EQUILIBRIA OF GENERALIZED GAMES WITH L-MAJORIZED CORRESPONDENCES

In this paper, we shall prove three equilibrium existence theorems for generalized games in Hausdorff topological vector spaces.

correspondences and KF-majorized correspondences and generalized Lemma 4 of Fan [5] to KF- majorized correspondences.Recently, Yannelis and Prabhakar [10] introduced the notions of L- majorized correspondences which generalize KF-majorized correspondences and they obtained an existence theorem of an equilibrium for a compact abstract economy but not with L-majorized preference correspondences.
In this paper, we shall prove existence theorems of equilibria for compact abstract economies with L-majorized correspondences in Hausdorff topological vector space.These results generalize the corresponding results of Borglin-Keiding ([1], Corollaries 2 and 3) with KF-majorized preference correspondences.

PRELIMINARIES.
If A is a set, we shall denote by 2 A the family of all subsets of A. If A is a subset of a topological space X, we denote by clxA the closure of A in X.If A is a subset of a vector space, we shall denote by coA the convex hull of A. Let E be a topological vector space and A,X be non-empty subsets of E. If T:A--2 E and S:A--2 x are correspondences, then coT:A-,2 E and cIS:A---2 x are correspondences defined by (coT)(x)=coT(x),(clS)(x)= clxS(X for each x E A, respectively. Let X be a non-empty subset of a topological vector space.A correspondence :X2 X is said to be of class L [10] if (i) for each x eX, x C_co(z), (ii) for each {z X:y (x)} is open in X.Let :X2 x be a given correspondence and x ff X; then a correspondence Cx:X2 x is said to be an L-majorant of at x [10] if Cx is of class L and there exists an open neighborhood Nx of x in X such that for each z E N,, (z) C ,(z).The correspondence is said to be L-majorized if for each x X with (x) } there exists an L- majorant of at x.
We remark here that the notions of a correspondence of class L and an L-majorized correspondence defined above by Yannelis-Prabhakar in [10] generalize the notions of a KF- correspondence and KF-majorized correspondence, respectively, introduced by Borglin-Keiding [1].These notions have been further generalized in ([2],[9]).
Let I be any set of agents.
We shall need the following which is essentially Lemma 5.1 of Yannelis-Prabhakar [10]" LEMMA 1.Let X be a topological space, Y be a vector space and :X--2 Y be a correspondence such that for each y Y, -'(y) is open in X. Define : X2 " by (x) co (x) for each x X.Then for each y q Y, -l(y) is open in X.
The following maximal element existence result is Theorem 5.1 of Yannelis-Prabhakar [10]" LEMMA 2. Let X be a non-empty compact convex subset of a Hausdorff topological vector space and : X-2 x be a correspondence of class L. Then there exists " X such that 3. EXISTENCE OF EQUILIBRXA FOR L-MAJORIZED PREFEINCE CORRESPONDENCES.
The following result is due to Yannelis-Prabhakar ([10], Corollary .1),which generalizes Lemma 2 to L-majorized correspondence; however they did not give a proof.For completeness, we shall give a proof.THEOREM 1.Let X be a non-empty compact convex subset of a Hausdorff topological vector space and :X-->2 X be an L-majorized correspondence.Then there exists a maximal element E X, i.e., &()-.
PROOF.Suppose that for each x X,(x)# @.Since is L-majorized for each x X, there exist a correspondence :X--2 x of class L and an open neighborhood N of x in X such that for each z N,(z)C (z).The family {N:x X} is an open covering of X, which by the compactness of X, contains a finite subcover {N,:i I}, where I is a finite set.Let {G.:i I} be a closed refinement of {N.,:i I}.For each i I, define a correspondence ,: X--,2 x by J $.,(z), ifzGx,, ,(z) X, if xCG,,.
Let : X---2 x be defined by (z) , i,(z) for each z e X.
The following simple example shows that Theorem is suitable for an L-majorized correspondence, which is not of class L, to assure the existence of a maximal element.EXAMPLE 1.Let X [0,1] and : X--,2 x be defined by {y X:0_< y_< x}, Then is not of class L since -(y)is not open in X for any y {5 (0,1).For any x {5 (0,1), let N, X, an open neighborhood of x in X, and define ,: X---,2 x by {y {5 X:0 _< y _< x}, if z {5 (0,1), (z) Then it is easy to see that . is an L-majorant of at x for each x {5 (0,1), and hence is an L- majorized correspondence.Therefore, by Theorem 1, there exists a maximal element.
As an application of Theorem 1, we shall prove the following existence theorem of equilibrium for an abstract economy with an L-majorized preference correspondence in a Hausdorff topological vector space.
THEOREM 2. Let X be a non-empty compact convex subset of a Hausdorff topological vector space (a choice set).Let A,B:X---2 x be constraint correspondences and P:X--2 x be a preference correspondence satisfying the following conditions: (1) P is L-majorized, (2) for each x {5 X,A(x) is non-empty and co A(x) C B(x), (3) for each y {5 X,A-'(y)is open in X, (4) the correspondence clB:X--,2 x is upper semicontinuous.
Suppose ,/,(x) # 0 for all x E X.Let x X be arbitrarily given.If x F, then Nx X\F is an open neighborhood of x in X such that z coA(z) for all z Nx.Define q.,,:X2 x by / }' if x F, (z) co A(z), Then z co4,(z) for all z X and, by (3) and Lemma 1, -l(y)= (X\F)3 (coA)-(y)is open in X for each y X.It follows that is of class L.Moreover, for each z N,:,,(z)=coA(z) x(z).Thus Cx is an L-majorant of at x. Now suppose that x F. Then (z)=coA(x)91P(x) so that P(x); then by the assumption (1), there exist Cx:X-2 x of class L and an open neighborhood N of x in X such that P(z) C (z) for all z X.
Note that as P(z) C Cx(z) for each z g, we have (z) C (z) for each z Y. Let z X; if z F, by (2), we have z co A(z)= coVe(z) and if z F, then (z) co A(z) Cx(z) C (z) so that z co (z) as z co (z).Hence z co (z) for all z X. Next, for each y X, (%)-(u) {z x: (z)} z f:y Cx(z)} U {z X\F:y (z)} {z F: y [co A(z) D ,(z)]} t.) {z X\F: is open in X by (3) and Lemma 1. Thus is also an L-majorant of at x. Therefore in both cases, is L-majorized.By Theorem 1, there exists a point ' X such that ('' )= }, which is a contradiction.
Hence there must exist a point " X such that () .By (2), we must have clxB(' and co A() 71P() so that A() P() }.This completes the proof.If A has an open graph in X X, then A-X(y) is open in X for each y X (see Corollary   4.1 in [10]).Hence we can obtain Corollary 2 of Borglin-Keiding [1] as an easy consequence of Theorem 2: COROLLARY 1.Let X be a non-empty compact convex subset of a Hausdorff topological vector space and let P,A: X--,2 x be two correspondences satisfying the following conditions: (1) P is L-majorized,