Nonlinear Analysis 62 (2005) 1127–1136www.elsevier.com/locate/na

Stability of solutions for Ky Fan’s section theoremwith some applications�

Yong-hui Zhoua,∗, Shu-wen Xiangb, Hui YangbaDepartment of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, ChinabDepartment of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China

Received 1 January 2005; accepted 20 January 2005

Abstract

In this paper, we prove that “most of” problems in Ky Fan’s section theorem (in the sense ofBaire category) are essential and that for any problem in Ky Fan’s section theorem, there exists atleast one essential component of its solution set. As applications, we deduce both the existence ofessential components of the set of Ky Fan’s points based on Ky Fan’s minimax inequality theoremand the existence of essential components of the set of Nash equilibrium points for generaln-personnon-cooperative games with non-concave payoffs.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Ky Fan’s section theorem; Ky Fan’s point; Nash equilibrium; Essential point; Essential component;Collective-better-reply

1. Introduction

In [6], Fort first introduced the notion of essential fixed points of a continuous mappingf from a compact metric spaceX into itself and proved that any mappingf can be approxi-mated closely by amapping whose fixed points are all essential. Owing to the non-existenceof essential fixed points even for the identity mapping, Kinoshita[9] then introduced thenotion of essential components of the set of fixed points and proved that for any continuous

� Supported by NSCF (10061002) and Science Foundations of Guizhou Province, China.∗ Corresponding author. Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China.E-mail address:Yonghuizhou@163.com(Y.-H. Zhou).

0362-546X/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2005.01.113

1128 Y.-H. Zhou et al. / Nonlinear Analysis 62 (2005) 1127–1136

mapping from the Hilbert cube into itself, there is at least one essential component of theset of its fixed points.Inspired by their work,Wu and Jiang[14] introduced the notion of essential Nash equilib-

rium points for finiten-person non-cooperative games (briefly, finite games) and proved thatany finite game can be closely approximated by a game whose Nash equilibrium points areall essential. Later, Jiang[8] introduced the notion of essential components of the set of Nashequilibrium points and proved that for any finite game, there is at least one essential compo-nent of the set of its Nash equilibrium points. Moreover, Kohlberg and Mertens[11] arguedthat for finite games a satisfactory solution concept should be therefore called an essentialcomponent of the set of Nash equilibrium points and they proved that for any finite gamethere are finite components of the set of Nash equilibrium points, at least one of which isessential.On the other hand, Tan et al.[12] first defined Ky Fan’s points which is based on Ky

Fan’s minimax inequality theorem of[4], and they studied the stability of Ky Fan’s points,especially, essential Ky Fan’s points. Yu and Luo[16] as well as Yu and Xiang[18] thenintroduced the notion of essential components of the set of Ky Fan’s points. By usingdifferent methods, they both established the existence of essential components of the set ofKy Fan’s points for concave functions and deduced the existence of essential componentsof the set of Nash equilibrium points for general non-cooperative games with concavepayoffs.As we know, Ky Fan’s minimax inequality theorem is equivalent to the following result

proved by Ky Fan[3,4], later called Ky Fan’s section theorem, which plays a very importantrole in non-linear analysis (e.g., seeYuan[20]):

Ky Fan’s section theorem. Let X be a non-empty compact convex subset of a Hausdorffvector space, E be a subset ofX ×X such that

(i) for each fixedx ∈ X, (x, x) ∈ E;(ii) for each fixedx ∈ X, {y ∈ X, (x, y) ∈ E} is closed;(iii) for each fixedy ∈ X, {x ∈ X, (x, y) /∈E} is convex(possibly empty).

Then the set⋂

x∈X{y ∈ X, (x, y) ∈ E} �= ∅

In this paper, we will study the stability of the solution set⋂

x∈X{y ∈ X, (x, y) ∈ E}with varyingE in a normed vector space, whereE is called a problem in Ky Fan’s sectiontheorem. We prove that “most of” problems in Ky Fan’s section theorem are essential (inthe sense of Baire category) and also that any problem in Ky Fan’s section theorem hasat least one essential component of the set of its solutions. As applications, we deduceboth the existence of essential components of the set of Ky Fan’s points (not only forconcave functions) and the existence of essential components of the set of Nash equilibriumpoints for generaln-person non-cooperative games with non-concave (including concave)payoffs. We also show that our results are independent of the results in[16,18]. See theauthors’other paper[21] on essential components of the set of solutions of Ky Fan’s Lemma(which asks for the closedness ofE in Ky Fan’s section theorem) with application to gametheory.

