Yuetal. Fixed Point Theory and Applications (2016) 2016:69 DOI 10.1186/s13663-016-0562-z

0 Fixed Point Theory and Applications

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Equivalence results between Nash equilibrium theorem and some fixed point theorems

Jian Yu1, Neng-Fa Wang1,2* a

"Correspondence: scinfwang@163.com

1 Department of Mathematics, Guizhou University, Guiyang, 550025, P.R. China

2Schoolof Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, P.R. China Full list of author information is available at the end of the article

nd Zhe Yang3,4 Abstract

We show that the Kakutani and Brouwer fixed point theorems can be obtained by directly using the Nash equilibrium theorem. The corresponding set-valued problems, such as the Kakutani fixed point theorem, Walras equilibrium theorem (set-valued excess demand function), and generalized variational inequality, can be derived from the Nash equilibrium theorem, with the aid of an inverse of the Berge maximum theorem. For the single-valued situation, we derive the Brouwer fixed point theorem, Walras equilibrium theorem (single-valued excess demand function), KKM lemma, and variational inequality from the Nash equilibrium theorem directly, without any recourse.

MSC: 47H10; 54C60; 91A10

Keywords: Brouwer fixed point theorem; Kakutani fixed point theorem; Nash equilibrium theorem; Walras equilibrium theorem; KKM lemma; variational inequality

1 Introduction

It is well known that fixed point theorems play an important role in game theory and mathematical economics [1-3]. Nash [4] firstly defined the best response correspondence and applied the Berge maximum theorem and Kakutani fixed point theorem to prove the existence of Nash equilibrium points in finite games, where finitely many players may choose from a finite number of pure strategies in finite-dimensional Euclidean spaces. Later, De-breu [5] extended finite games to noncooperative games with nonlinear payoff functions and obtained the following equilibrium theorem.

Theorem 1.1 (see [6, 7]) Let N = {1,..., n} be a finite set of players. For each i e N, Xi is a nonempty, convex, and compact subset of the ni-dimensional Euclidean space, f : X := WieNXi —► R is continuous, andfi(xi,x_i) is quasi-concave in xifor any x-i, where -i = N\{i}. Then, there exists x* e X such that

fi(x*,x-) = maxfi(ui,x-), Vi e N.

ui eXi

ft Spri

ringer

Such x* is called an equilibrium of the game T = (Xi,...,Xn;fi,...,fn).

© 2016 Yu et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use,distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

In recent years, a great deal of mathematical effort has been devoted to prove the equivalence between the KKM principle and several fixed point theorems or minimax inequalities. Park [8] showed a sequence of equivalent formulations for the KKM principle in abstract convex spaces. From the statements of [8, 9] we know that the fixed point theorem, minimax inequility, and Nash equilibrium theorem can be derived from the KKM principle. However, to the best of our knowledge, there is no proof for the Kakutani and Brouwer fixed point theorems via the Nash equilibrium theorem, although we can find in the previous literature many proofs or equivalent results for these two theorems [2, 8, 9]. In this paper, we fill these gaps. In Section 2, we show that the Kakutani fixed point theorem, Walras equilibrium theorem (set-valued excess demand function), and generalized variational inequality can be derived from the Nash equilibrium theorem with the aid of an inverse of the Berge maximum theorem [10,11]. In Section 3, for the single-valued situation, we derive the Brouwer fixed point theorem, Walras equilibrium theorem (single-valued excess demand function), KKM lemma, and variational inequality from the Nash equilibrium theorem directly, without any recourse.

2 Kakutani fixed point theorem via Nash equilibrium theorem

To obtain the Kakutani fixed point theorem from the Nash equilibrium theorem, we need an inverse of the Berge maximum theorem.

Theorem 2.1 (Berge maximum theorem) (see [2,6]) Let X be a subset of then-dimensional Euclidean space Rn, and Y be a subset of the m-dimensional Euclidean space Rm. Let u: X x Y —^ R be continuous, and letS: X ^ Y be continuous and nonempty compact-valued. Then, the correspondence K: X ^ Y defined by

is upper semicontinuous and compact-valued.

In 1997, Komiya [10] considered an inverse of the Berge maximum theorem, and Zhou [11] gave a simple alternative proof.

Theorem 2.2 (Inverse of Berge maximum theorem) LetX be a subset of the n-dimensional Euclidean space Rn, and K: X ^ Rm be a nonempty convex compact-valued and upper semicontinuous correspondence. Then there exists a continuous function v: X x Rm —^ [0,1] such that

(i) K(x) = {y e Rm : v(x,y) = maxzeRm v(x, z)}, Vx e X;

(ii) v(x, y) is quasi-concave in y for any x e X.

We begin by proving the following results. 2.1 Kakutani fixed point theorem

Komiya [10] showed that the Kakutani fixed point theorem can be derived from the existence theorem of maximal elements with the aid of Theorem 2.2. However, in this section, by using different methods, we derive the Kakutani fixed point theorem.

