Volume 2010, Article ID 234706,23pages doi:10.1155/2010/234706
Research Article
Generalizations of the Nash Equilibrium Theorem in the KKM Theory
Sehie Park
1, 21The National Academy of Sciences, Seoul 137-044, Republic of Korea
2Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea
Correspondence should be addressed to Sehie Park,shpark@math.snu.ac.kr Received 5 December 2009; Accepted 2 February 2010
Academic Editor: Anthony To Ming Lau
Copyrightq2010 Sehie Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. In this paper, we derive generalized forms of the Ky Fan minimax inequality, the von Neumann-Sion minimax theorem, the von Neumann-Fan intersection theorem, the Fan-type analytic alternative, and the Nash equilibrium theorem for abstract convex spaces satisfying the partial KKM principle. These results are compared with previously known cases forG-convex spaces. Consequently, our results unify and generalize most of previously known particular cases of the same nature. Finally, we add some detailed historical remarks on related topics.
1. Introduction
In 1928, John von Neumann found his celebrated minimax theorem 1 and, in 1937, his intersection lemma 2, which was intended to establish easily his minimax theorem and his theorem on optimal balanced growth paths. In 1941, Kakutani3obtained a fixed point theorem for multimaps, from which von Neumann’s minimax theorem and intersection lemma were easily deduced.
In 1950, John Nash4,5established his celebrated equilibrium theorem by applying the Brouwer or the Kakutani fixed point theorem. In 1952, Fan 6 and Glicksberg 7 extended Kakutani’s theorem to locally convex Hausdorfftopological vector spaces, and Fan generalized the von Neumann intersection lemma by applying his own fixed point theorem.
In 1972, Himmelberg8obtained two generalizations of Fan’s fixed point theorem6and applied them to generalize the von Neumann minimax theorem by following Kakutani’s method in3.
In 1961, Ky Fan9obtained his KKM lemma and, in 196410, applied it to another intersection theorem for a finite family of sets having convex sections. This was applied in 196611to a proof of the Nash equilibrium theorem. This is the origin of the application of the KKM theory to the Nash theorem. In 1969, Ma12extended Fan’s intersection theorem
10to infinite families and applied it to an analytic formulation of Fan type and to the Nash theorem for arbitrary families.
Note that all of the above results are mainly concerned with convex subsets of topological vector spaces; see Granas 13. Later, many authors tried to generalize them to various types of abstract convex spaces. The present author also extended them in our previous works 14–28 in various directions. In fact, the author had developed theory of generalized convex spaces simply, G-convex spaces related to the KKM theory and analytical fixed point theory. In the framework of G-convex spaces, we obtained some minimax theorems and the Nash equilibrium theorems in our previous works17,18,21,22, based on coincidence theorems or intersection theorems for finite families of sets, and in22, based on continuous selection theorems for the Fan-Browder maps.
In our recent works24–26, we studied the foundations of the KKM theory on abstract convex spaces. The partial KKM principle for an abstract convex space is an abstract form of the classical KKM theorem. A KKM space is an abstract convex space satisfying the partial KKM principle and its “open” version. We noticed that many important results in the KKM theory are closely related to KKM spaces or spaces satisfying the partial KKM principle.
Moreover, a number of such results are equivalent to each other.
On the other hand, some other authors studied particular types of KKM spaces and deduced some Nash-type equilibrium theorem from the corresponding partial KKM principle, for example,17,21,29–33, explicitly, and many more in the literature, implicitly.
Therefore, in order to avoid unnecessary repetitions for each particular type of KKM spaces, it would be necessary to state clearly them for spaces satisfying the partial KKM principle.
This was simply done in27.
In this paper, we study several stages of such developments from the KKM principle to the Nash theorem and related results within the frame of the KKM theory of abstract convex spaces. In fact, we clearly show that a sequence of statements from the partial KKM principle to the Nash equilibria can be obtained for any space satisfying the partial KKM principle.
This unifies previously known several proper examples of such sequences for particular types of KKM spaces. More precisely, our aim in this paper is to obtain generalized forms of the KKM space versions of known results due to von Neumann, Sion, Nash, Fan, Ma, and many followers. These results are mainly obtained by1fixed point method,2continuous selection method, or 3 the KKM method. In this paper, we follow method 3 and will compare our results to corresponding ones already obtained by method2.
In Section 2, we state basic facts and examples of abstract convex spaces in our previous works24–26.Section 3deals with a characterization of the partial KKM principle and shows that such principle is equivalent to the generalized Fan-Browder fixed point theorem. In Section 4, we deduce a general Fan-type minimax inequality from the partial KKM principle.Section 5deals with various von Neumann-Sion-type minimax theorems for abstract convex spaces.
InSection 6, a collective fixed point theorem is deduced as a generalization of the Fan- Browder fixed point theorem.Section 7deals with the Fan-type intersection theorems for sets with convex sections in product abstract convex spaces satisfying the partial KKM principle.
In Section 8, we deduce a Fan-type analytic alternative and its consequences. Section 9 is devoted to various generalizations of the Nash equilibrium theorem and their consequences.
Finally, inSection 10, some known results related to the Nash theorem and historical remarks are added.
This paper is a revised and extended version of22,27and a supplement to24–26, where some other topics on abstract convex spaces can be found.
2. Abstract Convex Spaces and the KKM Spaces
Multimaps are also called simply maps. LetDdenote the set of all nonempty finite subsets of a setD. Recall the following in24–26.
Definition 2.1. An abstract convex spaceE, D;Γconsists of a topological spaceE, a nonempty setD, and a multimapΓ:DEwith nonempty valuesΓA : ΓAforA∈ D.
