for Γ, i.e., yi ∈ clBi(x, y) and Ai(x, y)∩Pi(x, y) =∅for each i∈I.
Theorem 6. Let Γ = (Xi, Ai, Bi, Pi)i∈I be a generalized quasi-game where I is a (possibly uncountable) set of agents such that, for each i∈I,
(1) Xi is a non-empty convex subset of a Banach space and there exists a compact convex subset Di ofXi containing all values of the correspondences Ai, Bi andPi;
(2)Ai has open graph and convex values;
(3)Biis upper semicontinuous with non-empty convex values and Ai(x, y)
⊆Bi(x, y) for every (x, y)∈Z;
(4)Pi is Qθ-majorized with non-empty values;
(5)the set Wi :={(x, y)∈Z |(Ai∩Pi)(x, y)6=∅} is open.
Then there exists an equilibrium point (x, y) ∈D×D for Γ, i.e., xi ∈ Bi(x, y) and Ai(x, y)∩Pi(x, y) =∅ for each i∈I.
6. PROOFS
Proof of Theorem 4. For each i ∈ I, we first define a correspondence Φi :Z →2Di by
Φi(x, y) =
co(Ai∩Pi)(x, y) if (x, y)∈Wi,
∅ if (x, y)∈/Wi.
Note that Φi(x, y)⊂Di because co(Ai∩Pi)(x, y)⊂clBi(x))∩Di ⊂Di. The correspondenceAi∩Pi has open lower sections, therefore the corre- spondence co(Ai∩Pi) has open lower sections (see Lemma 5.1 in [19]); Φi has convex values included in Di and open lower sections.
The setWi = S
xi∈Xi
(Ai∩Pi)−1(xi) is open by assumption (4).
For eachxi ∈Di, Φ−1i (xi) = [co(Ai∩Pi)]−1(xi)∩Wi is open.
For each j ∈ I, define Ψj : Z → 2Dj by Ψj(x, y) := clBj(x, y)∩Dj if (x, y) ∈ Z. Then Ψj is upper semicontinuous with non-empty convex and closed values.
Since the setWi is paracompact, by applying Lemma 3 to the restriction of Φi on Wi, we deduce that Φi/Wi : Wi → 2Di has a continuous selection fi :Wi→2Di, i.e., fi(x, y)⊂Φi(x, y) for each (x, y)∈Wi.
For eachi∈I, define the correspondence Φ0i:Z →2Di, by Φ0i(x, y) :=
{fi(x, y)} if (x, y)∈Wi, Di if (x, y)∈/ Wi.
Then for each (x, y)∈Z, Φ0i(x, y) is a non-empty closed convex subset ofDi. Moreover, Φ0i is upper semicontinuous on Z.
LetV be an open subset ofDi. Then, U :={(x, y)∈Z|Φ0i(x, y)⊂V}
={(x, y)∈Wi |Φ0i(x, y)⊂V} ∪ {(x, y)∈Z\Wi|Φ0i(x, y)⊂V}
={(x, y)∈Wi |fi(x, y)∈V} ∪ {(x, y)∈Z\Wi |Di ⊂V}
={(x, y)∈Wi |fi(x, y)∈V}=fi−1(V)∩Wi.
U is an open set, becauseWi is open and fi is a continuous map onWi. Define Φ : Z → 2D by Φ(x, y) := Q
i∈I
Φ0i(x, y) × Q
j∈I
Ψj(x, y) for each (x, y)∈Z. By Lemma 3 in [7], Φ is an upper semicontinuous correspondence and also has non-empty convex closed values.
Since Z is a convex set and D is compact, by the Himmelberg fixed- point theorem, there exists (x, y) ∈ D×D such that (x, y) ∈ Φ(x, y), i.e., xi∈Φ0i(x, y) for eachi∈I, andyj ∈Ψj(x, y) for eachj ∈J. If (x, y)∈Wi for somei∈I, thenxi =fi(x, y)∈co(Ai∩Pi)(x, y), which contradicts assumption 5b). Therefore, (x, y) ∈/ Wi, hence Φ(x, y) = Φ and (Ai ∩Pi)(x, y) = Φ.
Also, for each i ∈I, we have yi ∈Ψi(x, y), and then yi ∈ clBi(x, y)∩Di ⊂ clBi(x, y).
Proof of Theorem 5. For eachi∈I, define Φi :Z →2Xi by Φi(x, y) =
( co(Ai∩Pi)(x, y) if (x, y)∈Wi,
∅ if (x, y)∈/Wi;
SinceZ is metrizable, it is perfectly normal and paracompact. HenceWi as an open set in Z is aFσ. Because everyFσ subset of a paracompact space is paracompact, Wi is paracompact.