Y.-H. Zhou et al. / Nonlinear Analysis 62 (2005) 1127–1136 1129

2. Preliminary

We first introduce the notions of essential points, essential sets and essential componentsof solution sets for some kind of problems (such as problems in Ky Fan’s section theorem,Ky Fan’s point problems, Nash equilibrium problems and fixed point problems) in a unifiedpattern; more details are provided in[17]. In this section,X is a non-empty subset of ametric space, 2X andK(X) denote the space of all non-empty subsets ofX and the spaceof all non-empty compact subsets ofX, respectively, both endowed with Vietoris topology(see[10]).Let (P, d) be a metric space of some kind of problems andX be the space of solutions

for the problems inP. For any problemp ∈ P , S(p) denotes the set of all solutions inXof the problemp. Thus, a set-valued mappingS:P → 2X, called solution mapping, is welldefined. Letp be inP.

Definition 2.1. A solutionx ∈ S(p) is called an essential point if for any open neighbor-hoodN(x) of x in X, there is a�>0 such that for anyp′ ∈ P with d(p, p′)< �, S(p′) ∩N(x) �= ∅. The problemp is called essential (resp., weakly essential) if all its solutions areessential (resp. there is a solution which is essential).

Definition 2.2. A non-empty closed subsete(p) ⊂ S(p) is called an essential set if for anyopensubsetO inXwithO ⊃ e(x), there isa�>0such that foranyp′ ∈ P withd(p, p′)< �,S(p′) ∩O �= ∅. An essential subsetm(p) ⊂ S(p) is called a minimal essential set if it is aminimal element of the family of essential sets ordered by set inclusion.

If S(p) ∈ K(X), then it can be written as

S(p)=⋃�∈�

C�(p),

where� is an index set,C�(p) is a non-empty compact connected component subset ofS(p) for any� ∈ � andC�(p) ∩ C�(p)= ∅, for any�,� ∈ � with � �= �, see[2, p. 365].

Definition 2.3. A connected componentC�(p) ⊂ S(p) is called an essential component iffor any open subsetO in XwithO ⊃ C�(p), there is a�>0 such that for anyp′ ∈ P withd(p, p′)< �, S(p′) ∩O �= ∅.

Recall thatS:P → 2X is called (i) upper (resp., lower) semicontinuous atp ∈ P if forany open subsetO in X with O ⊃ S(p) (resp.,O ∩ S(p) �= ∅), there is a�>0 such thatfor anyp′ ∈ P with d(p, p′)< �,O ⊃ S(p′) (resp.,O ∩ S(p′ �= ∅)); (ii) Sis called a uscomapping ifSis upper semicontinuous with non-empty compact values at eachp ∈ P . Also,recall that a subsetR ⊂ P is called a residual set if it contains a countable intersection ofopen dense subsets ofP. Then, ifP is complete, any residual subset inPmust be a densesubset inP (then for a dense residual, there are “most of” points inP in the sense of Bairecategory).

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Remark 2.1. It is easy to see that the problemp ∈ P is essential if and only if the solutionmappingS:P → 2X is lower semicontinuous atp.

We also need the following three results which are due to Fort[6, Theorem 2], Yu et al.[17, Theorem 3.1(1) where condition(c) is unnecessary] andYang et al.[15, Lemma 3.3],respectively:

Lemma 2.1. If P is complete andS:P → K(X) is a usco mapping, then the set of pointswhere S is lower semicontinuous is a dense residual set in P.

Lemma 2.2. If S:P → K(X) is a usco mapping, then for eachp ∈ P , there exists atleast one minimal essential set ofS(p).