For anyD⊂D, theΓ-convex hull ofDis denoted and defined by coΓD:
ΓA |A∈ D
⊂E. 2.1
A subsetXofEis called aΓ-convex subset ofE, D;Γrelative toDif for anyN∈ D we have thatΓN⊂X, that is, coΓD⊂X.
WhenD ⊂E, the space is denoted byE⊃D;Γ. In such case, a subsetXofEis said to beΓ-convex if coΓX∩D⊂X; in other words,XisΓ-convex relative toD:X∩D. In case ED, letE;Γ: E, E;Γ.
Example 2.2. The following are known examples of abstract convex spaces.
1A tripleΔn ⊃V; cois given for the original KKM theorem34, whereΔnis the standardn-simplex,V is the set of its vertices{ei}ni0, and co:V Δnis the convex hull operation.
2A tripleX ⊃ D;Γis given, whereXandD are subsets of a t.v.s.Esuch that co D⊂XandΓ:co. Fan’s celebrated KKM lemma9is forE⊃D; co.
3A convex spaceX⊃D;Γis a triple whereXis a subset of a vector space such that coD⊂X, and eachΓAis the convex hull ofA∈ Dequipped with the Euclidean topology.
This concept generalizes the one due to Lassonde forX D; see35. However he obtained several KKM-type theorems w.r.t.X⊃D;Γ.
4A tripleX ⊃D;Γ,is called anH-space ifXis a topological space andΓ {ΓA}is a family of contractibleor, more generally,ω-connectedsubsets ofX indexed byA ∈ D such thatΓA⊂ΓBwheneverA⊂B∈ D. IfDX, thenX;Γ: X, X;Γis called ac-space by Horvath36,37.
5Hyperconvex metric spaces due to Aronszajn and Panitchpakdi are particular cases ofc-spaces; see37.
6 Hyperbolic spaces due to Reich and Shafrir 38 are also particular cases of c-spaces. This class of metric spaces contains all normed vector spaces, all Hadamard manifolds, the Hilbert ball with the hyperbolic metric, and others. Note that an arbitrary product of hyperbolic spaces is also hyperbolic.
7Any topological semilatticeX,≤with path-connected interval is introduced by Horvath and Llinares39.
8A generalized convex space or a G-convex spaceX, D;Γ due to Park is an abstract convex space such that for each A ∈ D with the cardinality |A| n1 there exists a continuous functionφA:Δn → ΓAsuch thatJ∈ Aimplies thatφAΔJ⊂ΓJ.
Here,ΔJ is the face ofΔncorresponding toJ∈ A, that is, ifA{a0, a1, . . . , an}and J{ai0, ai1, . . . , aik} ⊂A, thenΔJ co{ei0, ei1, . . . , eik}.
For details, see references of17,21,22,40–42.
9 A φA-space X, D;{φA}A∈D consists of a topological space X, a nonempty set D, and a family of continuous functionsφA : Δn → X that is, singular n-simplexes for A ∈ Dwith |A| n1. EveryφA-space can be made into aG-convex space; see43.
Recently φA-spaces are calledGFC-spaces in44and FC-spaces43 or simplicial spaces 45whenX D.
10Suppose thatX is a closed convex subset of a completeR-treeH, and for each A∈ X,ΓA :convHA, where convHAis the intersection of all closed convex subsets of Hthat containA; see Kirk and Panyanak46. ThenH⊃X;Γis an abstract convex space.
11A topological spaceX with a convexity in the sense of Horvath47is another example.
12AB-space due to Briec and Horvath30is an abstract convex space.
Note that each of 2–12 has a large number of concrete examples and that all examples1–9areG-convex spaces.
Definition 2.3. LetE, D;Γbe an abstract convex space. If a multimapG:DEsatisfies ΓA⊂GA:
y∈A
G y
∀A∈ D, 2.2
thenGis called a KKM map.
Definition 2.4. The partial KKM principle for an abstract convex spaceE, D;Γis the statement that, for any closed-valued KKM map G : D E, the family {Gy}y∈D has the finite intersection property. The KKM principle is the statement that the same property also holds for any open-valued KKM map.
An abstract convex space is called a KKM space if it satisfies the KKM principle.
In our recent works24–26, we studied the foundations of the KKM theory on abstract convex spaces and noticed that many important results therein are related to the partial KKM principle.
Example 2.5. We give examples of KKM spaces as follows.
1EveryG-convex space is a KKM space18.
2A connected linearly ordered spaceX,≤can be made into a KKM space26.
3The extended long lineL∗is a KKM spaceL∗, D;Γwith the ordinal spaceD : 0,Ω; see26. ButL∗is not aG-convex space.
4For a closed convex subsetXof a completeR-treeH, andΓA :convHAfor each A∈ X, the tripleH ⊃X;Γsatisfies the partial KKM principle; see46. Later we found thatH⊃X;Γis a KKM space48.
5Horvath’s convex spaceX;Γwith the weak Van de Vel property is a KKM space, whereΓA: Afor eachA∈ X; see47,48.
6AB-space due to Briec and Horvath30is a KKM space.
Now we have the following diagram for triplesE, D;Γ:
simplex⇒convex subset of a t.v.s. ⇒Lassonde-type convex space ⇒H-space ⇒G-convex space⇐⇒φA-space⇒KKM space
⇒space satisfying the partial KKM principle ⇒abstract convex space.
2.3
It is not known yet whether there is a space satisfying the partial KKM principle that is not a KKM space.
3. The KKM Principle and the Fan-Browder Map
LetE, D;Γbe an abstract convex space.
Recall the following equivalent form of26, Theorem 8.2.
Theorem 3.1. Suppose thatE, D;Γsatisfies the partial KKM principle and a map G : D E satisfies the following.
