Generalizations of the Nash Equilibrium Theorem in the KKM Theory
© Sehie Park. 2010
Received: 5 December 2009
Accepted: 2 February 2010
Published: 12 April 2010
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 for -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.
In 1928, John von Neumann found his celebrated minimax theorem  and, in 1937, his intersection lemma , which was intended to establish easily his minimax theorem and his theorem on optimal balanced growth paths. In 1941, Kakutani  obtained a fixed point theorem for multimaps, from which von Neumann's minimax theorem and intersection lemma were easily deduced.
In 1950, John Nash [4, 5] established his celebrated equilibrium theorem by applying the Brouwer or the Kakutani fixed point theorem. In 1952, Fan  and Glicksberg  extended Kakutani's theorem to locally convex Hausdorff topological vector spaces, and Fan generalized the von Neumann intersection lemma by applying his own fixed point theorem. In 1972, Himmelberg  obtained two generalizations of Fan's fixed point theorem  and applied them to generalize the von Neumann minimax theorem by following Kakutani's method in .
In 1961, Ky Fan  obtained his KKM lemma and, in 1964 , applied it to another intersection theorem for a finite family of sets having convex sections. This was applied in 1966  to a proof of the Nash equilibrium theorem. This is the origin of the application of the KKM theory to the Nash theorem. In 1969, Ma  extended Fan's intersection theorem  to 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 . 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, -convex spaces) related to the KKM theory and analytical fixed point theory. In the framework of -convex spaces, we obtained some minimax theorems and the Nash equilibrium theorems in our previous works [17, 18, 21, 22], based on coincidence theorems or intersection theorems for finite families of sets, and in , based on continuous selection theorems for the Fan-Browder maps.
In our recent works [24–26], 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 in .
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 by fixed point method, continuous selection method, or the KKM method. In this paper, we follow method and will compare our results to corresponding ones already obtained by method
In Section 2, we state basic facts and examples of abstract convex spaces in our previous works [24–26]. Section 3 deals 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 5 deals with various von Neumann-Sion-type minimax theorems for abstract convex spaces.
In Section 6, a collective fixed point theorem is deduced as a generalization of the Fan-Browder fixed point theorem. Section 7 deals 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, in Section 10, some known results related to the Nash theorem and historical remarks are added.
2. Abstract Convex Spaces and the KKM Spaces
A triple is given for the original KKM theorem , where is the standard -simplex, is the set of its vertices , and co: is the convex hull operation.
A triple is given, where and are subsets of a t.v.s. such that co and co. Fan's celebrated KKM lemma  is for .
A convex space is a triple where is a subset of a vector space such that co , and each is the convex hull of equipped with the Euclidean topology. This concept generalizes the one due to Lassonde for ; see . However he obtained several KKM-type theorems w.r.t. .
A triple is called an -space if is a topological space and is a family of contractible (or, more generally, -connected) subsets of indexed by such that whenever . If , then is called a -space by Horvath [36, 37].
Hyperconvex metric spaces due to Aronszajn and Panitchpakdi are particular cases of -spaces; see .
Hyperbolic spaces due to Reich and Shafrir  are also particular cases of -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.
Any topological semilattice with path-connected interval is introduced by Horvath and Llinares .
A -space consists of a topological space , a nonempty set , and a family of continuous functions (that is, singular -simplexes) for with . Every -space can be made into a -convex space; see . Recently -spaces are called -spaces in  and -spaces  or simplicial spaces  when .
Suppose that is a closed convex subset of a complete -tree , and for each , , where is the intersection of all closed convex subsets of that contain ; see Kirk and Panyanak . Then is an abstract convex space.
A topological space with a convexity in the sense of Horvath  is another example.
A -space due to Briec and Horvath  is an abstract convex space.
Thepartial KKM principle for an abstract convex space is the statement that, for any closed-valued KKM map , the family 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.
Every -convex space is a KKM space .
A connected linearly ordered space can be made into a KKM space .
The extended long line is a KKM space with the ordinal space ; see . But is not a -convex space.
A -space due to Briec and Horvath  is a KKM space.
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
Recall the following equivalent form of [26, Theorem 8.2].
Conditions (i)–(iii) in are called compactness conditions or coercivity conditions. In this paper, we mainly adopt simply (i), that is, is compact. However, most of results can be reformulated to the ones adopting (ii) or (iii).
There are several equivalent formulations of the partial KKM principle; see . For example, it is equivalent to the Fan-Browder-type fixed point theorem as follows.
Theorem 3.4 (see ).
The following is known.
It is not known yet whether this holds for KKM spaces.
From now on, for simplicity, we are mainly concerned with compact abstract convex spaces satisfying the partial KKM principle. For example, any compact -convex space, any compact -space, or any compact convex space is such a space.
4. The Fan-Type Minimax Inequalities
From the partial KKM principle we can deduce a very general version of the Ky Fan minimax inequality as follows.