Lemma 2.3. LetX, Y,Z be three metric spaces, S1:Y → 2X and S2:Z → 2X be twoset-valued mappings. Suppose that there exists at least one essential component ofS1(y)

for eachy ∈ Y and there exists a continuous single-valued mappingT :Z → Y such thatS2(z) ⊃ S1(T (z)) for eachz ∈ Z. Then, there exists at least one essential component ofS2(z) for eachz ∈ Z.

3. Stability of solutions in Ky Fan’s section theorem

LetXbe a non-empty compact convex subset of a normed vector space,E be the set{E ⊂X × X:E satisfies the conditions (i), (ii) and (iii) of Ky Fan’s section theorem}. For eachE ∈ E, it may be regarded as a section mapping ofE such thatE(x)={y ∈ X, (x, y) ∈ E}for all x ∈ X. So the set

⋂x∈XE(x) = ⋂

x∈X{y ∈ X, (x, y) ∈ E}, denoted byS(E), isnon-empty and, in fact, compact. So a solution mappingS:E → K(X) is well defined. ForeachE1, E2 ∈ E we define

�s(E1, E2)= supx∈X

h(E1(x), E2(x)),

whereh is Hausdorff distance. Clearly,(E,�s) is a metric space. In fact, it is complete, thatis, we have the following lemma:

Lemma 3.1. (E,�s) is a complete metric space.

Proof. Let {En ∈ E} be any Cauchy sequence, then for any�>0, there is ann0 suchthat for anyn,m�n0, �s(E

n,Em)< �, or, supx∈X h(En(x), Em(x))< �. It follows thatfor eachx ∈ X, {En(x)} is a Cauchy sequence inK(X). SinceX is compact, it iscomplete, soK(X) is complete by[10, Theorem 4.3.9]; thus, there is a set-valued map-ping E:X → K(X) such thath(En(x), E(x)) → 0 for eachx ∈ X. Since for anyn,m�n0, supx∈X h(En(x), Em(x))< �, so h(En(x), Em(x))< � for eachx ∈ X, andh(En(x), E(x))< �by the continuity ofh; hence supx∈X h(En(x), E(x))< �. Solimn→+∞supx∈X h(En(x), E(x))< �,and thus limn→+∞ supx∈X h(En(x), E(x))=0 by the arbitraryof �. We also denoteE = {(x, y), y ∈ E(x), x ∈ X}, thenEn → E under the metric�s .

Y.-H. Zhou et al. / Nonlinear Analysis 62 (2005) 1127–1136 1131

If we have thatE ∈ E, then the proof is complete.For eachx ∈ X, since(x, x) ∈ En for eachn, x ∈ En(x); and sinceh(En(x), E(x)) →

0, x ∈ E(x) and so(x, x) ∈ E. Note that for eachx ∈ X,E(x) is compact and thenclosed, we only need to prove that for eachy ∈ X, {x ∈ X, (x, y) /∈E} is convex (possiblyempty).Suppose not, then there arey0, x1, x2 ∈ X and a real number� with 0< �<1 such

that x1, x2 ∈ {x ∈ X, (x, y0) /∈E} but x0 = �x1 + (1− �)x2 /∈ {x ∈ X, (x, y0) /∈E}, or,y0 ∈ E(x0). Then, for i = 1,2, y0 /∈E(xi), and soy0 /∈En(xi) for all n large enoughsinceh(En(xi), E(xi)) → 0. Hencexi ∈ {x ∈ X, (x, y0) /∈En}. Therefore,x0 is inthe convex set{x ∈ X, (x, y0) /∈En} for En ∈ E, so y0 /∈En(x0). Thus, y0 /∈E(x0)sinceh(En(x0), E(x0)) → 0, a contradiction. Hence,E must be inE, and (E,�) iscomplete. �

Lemma 3.2. The solution mappingS:E → K(X) is a usco mapping.