1.1Gis closed valued.
1.2Gis a KKM map (i.e.,ΓA⊂GAfor allA∈ D).
1.3There exists a nonempty compact subsetKofEsuch that one of the following holds:
iKE,
iiK {Gz|z∈M}for someM∈ D,
iiifor eachN ∈ D, there exists a compactΓ-convex subsetLNofErelative to some D⊂Dsuch thatN⊂Dand
LN∩
z∈D
Gz⊂K. 3.1
ThenK∩ {Gz|z∈D}/∅.
Remark 3.2. Conditionsi–iiiin1.3are called compactness conditions or coercivity conditions.
In this paper, we mainly adopt simplyi, that is,E, D;Γis compact. However, most of results can be reformulated to the ones adoptingiioriii.
Definition 3.3. For a topological spaceX and an abstract convex spaceE, D;Γ, a multimap T : X Eis called aΦ-map or a Fan-Browder map provided that there exists a multimap S:X Dsatisfying the follwing:
afor eachx∈X,coΓSx⊂Tx i.e.,N∈ Sximplies thatΓN ⊂Tx, bX
z∈MIntS−zfor someM∈ D.
Here, Int denotes the interior with respect toEand, for eachz∈D,S−z:{x∈X | z∈Sx}.
There are several equivalent formulations of the partial KKM principle; see26. For example, it is equivalent to the Fan-Browder-type fixed point theorem as follows.
Theorem 3.4see26. An abstract convex spaceE, D;Γsatisfies the partial KKM principle if and only if anyΦ-mapT :EEhas a fixed pointx0∈E, that is,x0∈Tx0.
The following is known.
Lemma 3.5. Let {Xi, Di;Γi}i∈I be any family of abstract convex spaces. Let X :
i∈IXi be equipped with the product topology andD
i∈IDi. For eachi ∈ I, let πi : D → Di be the projection. For eachA ∈ D, defineΓA:
i∈IΓiπiA. ThenX, D;Γis an abstract convex space.
Let{Xi, Di;Γi}i∈Ibe a family ofG-convex spaces. ThenX, D;Γis aG-convex space.
It is not known yet whether this holds for KKM spaces.
From now on, for simplicity, we are mainly concerned with compact abstract convex spacesE;Γsatisfying the partial KKM principle. For example, any compactG-convex space, any compactH-space, or any compact convex space is such a space.
4. The Fan-Type Minimax Inequalities
Recall that an extended real-valued functionf :X → R, whereX is a topological space, is lowerresp., uppersemicontinuousl.s.c. resp., u.s.c.if{x ∈X |fx > r}resp.,{x∈ X | fx< r}is open for eachr∈R.
For an abstract convex spaceE⊃D;Γ, an extended real-valued functionf :E → R is said to be quasiconcaveresp., quasiconvexif{x∈E|fx> r}resp.,{x∈E|fx< r}is Γ-convex for eachr ∈R.
From the partial KKM principle we can deduce a very general version of the Ky Fan minimax inequality as follows.
Theorem 4.1. LetX, D;Γ be an abstract convex space satisfying the partial KKM principle,f : D×X → R, g:X×X → Rextended real functions, andγ∈Rsuch that
3.1for eachz∈D,{y∈X |fz, y≤γ}is closed,
3.2for eachy∈X, coΓ{z∈D|fz, y> γ} ⊂ {x∈X |gx, y> γ}, 3.3the compactness condition (1.3) holds forGz:{y∈X |fz, y≤γ}.
Then either (i) there exists ax∈Xsuch thatfz,x ≤γfor allz∈Dor (ii) there exists anx0 ∈X such thatgx0, x0> γ.
Proof. LetG:DXbe a map defined byGz:{y∈X |fz, y≤γ}forz∈D. Then each Gzis closed by3.1.
Casei:Gis a KKM map.
ByTheorem 3.1, we have z∈DGz/∅. Hence, there exists ax∈Xsuch thatx∈Gz for allz∈D, that is,fz,x ≤γfor allz∈D.
Caseii:Gis not a KKM map.
Then there existsN ∈ Dsuch that ΓN/⊂
z∈NGz. Hence there exists anx0 ∈ ΓN
such thatx0/∈Gzfor eachz ∈ N, or equivalentlyfz, x0 > γ for eachz ∈N. Since{z ∈ D |fz, x0> γ}containsN, by3.2,we havex0∈ΓN ⊂ {x∈X |gx, x0> γ},and hence, gx0, x0> γ.
Corollary 4.2. Under the hypothesis ofTheorem 4.1, letγ :supx∈Xgx, x.Then
y∈Xinfsup
z∈Df z, y
≤sup
x∈Xgx, x. 4.1
Example 4.3. 1 For a compact convex subset X D of a t.v.s. and f g, if f·, y is quasiconcave, then3.2holds; and iffx,·is l.s.c., then3.1holds. Therefore,Corollary 4.2 generalizes the Ky Fan minimax inequality49.
2 For a convex spaceX D and f g, Corollary 4.2 reduces to Cho et al. 50, Theorem 9.
3There is a very large number of generalizations of the Fan minimax inequality for convex spaces,H-spaces,G-convex spaces, and others. These would be particular forms of Corollary 4.2. For example, see Park18, Theorem 11, whereX, D;Γis aG-convex space.
4Some particular versions ofCorollary 4.2were given in27.
5. The von Neumann-Sion-Type Minimax Theorems
LetX;Γ1andY;Γ2be abstract convex spaces. For their product, as in theLemma 3.5we can defineΓX×YA: Γ1π1A×Γ2π2AforA∈ X×Y.