For a compact convex subset of a t.v.s. and , if is quasiconcave, then (3.2) holds; and if is l.s.c., then (3.1) holds. Therefore, Corollary 4.2 generalizes the Ky Fan minimax inequality .
For a convex space and , Corollary 4.2 reduces to Cho et al. [50, Theorem 9].
There is a very large number of generalizations of the Fan minimax inequality for convex spaces, -spaces, -convex spaces, and others. These would be particular forms of Corollary 4.2. For example, see Park [18, Theorem 11], where is a -convex space.
Some particular versions of Corollary 4.2 were given in .
5. The von Neumann-Sion-Type Minimax Theorems
For convex spaces , and , Theorem 5.1 reduces to that by Cho et al. [50, Theorem 8].
Corollary 5.5 (see ).
We list historically well-known particular forms of Corollary 5.5 in chronological order as follows.
(2)Nikaidô . Euclidean spaces above are replaced by Hausdorff topological vector spaces, and is continuous in each variable.
(3)Sion . and are compact convex subsets of topological vector spaces in Corollary 5.5.
(4)Komiya [55, Theorem 3]. and are compact convex spaces in the sense of Komiya.
(5)Horvath [36, Proposition 5.2]. and are -spaces with being compact and without assuming the compactness of .
(6)Bielawski [29, Theorem (4.13)]. and are compact spaces having certain simplicial convexities.
(7)Park [17, Theorem 5]. and are -convex spaces.
Theorem 5.7 (see ).
6. Collective Fixed Point Theorems
We have the following collective fixed point theorem.
Therefore, we have
If , is a convex space, and , then Theorem 6.1 reduces to the well-known Fan-Browder fixed point theorem; see Park .
Theorem 6.3 (see ).
In case when are all -spaces, Theorem 6.3 reduces to Tarafdar [58, Theorem 2.3]. This is applied to sets with -convex sections [58, Theorem 3.1] and to existence of equilibrium point of an abstract economy [58, Theorem 4.1 and Corollary 4.1]. These results also can be extended to -convex spaces and we will not repeat then here.
Each of Theorems 6.1, 7.1, 8.1, 9.1, and 9.4, 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 Theorems 6.3, 7.3, 8.3, 9.2 and 9.6, respectively, based on continuous selection method for infinite families of Hausdorff -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 work , from a -convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem for subsets of a cartesian product of compact -convex spaces. This was applied to obtain a von Neumann-sion-type minimax theorem and a Nash-type equilibrium theorem for -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 in Section 6 can be reformulated to a generalization of various Fan-type intersection theorems for sets with convex sections as follows.
If and , then let denote the th coordinate of . If and , then let be defined as follows: its th coordinate is and for the th coordinate is . Therefore, any can be expressed as for any , where denotes the projection of in .
Fan [10, Théeorème 1]. for all .
Fan [11, Theorem ]. and for .
From these results, Fan  deduced an analytic formulation, fixed point theorems, extension theorems of monotone sets, and extension theorems for invariant vector subspaces.
Bielawski [29, Proposition and Theorem ]. have the finitely local convexity.
Kirk et al. [32, Theorem 5.2]. are hyperconvex metric spaces.
Park [17, Theorem 4], [18, Theorem 19]. In , from a -convex space version of the Fan-Browder fixed point theorem, we deduced a Fan-type intersection theorem for subsets of a cartesian product of compact -convex spaces. This was applied to obtain a von Neumann-Sion-type minimax theorem and a Nash-type equilibrium theorem for -convex spaces.
Park [27, Theorem 4]. We gave a different proof.
In , 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 7.3 (see ).
Ma [12, Theorem 2]. The case for all with a different proof is given.
Park [19, Theorem 4.2]. are convex spaces.
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 Fan [9, 10] and Ma .
Fan [9, Théorème 2], [10, Theorem 3]. are convex subsets of t.v.s., and for all . From this, fan [9, 10] deduced Sion's minimax theorem , the Tychonoff fixed point theorem, solutions to systems of convex inequalities, extremum problems for matrices, and a theorem of Hardy-Littlewood-Pólya.
From the intersection Theorem 7.3, we can deduce the following equivalent form of a generalized Fan-type minimax inequality or analytic alternative.
Theorem 8.3 (see ).
We obtained Theorem 8.1 from Theorem 7.1. As was pointed out by Fan  for his case, we can deduce Theorem 7.1 from Theorem 8.1 by considering the characteristic functions of the sets and .
The conclusion of Theorems 8.1 and 8.3 can be stated as follows
9. The Nash-Type Equilibrium Theorems
From Theorem 8.1, we obtain the following form of the Nash-Fan-type equilibrium theorems in  with different proofs.
Since each is compact, by for any , exists for all and all . Hence Theorem 8.1(a) does not hold. Then by Theorem 8.1(b), there exists an such that for all . Since is arbitrary, the conclusion follows.
This is not comparable to the following generalized Nash-Ma type theorem:
Theorem 9.2 (see ).
Park [19, Theorem 8.2]. are convex spaces.