Proof. SinceS(E) is compact for allE ∈ E, we only need to prove thatS is uppersemicontinuous.SupposeSwere not upper semicontinuous at someE ∈ E, then there exist an open

subsetO of X with O ⊃ S(E) and a sequence{En ∈ E} with En −→ E such that foreveryn = 1,2, . . ., there existsyn ∈ S(En) with yn /∈O. Without loss of generality, wemay assume thatyn −→ y ∈ X sinceX is compact. Soy /∈O, andy /∈ S(E).On the other hand, for eachx ∈ X, yn ∈ En(x), yn ∈ S(En). SinceEn −→ E,

h(En(x), E(x)) → 0, and there is a cluster pointy ∈ E(x) by [12, Lemma 3.3]. Therefore,y = y by the uniqueness of limit. Hence,y ∈ E(x) for eachx ∈ X, i.e., y ∈ S(E), acontradiction. �

Theorem 3.1. There exists a dense residual subsetR of E such that for eachE ∈ E, Eis essential. In other words, there are“most of” the problems in Ky Fan’s section theoremwhose solutions are all essential.

Proof. Since the metric space(E,�s) is complete by Lemma 3.1 and the solution mappingS:E → K(X) is usco by Lemma 3.2, by Lemma 2.1 there is a dense residual subsetRof E whereS is lower semicontinuous, and thus by Remark 2.1 for eachE ∈ R, E isessential. �

Remark 3.1. (1) By the proof of Theorem 3.1,S is continuous on the dense residualRof E. Let E ∈ R, then for any�>0 there is a�>0 such that for anyE′ ∈ E with�s(E,E

′)< �, h(S(E), S(E′))< �, henceE is stable. (2) SinceR is dense inE, anyproblemp ∈ E can be closely approximated by a problem inE whose solutions are allessential.

However,R �= E. Consider the following example from[19].

Example 3.1. LetXbe [0,1] andEbeX×X. ThenE ∈ E andS(E)=[0,1]. Let us checkthat anyy ∈ S(E) is not essential. (1) IfEn={(x, y) : y ∈ [0,1]\(x, x+1/n), x ∈ [0,1]},

1132 Y.-H. Zhou et al. / Nonlinear Analysis 62 (2005) 1127–1136

thenEn ∈ E with En −→ E andS(En)={0}, so anyy ∈ (0,1] is not an essential solutionof E (2) If En={(x, y) : y ∈ [0,1]\(x−1/n, x), x ∈ [0,1]}, thenEn ∈ EwithEn −→ E

andS(En)={1}, so anyy ∈ [0,1) is not an essential solution ofE. Therefore, any solutiony ∈ S(E) is not essential.

Instead of essential points, let us consider essential sets.

Theorem 3.2. For eachE ∈ E, there exists at least one connectedminimal essential subsetof S(E).

Proof. By Lemma 3.2,S:E → K(X) is a usco mapping, so by Lemma 2.2, there exists aminimal essential subset ofS(E) for eachE ∈ E, and let us denote it bym(E).Supposem(E) were not connected. Then, there exist two non-empty compact subsets

c1(E), c2(E) withm(E)= c1(E)∪ c2(E), and there exist two disjoint open subsetsV1, V2in X such thatV1 ⊃ c1(E), V2 ⊃ c2(E) and inf{d(x, y): x ∈ V1, y ∈ V2} = �>0.On the one hand, sincem(E) is essential, forV1∪V2 ⊃ m(E) there exists�∗ >0(�∗ < �)

such thatS(E′) ∩ (V1 ∪ V2) �= ∅ with �s(E,E′)< �∗ for anyE′ ∈ E.

On the other hand, sincem(E) is a minimal essential subset ofS(E), neitherc1(E)nor c2(E) is essential. So for�∗/4>0, there exist two elements,E1, E2 ∈ E such that�s(E,E1)< �∗/4, �s(E,E2)< �∗/4, butS(E1) ∩ V1 = ∅ andS(E2) ∩ V2 = ∅. Clearly,�s(E1, E2)< �∗/2.We define

E′ = {(x, y): y ∈ (E1(x)\V2) ∪ (E2(x)\V1), for all x ∈ X}.Then, if we prove that (I)E′ ∈ E; (II) �s(E,E

′)< �∗ and (III) S(E′)∩ (V1∪V2)= ∅, thenwe get a contradiction with the fact thatm(E) is essential, and the proof is complete.