Theorem 5.1. LetE;Γ : X×Y;ΓX×Ybe the product abstract convex space, and letf, s, t, g : X×Y → Rbe four functions, then
μ:inf
y∈Ysup
x∈Xf x, y
, ν:sup
x∈Xinf
y∈Yg x, y
. 5.1
Suppose that
4.1fx, y≤sx, y≤tx, y≤gx, yfor eachx, y∈X×Y,
4.2for eachr < μandy ∈Y,{x∈X |sx, y> r}isΓ1-convex; for eachr > νandx∈X, {y∈Y |tx, y< r}isΓ2-convex,
4.3for eachr > ν, there exists a finite set{xi}mi1⊂Xsuch that
Y m
i1
Int
y∈Y |f xi, y
> r
, 5.2
4.4for eachr < μ, there exists a finite set{yj}nj1⊂Y such that
X n
j1
Int
x∈X|g x, yj
< r
. 5.3
IfE;Γsatisfies the partial KKM principle, then μinf
y∈Ysup
x∈Xf x, y
≤sup
x∈Xinf
y∈Yg x, y
ν. 5.4
Proof. Suppose that there exists a realcsuch that νsup
x∈Xinf
y∈Yg x, y
< c <inf
y∈Ysup
x∈Xf x, y
μ. 5.5
For the abstract convex space
E, D;Γ:
X×Y, xi, yj
i,j; ΓX×Y
, 5.6
define two mapsS:ED, T :EEby S−
xi, yj
:Int
x∈X |g x, yj
< c
×Int
y∈Y |f xi, y
> c
, 5.7
T x, y
:
x∈X|s x, y
> c
×
y∈Y |t x, y
< c
, 5.8
forxi, yj∈Dandx, y∈E, respectively. Then eachTx, yis nonempty andΓ-convex and Eis covered by a finite number of open setsS−xi, yj’s. Moreover,
S x, y
⊂ xi, yj
|g x, yj
< c, f xi, y
> c
⊂ x, y
|s x, y
> c, t x, y
< c
⊂T x, y
. 5.9
This implies that coΓSx, y ⊂ Tx, yfor allx, y ∈ E. ThenT is aΦ-map. Therefore, by Theorem 3.4, we havex0, y0∈X×Ysuch thatx0, y0∈Tx0, y0. Therefore,c < sx0, y0≤ tx0, y0< c, a contradiction.
Example 5.2. For convex spacesX,Y,andf s tg,Theorem 5.1reduces to that by Cho et al.50, Theorem 8.
Corollary 5.3. LetX;Γ1andY;Γ2be compact abstract convex spaces, letE;Γ: X×Y;ΓX×Y be the product abstract convex space, and letf, g:X×Y → Rbe functions satisfying the following:
1fx, y≤sx, y≤tx, y≤gx, yfor eachx, y∈X×Y, 2for eachx∈X, fx,·is l.s.c. andtx,·is quasiconvex onY, 3for eachy∈Y, s·, yis quasiconcave andg·, yis u.s.c. onX.
IfE;Γsatisfies the partial KKM principle, then
miny∈Ysup
x∈X
f x, y
≤max
x∈Xinf
y∈Yg x, y
. 5.10
Proof. Note that y → supx∈Xfx, y is l.s.c. on Y and x → infy∈Ygx, y is u.s.c. on X.
Therefore, both sides of the inequality exist. Then all the requirements ofTheorem 5.1are satisfied.
Example 5.4. 1Particular or slightly different versions ofCorollary 5.3are obtained by Liu 51, Granas13, Th´eor`emes 3.1 et 3.2, and Shih and Tan52, Theorem 4for convex subsets of t.v.s.
2Forfs, gt,Corollary 5.3reduces to27, Theorem 3.
For the casefstg,Corollary 5.3reduces to the following.
Corollary 5.5 see27. LetX;Γ1and Y;Γ2 be compact abstract convex spaces and letf : X×Y → Rbe an extended real function such that
1for eachx∈X, fx,·is l.s.c. and quasiconvex onY, 2for eachy∈Y, f·, yis u.s.c. and quasiconcave onX.
IfX×Y;ΓX×Ysatisfies the partial KKM principle, then ifhas a saddle pointx0, y0∈X×Y,
iione has
maxx∈X min
y∈Yf x, y
min
y∈Y max
x∈Xf x, y
. 5.11
Example 5.6. We list historically well-known particular forms ofCorollary 5.5in chronological order as follows.
1von Neumann1, Kakutani3.XandY are compact convex subsets of Euclidean spaces andfis continuous.
2Nikaid ˆo53. Euclidean spaces above are replaced by Hausdorfftopological vector spaces, andfis continuous in each variable.
3Sion 54. X and Y are compact convex subsets of topological vector spaces in Corollary 5.5.
4Komiya 55, Theorem 3. X and Y are compact convex spaces in the sense of Komiya.
5Horvath 36, Proposition 5.2. X and Y are c-spaces with Y being compact and without assuming the compactness ofX.
In these two examples, Hausdorffness ofY is assumed since they used the partition of unity argument.
6Bielawski 29, Theorem4.13. X and Y are compact spaces having certain simplicial convexities.
7Park17, Theorem 5.XandY areG-convex spaces.
In 1999, we deduced the following von Neumann–Sion type minimax theorem forG- convex spaces based on a continuous selection theorem:
Theorem 5.7 see17. LetX,Γ1 and Y,Γ2 be G-convex spaces, Y Hausdorff compact, f : X×Y → Ran extended real function, andμ:supx∈Xinfy∈Yfx, y. Suppose that
5.1fx,·is l.s.c. onYand{y∈Y |fx, y< r}isΓ2-convex for eachx∈Xandr > μ, 5.2f·, yis u.s.c. onXand{x∈X |fx, y> r}isΓ1-convex for eachy∈Y andr > μ.