K(x)= | y e S(x):u(x, y) = max u(x, , Vx e X,

Theorem 2.3 (Kakutani fixed point theorem) LetX be a nonempty, convex, bounded, and closed subset of Rn, and F: X ^ X be a nonempty convex compact-valued and upper semi-continuous correspondence. Then, there exists x* e X such thatx* e F(x*).

Proof We apply Theorem 2.2 to find a continuous functionf: X x Rn —^ [0,1] such that

F (x) = {y e Rn : f (x, y) = maxf (x, z)}, Vx e X, and f (x, y) is quasi-concave in y for any x e X. Since F(x) c X,

F (x) = {y e X :f (x, y) = maxf (x, z)}. Next, define the mapping g: X x X —► R by g(x,y) = — ||x -y||.

Obviously, g is continuous on X x X, and g(x, •) is concave on X for any x e X. For the game T = (X,X;f,g), by Theorem 1.1 there exists (x*,y*) e X x X such that

f{x* f) = mX/^ y),

g(x*,y*) = —1|x* -y* || = max[-||x -y* |] = -min|x -y* || = 0.

xeX xeX

Therefore, y* e F(x*) and x* = y*, which implies x* e F(x*). This completes the proof. □

2.2 Walras equilibrium theorem (set-valued excess demand function)

Walras equilibrium may be formulated as follows. Let there be n commodities, and P c Rn be the set of all price vectors,

P = (pi,...p ) e Rn : Pi > 0,£ Pi = 1 .

I i=1 >

The excess demand function Z(p) = (Z1(p),...,Zn(p)) is a correspondence from P to Rn. A price vector p* e P is an equilibrium if there exists z* e Z (p*) such that

z*< 0, Vi = 1,..., n.

Theorem 2.4 (Walras equilibrium theorem) Let an excess demand function Z (p) satisfy the following conditions:

(i) Z : P ^ Rn is a nonempty convex compact-valued and upper semicontinuous correspondence;

(ii) the weak Walras law holds:

<p, z}<0, Vp e P, Vz e Z (p). Then there exists at least one equilibrium p*, that is, there exists z* e Z (p*) such that

z*< 0, Vi = 1,..., n.

Proof Let Z = coZ (P), where coZ (P) is the convex hull. Corollary 5.33 and Lemma 17.84 of [12] yield that Z is a nonempty, convex, and compact subset of Rn. We apply Theorem 2.2 to find a continuous function f: P x Z —► [0,1] such that

Z (p) = | z e Z: f (p, z) = maxf (p, y) |, Vp e P,

V yeZ J

and f (p, z) is quasi-concave in z for any p e P. Next, define the mapping g: P x Z —► R by

g(p, z) = <p, z>.

Obviously, g is continuous on P x Z, and g(p, ■) is concave for any p e P. For the game T = (Z, P;f,g), by Theorem 1.1 there exists (z*,p*) e Z x P such that

z*) = maxf(p*, z),

g(p*, z**) = (p*, z*) = max(p, z*).

Therefore, z* e Z(p*). From the weak Walras law we have 0 > (p*, z*) = maxip, z*),

that is,

(p, z*) < 0, Vp e P. We conclude that

z*< 0, Vi = 1,..., n.

Otherwise, there is i0 e{1,..., n} such that z*0 > 0. Let q e P with qio = 1 and qi = 0 for any i = i0. Then

(q, z*) = z*0 > 0,

which is a contradiction. □

2.3 Generalized variational inequality

In 1968, Browder [13] first gave the generalized variational inequality, which plays a very important role in game theory and nonlinear analysis (see, for example, [6] and the references therein). Here we show that the generalized variational inequality can be derived from the Nash equilibrium theorem with the aid of Theorem 2.2 as follows.

Theorem 2.5 (Generalized variational inequality) LetX be a nonempty, convex, bounded, and closed subset of Rn, and F: X ^ Rn be a nonempty convex compact-valued and upper semicontinuous correspondence. Then, there existx* e X and u* e F(x*) such that

[u\y - x*)> 0, Vy e X.

Proof Let U = coF(X), where coF(X) is the convex hull. Corollary 5.33 and Lemma 17.8 of [12] yield that U is a nonempty, convex, and compact subset of Rn. We apply Theorem 2.2 to find a continuous function f: X x U —► [0,1] such that

F (x) =| u e U: f (x, u) = maxf (x, z) |, Vx e X,

I zeU I

and f (x, u) is quasi-concave in u for any x e X. Next, define two mappings g, h : U x X x X —► R by

g(x,y) = -||x -y||, V(x,y) e X x X,

h(u,x,y) = <u,x -y}, V(u,x,y) e U x X x X.

Obviously, g, h are continuous on U x X x X, andg(-,y) and h(u,x, •) are concave for any x e X and any u e U.