Checking (I)E′ ∈ E:For eachx ∈ X, sinceEi ∈ E, x ∈ Ei(x) andEi(x) is closed fori = 1,2; and since the

two open setsV1 andV2 are disjoint,x ∈ E′(x), i.e.,(x, x) ∈ E′, andE′(x)=(E1(x)\V2)∪(E2(x)\V1) is closed.Now, we prove that for eachy ∈ X, {x ∈ X : y /∈E′(x)} is convex. Note thatV1∩V2=∅

and (1) ify ∈ V1, then{x ∈ X: y /∈E′(x)} = {x ∈ X: y /∈E1(x)}; (2) if y ∈ V2, then{x ∈X: y /∈E′(x)}={x ∈ X: y /∈E2(x)}; (3) if y ∈ X\(V1∪V2), then{x ∈ X: y /∈E′(x)}={x ∈X: y /∈E1(x)} ∩ {x ∈ X: y /∈E2(x)}, so{x ∈ X: y /∈E′(x)} is convex, forE1, E2 ∈ E suchthat{x ∈ X: y /∈E1(x)} and{x ∈ X: y /∈E2(x)}are both convex.Checking (II)�s(E,E

′)< �∗:Note thatE′(x)= (E1(x)\V2) ∪ (E2(x)\V1) for all x ∈ X.First, we have that sup{d(y,E1(x)): y ∈ E′(x))}< �∗/2, since (1) ify ∈ E1(x)\V2, then

d(y,E1(x))=0; (2) if y ∈ E2(x)\V1, thend(y,E1(x))�h(E2(x), E1(x))��s(E1, E2)<

�∗/2.Second, we have that sup{d(y,E′(x)) : y ∈ E1(x)}< �∗/2, since (1) ify ∈ E1(x)\V2

thend(y,E′(x))=0; (2) if y ∈ E1(x)∩V2, theny ∈ V2, for eachz ∈ V1,d(y, z)��> �∗/2,observing thatd(y,E2(x))�h(E1(x), E2(x))��s(E1, E2)< �∗/2, sod(y,E2(x)\V1)=d(y,E2(x)), and thus,d(y,E′(x))�d(y,E2(x)\V1)< �∗/2.

Y.-H. Zhou et al. / Nonlinear Analysis 62 (2005) 1127–1136 1133

By the two steps above,h(E1(x), E′(x))=max{sup{d(y,E1(x)): y ∈ E′(x)}, sup{d(y,

E′(x)): y ∈ E1(x)}}< �∗/2.By the arbitrary ofx, �s(E1, E

′)< �∗/2, we have�s(E,E′)��s(E,E1)+�s(E1, E′)<

�∗/4+ �∗/2< �∗.Checking (III)S(E′) ∩ (V1 ∪ V2)= ∅:Suppose thatS(E′)∩(V1∪V2) �= ∅. SinceV1∩V2=∅, wemay suppose thatS(E′)∩V1 �=

∅. Then, there is somey ∈ S(E′)∩V1, so for eachx ∈ X, y ∈ [(E1(x)\V2)∪(E2(x)\V1)]∩V1 = E1(x) ∩ V1, y ∈ S(E1) ∩ V1, which contradicts the fact thatS(E1) ∩ V1 = ∅. �

Theorem 3.3. For eachE ∈ E, there exists at least one essential component ofS(E).

Proof. ByTheorem3.2, thereexistsat least oneconnectedminimal essential subsetm(E)ofS(E). So by[2, Theorem 3.2, p. 112], there is a componentCof S(E) such thatm(E) ⊂ C.It is obvious thatC is essential. �

A proof analogous to that of[18, Theorem 3.4]gives us the following results and isthus omitted.

Theorem 3.4. (1) If E ∈ E such thatS(E) is a totally disconnected set, then E is weaklyessential.(2) If E ∈ E such thatS(E) is a singleton set, then E is essential.