Then
sup
x∈Xmin
y∈Yf x, y
min
y∈Y sup
x∈Xf x, y
. 5.12
Example 5.8. 1Komiya55, Theorem 3.XandY are compact convex spaces in the sense of Komiya.
2Slightly different form ofTheorem 5.7can be seen in17with different proof.
6. Collective Fixed Point Theorems
We have the following collective fixed point theorem.
Theorem 6.1. Let{Xi;Γi}ni1be a finite family of compact abstract convex spaces such thatX;Γ n
i1Xi;Γsatisfies the partial KKM principle, and for eachi, Ti :X Xiis aΦ-map. Then there
exists a point x ∈ X such that x ∈ Tx : n
i1Tix, that is, xi πix ∈ Tix for each i1,2, . . . , n.
Proof. LetSi:XXibe the companion map corresponding to theΦ-mapTi. DefineS:X Xby
Sx:n
i1
Six for eachx∈X. 6.1
We show thatT is aΦ-map with the companion mapS. In fact, we have x∈S−
y
⇐⇒y∈Sx⇐⇒yi∈Six for eachi⇐⇒x∈S−i yi
for eachi, 6.2
wherey{y1, . . . , yn}. Since eachS−iyiis open, we have afor eachy∈X,S−y ni1S−iyiis open.
Note that
M∈ Sx⇒πiM∈ Six⇒ΓiπiM⊂Tix, 6.3
and hence,
ΓMn
i1
ΓiπiM⊂n
i1
Tix Tx. 6.4
Therefore, we have
bfor eachx∈X, M∈ Sximplies thatΓM⊂Tx.
Moreover, letx ∈X. SinceSi : X Xiis the companion map corresponding to the Φ-mapTi, for eachi, there existsjjisuch that
x∈S−i yi,j
⇒yi,j ∈Six⇒y∈n
i1
Six Sx⇒x∈S− y
, 6.5
wherey: y1,j1, . . . , yn,jn.SinceXis compact, we have cX
z∈MS−zfor someM∈ X.
SinceX;Γ satisfies the partial KKM principle, byTheorem 3.4, theΦ-mapT has a fixed point.
Example 6.2. 1Ifn 1,X is a convex space, and S T, thenTheorem 6.1reduces to the well-known Fan-Browder fixed point theorem; see Park56.
2 For the casen 1, Theorem 6.1for a convex spaceX was obtained by Ben-El- Mechaiekh et al.69, Theorem 1and Simons57, Theorem 4.3. This was extended by many authors; see Park56.
We have already the following collective fixed point theorem for arbitrary family of G-convex spaces.
Theorem 6.3see40. Let{Xi;Γi}i∈I be a family of compact HausdorffG-convex spaces,X
i∈IXi, and for eachi∈I,letTi :X Xibe aΦ-map. Then there exists a pointx∈X such that x∈Tx:
i∈ITix,that is,xiπix∈Tixfor eachi∈I.
Example 6.4. In case when Xi;Γi are all H-spaces, Theorem 6.3 reduces to Tarafdar 58, Theorem 2.3. This is applied to sets with H-convex sections 58, Theorem 3.1 and to existence of equilibrium point of an abstract economy58, Theorem 4.1 and Corollary 4.1.
These results also can be extended toG-convex spaces and we will not repeat then here.
Remark 6.5. Each of Theorems6.1,7.1,8.1,9.1, and9.4, respectively, in this paper is based on the KKM method and concerns with finite families of abstract convex spaces such that their product satisfies the partial KKM principle. Each of them has a corresponding Theorems6.3, 7.3,8.3,9.2and9.6, respectively, based on continuous selection method for infinite families of HausdorffG-convex spaces. Note that for finite families the Hausdorffness is redundant in these corresponding theorems.
7. Intersection Theorems for Sets with Convex Sections
In our previous work17, from a G-convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem fornsubsets of a cartesian product of ncompactG-convex spaces. This was applied to obtain a von Neumann-sion-type minimax theorem and a Nash-type equilibrium theorem forG-convex spaces.
In the present section, we generalize the abovementioned intersection theorem to product abstract convex spaces satisfying the partial KKM principle.
The collective fixed point theorem inSection 6can be reformulated to a generalization of various Fan-type intersection theorems for sets with convex sections as follows.
Let{Xi}i∈Ibe a family of sets, and leti∈Ibe fixed. Let
X
j∈I
Xj, Xi
j∈I\{i}
Xj. 7.1
Ifxi∈Xiandj∈I\ {i}, then letxjidenote thejth coordinate ofxi. Ifxi∈Xiandxi∈Xi, then letxi, xi∈X be defined as follows: itsith coordinate isxiand forj /ithejth coordinate is xij. Therefore, anyx∈Xcan be expressed asx xi, xifor anyi∈I, wherexidenotes the projection ofxinXi.
Theorem 7.1. Let{Xi;Γi}ni1be a finite family of compact abstract convex spaces such thatX;Γ n
i1Xi;Γsatisfies the partial KKM principle and, for eachi, letAiandBibe subsets ofXsatisfying the following.
7.1For eachxi∈Xi,∅/coΓi Bixi⊂Aixi:{yi∈Xi|xi, yi∈Ai}.
7.2For eachyi∈Xi, Biyi:{xi∈Xi |xi, yi∈Bi}is open inXi. Then ni1Ai/∅.