(1)Nash [5, Theorem 1] where are subsets of Euclidean spaces,
(2)Nikaido and Isoda [60, Theorem 3.2],
(3)Fan [10, Theorem 4],
(4)Tan et al. [61, Theorem 2.1].
(5)Bielawski [29, Theorem 4.16]. have the finitely local convexity.
(6)Kirk et al. [32, Theorem ]. are hyperconvex metric spaces.
(8)Park [21, Theorem 4.7]. A variant of Theorem 9.4 is under the hypothesis that is a compact -convex space and are continuous functions.
(9)González et al. . Each is a compact, sequentially compact -space and each is continuous as in 8.
(10)Briec and Horvath [30, Theorem ]. Each is a compact -convex set and each is continuous as in 8.
Theorem 9.6 (see ).
For continuous functions and for convex subsets of Hausdorff topological vector spaces, Theorem 9.6 was due to Ma [12, Theorem 4].
Let be equipped with the weak topology. Then, by the Hahn-Banach theorem, is still u.s.c. because is quasiconcave, and is still closed. Being bounded, is contained in some closed ball which is weakly compact. Since any closed subset of a compact set is compact, is (weakly) compact. Now, by Theorem 9.4 for a single family, we have the conclusion.
Proof of Corollary 5.5 from Theorem 9.4.
Therefore, we have the conclusion.
10. Historical Remarks on Related Results
As we have seen in Sections 1–3, we have three methods in our subject as follows:
fixed point method—applications of the Kakutani theorem and its various generalizations (e.g., acyclic-valued multimaps, admissible maps, or better admissible maps in the sense of Park); see [3–8, 10, 12, 14–16, 19, 20, 23, 28, 42, 53, 64–68] and others,
continuous selection method—applications of the fact that Fan-Browder-type maps have continuous selections under certain assumptions like Hausdorffness and compactness of relevant spaces; see [17, 22, 36, 39, 40, 58, 69, 70] and others,
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, 72] and others.
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  in 1972 for compact Kakutani maps instead of assuming compactness of domains. In 1990, Lassonde  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 theorem .
Since 1996 , many authors have published some results of the present paper for hyperconvex metric spaces. For example, Kirk et al. in 2000  established the KKM theorem, its equivalent formulations, fixed point theorems, and the Nash theorem for hyperconvex metric spaces. However, already in 1993, Horvath  found that hyperconvex metric spaces are a particular type of -spaces.
In 1998 , 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 10.1 (see ).
is empty or acyclic.
In 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"); see .
In 2000  and 2002 , we applied our fixed point theorem for compact compositions of acyclic maps on admissible (in the sense of Klee) convex 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 10.2 (see ).
is l.s.c., and
Theorem 10.3 (see ).
is l.s.c., and
In 2001 , we obtained generalized forms of the von Neumann-Sion-type minimax theorem, the Fan-Ma intersection theorem, the Fan-Ma type analytic alternative, and the Nash-Ma-equilibrium theorem for -convex spaces. In , all -convex spaces were assumed to be Hausdorff because the results are based on a selection theorem in , where Hausdorffness was indispensable.
In 2001, for any topological semilattice with path-connected interval introduced by Horvath and Llinares , the KKM theorem, the Fan-Browder theorem, and the Nash theorem are shown by Luo . Note that such semilattice is known to be a -convex space.
Cain and González  considered relationship among some subclasses of the class of -convex spaces and introduced a subclass of the so-called -spaces. In 2007, González et al.  repeated to show that -convex spaces and -spaces satisfy the partial KKM principle. They added that -spaces satisfy the properties of the Fan type minimax inequality, Fan-Browder-type fixed point, and the Nash-type equilibrium. All of such results are already known for more general -convex spaces.
In 2008, Kulpa and Szymanski  introduced a series of theorems called Infimum Principles in simplicial spaces. As for applications, they derive fixed point theorems due to Schauder, Tychnoff, Kakutani, and Fan-Browder: minimax theorems, the Nash equilibrium theorem, the Gale-Nikaido-Debreu theorem, and the Ky Fan minimax inequality. Their study is based on and utilizes the techniques of simplicial structure and the Fan-Browder map. Recall that for any abstract convex spaces satisfying abstract KKM principle we can deduce such classical theorems without using any Infimum Principles. Moreover, we note that the newly defined -spaces in  are particular types of abstract convex spaces satisfying the abstract KKM principle.
In 2008, for -spaces, Briec and Horvath  showed that some theorems mentioned in this paper hold, that is, Fan-Browder fixed point theorem, Himmelberg-type (in fact, Browder-type and Kakutani-type) fixed point theorems, Fan type minimax inequality, existence of Nash equilibria, and others. Note that -spaces are KKM spaces [30, Corollary 2.2], and their authors depend the Pelegs type multiple KKM theorem.
Finally, recall that there are several hundred published works on the KKM theory and we can cover only a part of them. For more historical background for the related fixed point theory and for more involved or related results in this paper, see the references [24–27, 41, 43, 48] and the literature therein.
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