4. Application (I): Essential component of the set of Ky Fan’s points

LetXbeanon-empty compact convex subset of a normedvector space,�be the collectionof all functions:X × X → R such that (i) for each fixedx ∈ X, y → (x, y) is lowersemicontinuous; (ii) for each fixedy ∈ X, x → (x, y) is quasi-concave; (iii)(x, x)�0for all x ∈ X. For each ∈ �, we denoteS1() = {y ∈ X,(x, y)�0 for allx ∈X}, thenS1() is non-empty and compact, and the points inS1() are called Ky Fan’spoints of, see Tan et al.[12]. And then a solution mappingS1:� → K(X) is welldefined.LetE ={(x, y) ∈ X×X:(x, y)�0} for each ∈ �. Then it is easy to check thatE

is inE, the definition ofE in Section 3. So there exists a single-valued mappingT1:� → Esuch thatT1()= E.For, ∈ �, we define

�1(,)= �s(E, E).

Theorem 4.1. For each ∈ �, there exists at least one essential component of the setS1() of Ky Fan’s points.

Proof. SinceT1:� → E is an isometric mapping such thatT1()= E, it is continuous.It is easy to see thatS1()= S(E)= S(T1()). By Theorem 3.3, there exists at least oneessential component ofS(E) for eachE ∈ E, so by Lemma 2.3, there exists at least oneessential component ofS1(), for each ∈ �. �

1134 Y.-H. Zhou et al. / Nonlinear Analysis 62 (2005) 1127–1136

Similar to Theorem 3.4, we have the following results:

Theorem 4.2. (1) If ∈ � such thatS1() is a totally disconnected set, then ∈ � isweakly essential. (2) If ∈ � such thatS1() is a singleton set, then is essential.

Remark 4.1. Here Theorem 4.1 is independent of[16, Theorem 2.2]and[18, Theorem3.3]. There are two reasons

(1) The perturbation spaceM in [16, Theorem 2.2]and[18, Theorem 3.3]is the collectionof all functions:E × E → R such that (i) for each fixedx ∈ X, y → (x, y) is lowersemicontinuous; (ii) for each fixedy ∈ X, x → (x, y) is concave; (iii)(x, x)�0for all x ∈ X and (iv) sup(x,y)∈E×E |(x, y)|< + ∞. Clearly,M ⊂ �, in other words,Theorem 4.1, here, holds for the functions onX×X such that for each fixedy ∈ X, x →(x, y) is quasi-concave, not only concave, and that the uniform boundary condition (iv)is unnecessary.(2) Our notion of essential components based on the metric�1 is different from those in

[16,18]whicharebothbasedon theuniformmetric�, i.e.,�(,)=sup(x,y)∈X×X|(x, y)−(x, y)| for , ∈ M. In fact, the metric�1 is neither stronger nor weaker than the metric� even in the same spaceM. Consider the following example.

Example 4.1. LetX=[0,1],(x, y)=0 for all (x, y) ∈ X×X. Then ∈ M,E=X×X.

On the one hand, forn = 1,2, . . . , definen(x, y) = 1 for all (x, y) ∈ X × X. Thenn ∈ M andEn = X × X, and so{n} converges to under the metric�1 while it doesnot converge to under�.On the other hand, forn= 1,2, . . . , definen(x, y)= (1/n)x − (1/n)y for all (x, y) ∈

X × X. Then,n ∈ M andEn = {(x, y) ∈ X × X : y ∈ [x,1], x ∈ [0,1]}, and so{n}converges to under� while it does not converge to under�1.

5. Application (II): Essential component of the set of Nash equilibrium points

Let I = {1, . . . , n} be a set of players, ann-person non-cooperative game� is a 2n-tuple(X1, X2, . . . , Xn; f1, f2, . . . , fn), where for each playeri ∈ I the non-empty setXi

is his(her) strategy set andfi :X = ∏ni=1Xi → R is his(her) payoff function. For each

i ∈ N , let us denoteX−i = ∏j∈I\{i}Xj . If x = (x1, . . . , xn) ∈ X, we shall notex−i =

(x1, . . . , xi−1, xi+1, . . . , xn) ∈ X−i . x∗ = (x∗1, . . . , x

∗n) ∈ X is called a Nash equilibrium

point if for eachi ∈ I ,

fi(x∗i , x

∗−i )= maxzi∈Xi

fi(zi, x∗−i ).

Let us introduce collective-better-reply correspondence for the game�, which is basedon the aggregate function,U :X ×X → R given by

U(x, y)=n∑i=1

fi(xi, y−i ).