Proof. We applyTheorem 6.1with multimapsSi, Ti : X Xi given by Six : Bixiand Tix:Aixifor eachx∈X. Then for eachiwe have the following.
aFor eachx∈X, we have∅/coΓi Six⊂Tix.
bFor eachyi∈Xi, we have
x∈S−i yi
⇐⇒yi∈Six Bi xi
⇐⇒
xi, yi
∈Bi⊂Xi×XiX. 7.2
Hence,
S−i yi
x
xi, xi
∈X|xi∈Bi yi
, xi ∈Xi Bi
yi
×Xi. 7.3 Note thatS−iyiis open inX Xi×Xi and thatTiis aΦ-map. Therefore, byTheorem 6.1, there existsx∈Xsuch thatxi∈Tix Aixifor alli. Hencex xi,xi∈ ni1Ai/∅.
Example 7.2. For convex spacesXi, particular forms ofTheorem 7.1have appeared as follows:
1Fan10, Th´eeor`eme 1.AiBifor alli.
2Fan11, Theorem 1.n2 andAi Bifori1,2.
From these results, Fan11deduced an analytic formulation, fixed point theorems, extension theorems of monotone sets, and extension theorems for invariant vector subspaces.
For particular types ofG-convex spaces,Theorem 7.1was known as follows.
3Bielawski 29, Proposition 4.12 and Theorem 4.15. Xi have the finitely local convexity.
4Kirk et al.32, Theorem 5.2.Xiare hyperconvex metric spaces.
5Park17, Theorem 4,18, Theorem 19. In17, from aG-convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem fornsubsets of a cartesian product ofncompactG-convex spaces. This was applied to obtain a von Neumann-Sion-type minimax theorem and a Nash-type equilibrium theorem forG-convex spaces.
6Park27, Theorem 4. We gave a different proof.
In22, a collective fixed point theorem was reformulated to a generalization of various Fan-type intersection theorems for arbitrary number of sets with convex sections as follows.
Theorem 7.3see22. Let{Xi;Γi}i∈Ibe a family of HausdorffcompactG-convex spaces and, for eachi∈I, letAiandBibe subsets ofX
i∈IXisatisfying the following.
7.1For eachxi ∈Xi,∅/coΓi Bixi⊂Aixi:{yi∈Xi |xi, yi∈Ai}.7.2For each yi∈Xi, Biyi:{xi∈Xi |xi, yi∈Bi}is open inXi.
Then i∈IAi/∅.
Example 7.4. For convex subsets Xi of topological vector spaces, particular forms of Theorem 7.3have appeared as follows.
1Ma12, Theorem 2. The caseAiBifor alli∈Iwith a different proof is given.
2Chang59, Theorem 4.2obtainedTheorem 7.3with a different proof. The author also obtained a noncompact version ofTheorem 7.3as in59, Theorem 4.3.
3Park19, Theorem 4.2.Xiare convex spaces.
Note that ifIis finite inTheorem 7.3, the Hausdorffness is redundant byTheorem 7.1.
8. The Fan-Type Analytic Alternatives
From the intersection Theorem 7.1, we can deduce the following equivalent form of a generalized Fan-type minimax inequality or analytic alternative. Our method is based on that of Fan9,10and Ma12.
Theorem 8.1. Let{Xi;Γi}ni1be a finite family of compact abstract convex spaces such thatX;Γ n
i1Xi;Γsatisfies the partial KKM principle and, for eachi, letfi, gi :X Xi×Xi → Rbe real functions satisfying
8.1fix≤gixfor eachx∈X,
8.2for eachxi∈Xi, xi→gixi, xiis quasiconcave onXi, 8.3for eachxi∈Xi, xi→fixi, xiis l.s.c. onXi.
Let{ti}ni1be a family of real numbers. Then either athere exist aniand anxi∈Xisuch that
fi
xi, yi
≤ti ∀yi∈Xi, 8.1
orbthere exists anx∈Xsuch that
gix> ti ∀i1,2, . . . , n. 8.2
Proof. Suppose thatadoes not hold, that is, for anyiand anyxi∈Xi, there exists anxi∈Xi
such thatfixi, xi> ti. Let Ai:
x∈X|gix> ti
, Bi
x∈X|fix> ti
8.3
for eachi. Then
1for eachxi∈Xi,∅/Bixi⊂Aixi, 2for eachxi∈Xi,AixiisΓi-convex, 3for eachyi∈Xi,Biyiis open inXi.
Therefore, byTheorem 7.1, there exists anx∈ ni1Ai. This is equivalent tob.
Example 8.2. Fan9, Th´eor`eme 2,10, Theorem 3.Xiare convex subsets of t.v.s., andfi gi
for alli. From this, fan9, 10 deduced Sion’s minimax theorem 54, the Tychonoff fixed point theorem, solutions to systems of convex inequalities, extremum problems for matrices, and a theorem of Hardy-Littlewood-P ´olya.
From the intersectionTheorem 7.3, we can deduce the following equivalent form of a generalized Fan-type minimax inequality or analytic alternative.
Theorem 8.3see22. Let{Xi;Γi}i∈Ibe a family of compact HausdorffG-convex spaces and, for eachi∈I, letfi, gi :X Xi×Xi → Rbe real functions as inTheorem 8.1. Then the conclusion of Theorem 8.1holds.
Example 8.4. 1Ma12, Theorem 3.Xiare convex subsets of t.v.s. andfigifor alli∈I. 2Park19, Theorem 8.1.Xiare convex spaces.
Remark 8.5. 1We obtainedTheorem 8.1fromTheorem 7.1. As was pointed out by Fan9 for his case, we can deduceTheorem 7.1fromTheorem 8.1by considering the characteristic functions of the setsAiandBi.
2The conclusion of Theorems8.1and8.3can be stated as follows
xmini∈Xisup
xi∈Xi
fi
xi, xi
> ti ∀i, 8.4
thenbholds; see Fan9,10.