Xi is a Banach space, the restriction Pi/Wi :Wi → 2Xi has open graph, Ai/Wi : Wi → 2Xi is lower semicontinuous with closed convex values and Ai(x, y)∩Pi(x, y)6= Φ for each (x, y)∈Wi. Then, by applying Lemma 2 to the restrictionsAiandPionWi, we deduce that there exists a continuous selection fi :Wi→Xi such that fi(x, y)∈co(Ai∩Pi)(x, y) for each (x, y)∈Wi.
For each j ∈ I, define Ψj :Z → 2Xi by Ψj(x, y) = clBj(x, y) for each (x, y)∈Z. Then Ψjis an upper semicontinuous correspondence and Ψj(x, y) is a non-empty, convex, closed subset of Xj for each (x, y)∈Z. By Theorem 2, there exists (x, y) ∈ Z such that, for each i ∈ I, either Φi(x, y) = ∅ or xi∈Φi(x, y) and yj ∈Ψj(x, y) for each j∈J.
If xi ∈ Φi(x, y) for some i∈I, then xi ∈Φi(x, y) = co(Ai∩Pi)(x, y)⊂ coPi(x, y) which contradicts assumption (6). Therefore, for each i ∈ I,
Φi(x, y) =∅and then (x, y)∈/ Wi. Hence (Ai∩Pi)(x, y) =∅andy∈Ψi(x, y) = clBi(x, y) for each i∈I.
Proof of Theorem 6. Every metrizable space is perfectly normal and paracompact ([14]). Therefore, we can apply Lemma 1 to the Qθ-majorized correspondences Pi:Z →2Di,i∈I.
Let θ be pr: Z → X. By Lemma 1, for each i ∈ I there exists a corre- spondence Fi : Z → 2Di of class Qθ such that Pi (x, y) ⊂ Fi(x, y) for each (x, y)∈Z. SinceFi is of class Qθ, we have
(a) for each (x, y)∈Z,xi∈clF/ i(x, y);
(b)Fi is l.s.c. with open and convex values in Di.
For each i∈ I, define the correspondence Ti : Wi → 2Di by Ti(x, y) = Ai(x, y)∩clFi(x, y) for each (x, y)∈Wi.
The correspondence clFi is lower semicontinuous because Fi is lower semicontinuous.
We know thatAi :Wi →2Di has open graph and convex values, clFi : Wi →2Di is a lower semicontinuous, closed convex valued correspondence. For each (x, y)∈Wi we haveAi(x, y)∩Pi(x, y)6=∅.Since Pi (x, y)⊂Fi(x, y) for each (x, y)∈Z, we deduce thatAi(x, y)∩clFi(x, y)6=∅for each (x, y)∈Wi. Because every Fσ subset of a paracompact space is paracompact (see Proposition 3 in [14]) andWi is open in a perfectly normal space,Wi is para- compact.
By applying Lemma 2 to Ai and clFi on Wi, we deduce that there exists a continuous selection fi :Wi → Di such that fi(x, y) ∈ co(Ai(x, y)∩ clFi(x, y) =Ai(x, y)∩clFi(x, y)⊂Ai(x, y) for each (x, y)∈Wi.
For j ∈I define Ψj :Z → 2Dj, by Ψj(x, y) = clBj(x, y) for (x, y) ∈Z. Then Ψj is an upper semicontinuous correspondence such that Ψj(x, y) is a non-empty, convex closed set for each (x, y)∈Z.
For eachi∈I define Φ0i:Z →2Di, by Φ0i(x, y) :=
( {fi(x, y)} if (x, y)∈Wi, Di if (x, y)∈/ Wi.
For each (x, y) ∈Z, Φ0i(x, y) is nonempty, convex, closed in Di and Φ0i is an upper semicontinuous correspondence. Denote D := Q
i∈I
Di and define the correspondence Φ : Z → 2D, by Φ(x, y) = Q
i∈I
Φ0i(x, y)× Q
i∈I
Ψj(x, y) for each (x, y)∈Z. Since Φ :Z →2D is upper semicontinuous by Lemma 3 in [7] and is non-empty convex closed valued, we can apply the Himmelberg theorem and obtain that there exists (x, y)∈Φ(x, y). Thenxi ∈Φ0i(x, y) for eachi∈I and yj ∈Ψj(x, y) = clBj(x, y) for eachj∈J. Sincefi(x, y)⊂Ai(x, y)∩clFi(x, y)
and xi ∈/ clFi(x, y), we have xi ∈/ fi(x, y). It follows that xi ∈ Di and (x, y) ∈/ Wi, for each i ∈ I. Then Ai(x, y)∩Pi(x, y) = ∅. We also have yj ∈clBj(x, y) for eachj ∈I. Then (x, y) is an equilibrium point for Γ.
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Received 11 December 2007 University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest, Romania monica.patriche@yahoo.com