Y.-H. Zhou et al. / Nonlinear Analysis 62 (2005) 1127–1136 1135

SinceU(y, y) = ∑ni=1fi(yi, y−i ) may be regarded as the payoff of an agent for the

collective consisting of all players when the agent takes a strategy profiley ∈ X, andU(x, y) = ∑n

i=1fi(xi, y−i ) may be regarded as the payoff of the agent for the collectivewhen each player unilaterally deviates fromyi to xi giveny−i , so if

U(x, y)�U(y, y),

thenwesay thaty is a collective-better-reply strategy againstx ∈ X (it is said that a deviationprofilex upsets a candidatey if U(x, y)<U(y, y)in [1]).For everyx ∈ X, CBR�(x) denotes the set of all collective-better-reply strategies of the

agent for the collective against the deviation profilex in the game�. Thus, a set-valuedmappingCBR� fromX into itself such that for allx ∈ X,

CBR�(x)= {y ∈ X:U(x, y)�U(y, y)}is well defined; and just in this sense, we call it collective-better-reply correspondence of thegame�.We denote the graph of collective-better-reply correspondence for the game� alsoby CBR�, i.e.,CBR� = {(x, y) ∈ X × X:U(x, y) − U(y, y)�0, x ∈ X}. The followinglemma is obvious, or see[18, Lemma 4.2]:

Lemma 5.1. y∗ is a Nash equilibrium of game� if and only ify∗ ∈ ⋂x∈XCBR�(x).

Now we establish the existence of essential connected components of Nash equilibriumsets for general non-cooperative games with non-concave payoffs.LetGbe the set of all games satisfying the following (1) for eachi ∈ I ,Xi beanon-empty

compact convex subset of a normed vector spaceLi ; (2)∑n

i=1fi is upper semicontinuousonX; (3) for eachi ∈ I and each fixedui ∈ Xi, fi(ui, ·) is lower semicontinuous onX−i ;(4) for each fixedu ∈ X, �fi(·, u−i ) is quasi-concave onX. Then, by Ky Fan’s sectiontheorem (or see[1,5,13]), the set of Nash equilibrium points of� is non-empty and compactfor each� ∈ G, denoted byN(�). And also a solution mappingN :G → K(X) is welldefined.It is easy to check that for each� ∈ G, the graph of the collective-better-reply corre-

spondenceCBR� is in E in Section 3. There is a single-valued mappingT2:G → E suchthatT2(�)= CBR�, and thenN(�)= S(CBR�)=N(T2(�) by Lemma 5.1.For�1,�2 ∈ G, we define

�2(�1,�2)= �s(CBR�1,CBR�2)= supx∈X

h(CBR�1(x),CBR�2(x)).

Similar to the discussion of the existence of essential components of the set of Ky Fan’spoints in Section 4, we have the following results:

Theorem 5.1. For each game� ∈ G, there exists at least one essential component ofN(�).

Theorem 5.2. (1) If � ∈ G such thatN(�) is a totally disconnected set, then� is weaklyessential. (2) If � ∈ G such thatN(�) is a singleton set, then� is essential.

1136 Y.-H. Zhou et al. / Nonlinear Analysis 62 (2005) 1127–1136

Remark 5.1. The two reasons for us to introduce the notion of collective-better-reply cor-respondences for the games inG are as follows:

(1) Both the methods in[16,18], which are only suitable for the game spaceC whichasks for games with concave and uniform bounded payoffs, are invalid for the games inGin Theorems 5.1 and 5.2; in fact,C ⊂ G, also see Remark 4.1(2) Although for finiten-person games, Hillas[7] took advantage of perturbations of the

best reply correspondencesbyusco convex-valued correspondences to defineasolution con-cept which satisfies all the requirements Kohlberg and Mertens[11] proposed, his methodis invalid for the infiniten-person games� in G here whose best-reply correspondences,i.e., for allx ∈ X

BR(x)= n×i=1BRi (x)= n×

i=1

{yi ∈ Xi : fi(yi, x−i )= max

zi∈Xi

fi(zi, x−i )},

need not be usco mappings.

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