3ForI{1,2}, Theorems8.1and8.3imply the Fan minimax inequality.
9. The Nash-Type Equilibrium Theorems
FromTheorem 8.1, we obtain the following form of the Nash-Fan-type equilibrium theorems in27with different proofs.
Theorem 9.1. Let{Xi;Γi}ni1be a finite family of compact abstract convex spaces such thatX;Γ n
i1Xi;Γsatisfies the partial KKM principle and, for eachi, letfi, gi :X Xi×Xi → Rbe real functions such that
9.0fix≤gixfor eachx∈X,
9.1for eachxi∈Xi,xi→gixi, xiis quasiconcave onXi, 9.2for eachxi∈Xi,xi→fixi, xiis u.s.c. onXi, 9.3for eachxi∈Xi,xi→fixi, xiis l.s.c. onXi. Then there exists a pointx∈Xsuch that
gix≥max
yi∈Xifi
xi, yi
∀i1,2, . . . , n. 9.1
Proof. Since eachXiis compact, by9.2,for anyε >0,ti:maxyi∈Xifixi, yi−εexists for all xi ∈Xiand alli. HenceTheorem 8.1adoes not hold. Then byTheorem 8.1b, there exists anx ∈Esuch thatgix > ti maxyi∈Xifixi, yi−εfor alli∈I. Sinceε >0 is arbitrary, the conclusion follows.
This is not comparable to the following generalized Nash-Ma type theorem:
Theorem 9.2see22. Let{Xi;Γi}i∈Ibe a family of compact HausdorffG-convex spaces and, for eachi∈I, letfi, gi :X Xi×Xi → Rbe real functions satisfying (9.0)–(9.3). Then there exists a pointx∈Xsuch that
gix ≥max
yi∈Xi
fi
xi, yi
∀i∈I. 9.2
Example 9.3. Park19, Theorem 8.2.Xiare convex spaces.
From Theorem 9.1 for fi gi, we obtain the following form of the Nash-Fan-type equilibrium theorem for abstract convex spaces.
Theorem 9.4. Let{Xi;Γi}ni1be a finite family of compact abstract convex spaces such thatX;Γ n
i1Xi;Γsatisfies the partial KKM principle and, for eachi∈I, letfi:X → Rbe a function such that
10.1for eachxi∈Xi,xi→fixi, xiis quasiconcave onXi, 10.2for eachxi∈Xi,xi→fixi, xiis u.s.c. onXi, 10.3for eachxi ∈Xi,xi→fixi, xiis l.s.c. onXi. Then there exists a pointx∈Xsuch that
fix max
yi∈Xifi
xi, yi
∀i1,2, . . . , n. 9.3
Example 9.5. For continuous functionsfi, a number of particular forms ofTheorem 9.4have appeared for convex subsetsXiof Hausdorfftopological vector spaces as follows:
1Nash5, Theorem 1whereXiare subsets of Euclidean spaces, 2Nikaido and Isoda60, Theorem 3.2,
3Fan10, Theorem 4, 4Tan et al.61, Theorem 2.1.
For particular types of G-convex spaces Xi and continuous functions fi, particular forms ofTheorem 9.4have appeared as follows.
5Bielawski29, Theorem 4.16.Xihave the finitely local convexity.
6Kirk et al.32, Theorem 5.3.Xiare hyperconvex metric spaces.
7Park17,Theorem 6.1,18, Theorem 20.XiareG-convex spaces.
8Park21, Theorem 4.7. A variant ofTheorem 9.4is under the hypothesis thatX;Γ is a compactG-convex space andf1, . . . , fn:X → Rare continuous functions.
9Gonz´alez et al.31. EachXiis a compact, sequentially compactL-space and each fiis continuous as in 8.
10Briec and Horvath30, Theorem 3.2. EachXiis a compactB-convex set and eachfi
is continuous as in 8.
From Theorem 9.2, we obtain the following generalization of the Nash-Ma-type equilibrium theorem forG-convex spaces.
Theorem 9.6see22. Let{Xi;Γi}i∈Ibe a family of compact HausdorffG-convex spaces and, for eachi∈I, letfi:X → Rbe a function satisfying conditions (10.1)–(10.3). Then there exists a point
x∈Xsuch that
fix max
yi∈Xi
fi
xi, yi
∀i∈I. 9.4
Example 9.7. For continuous functionsfiand for convex subsetsXiof Hausdorfftopological vector spaces,Theorem 9.6was due to Ma12, Theorem 4.
The pointxin the conclusion of Theorems9.4or9.6is called a Nash equilibrium. This concept is a natural extension of the local maxima and the saddle point as follows.
In caseIis a singleton, we obtain the following.
Corollary 9.8. LetXbe a closed bounded convex subset of a reflexive Banach spaceEandf :X → R a quasiconcave u.s.c. function. Thenfattains its maximum onX, that is, there exists anx∈Xsuch thatfx ≥fxfor allx∈X.
Proof. LetEbe equipped with the weak topology. Then, by the Hahn-Banach theorem,f is still u.s.c. because f is quasiconcave, and X is still closed. Being bounded,X is contained in some closed ball which is weakly compact. Since any closed subset of a compact set is compact,X isweaklycompact. Now, by Theorem 9.4for a single family, we have the conclusion.
Corollary 9.8 is due to Mazur and Schauder in 1936. Some generalized forms of Corollary 9.8were known by Park et al.62,63.
ForI{1,2},Theorem 9.4reduces toCorollary 5.5as follows.
Proof ofCorollary 5.5fromTheorem 9.4. Letf1x, y : −fx, yand f2x, y : fx, y. Then all of the requirements ofTheorem 9.4are satisfied. Therefore, byTheorem 9.4, there exists a pointx0, y0∈X×Y such that
f1 x0, y0
max
y∈Y f1 x0, y
, f2 x0, y0
max
x∈Xf2 x, y0
. 9.5
Therefore, we have
−f x0, y0
f1 x0, y0
≥f1 x0, y
−f x0, y
∀y∈Y, 9.6
f x0, y0
f2 x0, y0
≥f2 x, y0
f x, y0
∀x∈X. 9.7
Hence
f x, y0
≤f x0, y0
≤f x0, y
∀ x, y
∈X×Y. 9.8
Therefore
maxx∈Xf x, y0
≤f x0, y0
≤min
y∈Xf x0, y
. 9.9
This implies that
miny∈Xmax
x∈Xf x, y
≤f x0, y0
≤max
x∈X min
y∈Xf x, y
. 9.10
On the other hand, we have trivially
miny∈Xf x, y
≤max
x∈Xf x, y
, 9.11
and hence
maxx∈X min
y∈Xf x, y
≤min
y∈Xmax
x∈Xf x, y
. 9.12
Therefore, we have the conclusion.
10. Historical Remarks on Related Results
IAs we have seen in Sections1–3, we have three methods in our subject as follows:
1 fixed point method—applications of the Kakutani theorem and its various generalizationse.g., acyclic-valued multimaps, admissible maps, or better admissible maps in the sense of Park; see3–8,10,12,14–16,19,20,23,28,42,53,64–68and others,
2continuous selection method—applications of the fact that Fan-Browder-type maps have continuous selections under certain assumptions like Hausdorffness and compactness of relevant spaces; see17,22,36,39,40,58,69,70and others,
3 the KKM method—as for the Sion theorem, direct applications of the KKM theorem, or its equivalents like the Fan-Browder fixed point theorem for which we do not need the Hausdorffness; see 9, 11, 17, 21, 24–27, 30, 31, 33, 35–37, 39, 43, 45, 47–
50,54,55,57,59,71,72and others.
For Case1, we will study it elsewhere and, in this paper, we are mainly concerned with Cases2and3.
II An upper semicontinuous u.s.c. multimap with nonempty compact convex values is called a Kakutani map. The Fan-Glicksberg theorem was extended by Himmelberg 8in 1972 for compact Kakutani maps instead of assuming compactness of domains. In 1990, Lassonde 67 extended the Himmelberg theorem to multimaps factorizable by Kakutani maps through convex sets in Hausdorff topological vector spaces. Moreover, Lassonde applied his theorem to game theory and obtained a von Neumann-type intersection theorem for finite number of sets and a Nash-type equilibrium theorem comparable to Debreu’s social equilibrium existence theorem66.
Fixed point theorems extending the Kakutani theorem can be applied to particular forms of results in this paper. Since such extended theorems usually assume Hausdorffness and certainabstractlocal convexity of the related space, their applicability is restrictive.
IIIIn 1946, the Kakutani theorem was extended for acyclic maps by Eilenberg and Montgomery73. This result was applied by Debreu66to the social equilibrium existence theorem and related results.
IV Since 1996 72, many authors have published some results of the present paper for hyperconvex metric spaces. For example, Kirk et al. in 200032 established the KKM theorem, its equivalent formulations, fixed point theorems, and the Nash theorem for hyperconvex metric spaces. However, already in 1993, Horvath37found that hyperconvex metric spaces are a particular type ofc-spaces.
V In 1998 16, an acyclic version of the social equilibrium existence theorem of Debreu is obtained. This is applied to deduce acyclic versions of theorems on saddle points, minimax theorems, and the following Nash equilibrium theorem.
Corollary 10.1see16. Let{Xi}ni1be a family of acyclic polyhedra,X n
i1Xi, and for eachi, fi:X → Ra continuous function such that
0for eachxi∈Xiand eachα∈R, the set
xi ∈Xi|fi
xi, xi
≥α
10.1
is empty or acyclic.
Then there exists a pointa∈Xsuch that
fia max
yi∈Xifi
ai, yi
∀i∈I. 10.2
VIIn the present paper, for abstract convex spaces, we notice that the partial KKM principle⇒the Fan-Browder fixed point theorem⇒the Nash equilibrium theorem, with or without additional intermediate steps. This procedure can be called “from the KKM principle to the Nash equilibria”simply, “K to N”; see27.
In 199917, we obtained a “K to N” forG-convex spaces. These results extended and unified a number of known results for particular types of G-convex spaces; see also18–
21,42. Therefore, the procedure also holds for Lassonde type-convex spaces, Horvath’sc- space, hyperconvex metric spaces, and others.
VII In 200020 and 2002 23, we applied our fixed point theorem for compact compositions of acyclic maps on admissiblein the sense of Kleeconvex subsets of a t.v.s.
to obtain a cyclic coincidence theorem for acyclic maps, generalized von Neumann-type intersection theorems, the Nash type equilibrium theorems, and the von Neumann minimax theorem.
The following examples are generalized forms of quasi equilibrium theorem or social equilibrium existence theorems which directly imply generalizations of the Nash-Ma-type equilibrium existence theorem.
Theorem 10.2see20. Let{Xi}ni1be a family of convex sets, each in a t.v.s.Ei,Kia nonempty compact subset ofXi,Si :XKia closed map, andfi, gi :X Xi×Xi → Ru.s.c. functions for eachi.
Suppose that, for eachi,
igix≤fixfor eachx∈X, iithe functionMidefined onXby
Mix max
y∈Sixgi
xi, y
forx∈X 10.3
is l.s.c., and