For the game T = (U,X,X;f,g, h), by Theorem 1.1 there exists (u*,x*,y*) e U x X x X such that

f (x*, u*) = maxf (x*, z),

g(x*,y*) = -||x* -y* || = max[-||x -y* ||] = -min||x -y* || = 0,

xeX xeX

h(u*,x*,y*) = (u*,x* -y*) = max(u*,x* -y). Therefore, u* e F(x*), x* = y* and

0 = (u*,x* - x*) = (u*,x* -y*) = u*,x* -y), that is, u* e F(x*) and

(u*,y - x*) > 0, Vy e X. This completes the proof. □

3 Brouwer fixed point theorem via Nash equilibrium theorem

In this section, we apply only the Nash equilibrium theorem to conclude the Brouwer fixed point theorem and related problems, without recourse to the inverse of the Berge maximum theorem.

3.1 Brouwer fixed point theorem

Theorem 3.1 (Brouwer fixed point theorem) LetX be a nonempty, convex, bounded, and closed subset of Rn, and y be a continuous function from X to itself. Then, there exists x* e X such thatx* = y(x*).

Proof Define two mappings f,g: X x X —► R by f (x, y) = -||x - y||,

g(x, y) = —||y - y(x)\.

Obviously, f, g are continuous on X x X, andf (■,y) and g(x, ■) are concave for any x e X and any y e X.

For the game T = (X,X;f,g), by Theorem 1.1 there exists (x*,y*) e X x X such that f(x*, y*) = —I x* — y*|| = maxl" —II x — y*||l = — mini x — y*|| = 0,

J v ' 11 11 xeX xeX 11

g(x*,y*) = — ||y* — p(x*) \ = max[ — \y — p(x*) \] = — min||y — p(x*) \ = 0.

Therefore, x* = y* and y* = <p(x*), that is, x* = <p(x*). This completes the proof. □

3.2 Walras equilibrium theorem (single-valued excess demand function)

Following the statement of Section 2.2, the Walras equilibrium theorem for a single-valued excess demand function can be obtained from the Nash equilibrium theorem. The excess demand function Z (p) = (Z1(p),...,Zn(p)) is a function from P to Rn .A price vector p* e P is an equilibrium if

Zip) < 0, Vi = 1,...,n.

Theorem 3.2 (Walras equilibrium theorem) Let an excess demand function Z (p) satisfy the following conditions:

(i) Z (p) is a continuous function from P to Rn;

(ii) The Weak Walras law holds:

(Z (p),p) < 0, Vp e P. Then there exists at least one equilibrium p*, that is, there exists p* e P such that

Zip) < 0, Vi = 1,...,n. Proof Define two mappings f,g: P x P —^ R by

f (p, q) = —\\p — qll,

g(p, q)=( q, Z (p).

Obviously,f, g are continuous on P x P, andf (■, q) andg(p, ■) are concave for any p e P and any q e P.

For the game T = (P, P;f,g), by Theorem 1.1 there exists (p*, q*) e P x P such that

f q*) =— Hp* — q* || = maPx[— \p — q* \] =— g \p — q* \ = 0

gp, q*)=(q*, z (p*))=mmax(q, z p)).

Therefore, p* = q*, and by the weak Walras law we have

0 > p*. Z (p*)) = [it, Z CP*)) = maXq, Z (p*)),

that is,

(q, Z (p*)> < 0, Vq e P.

We conclude

Zip) < 0, Vi = l,...,n.

Otherwise, there is i0 e{l,..., n} such that Zi0 (p*) > 0. Let q e P with qio = l and q = 0 for any i = i0. Then

(q, Z (p*)> = Zio (p*) >0, which is a contradiction. □

3.3 KKM lemma

The KKM lemma is a very basic theorem, and the Brouwer fixed point theorem can be obtained by this lemma. The proof can be found in [6, 7]. We still derive the KKM lemma from the Nash equilibrium theorem.

Theorem 3.3 (KKM lemma) Let

A = co{e0,...,em} c Rm+\ and let {F0,..., Fm} be a family of closed subsets of A such that, for any A c {0,..., m}, co{ei: i e A} cUF.

PiF = 0.

Proof For any x = J2m=0 Xiei e A, y = J2m=0yiei e A, where Xi > 0^m=0 Xi = l, yi > 0, YT=0 yi = l, define two mappingsf,g: A x A —► R by

f (x,y) = — ||x -y||,

g(x, y) = J2 yid(x, Fi),

where d(x, Fi) is the distance from a point x to the set Fi. Obviously,f, g are continuous on A x A, andf (•,y) andg(x, •) are concave for any x e A and any y e A. For the game T = (A, A;f,g), by Theorem l.l there exists (x*,y*) e A x A such that

f(x*, y*) = —Ix* - y*|| = max[-IIx - y^l = - minix - y*|| = 0,

J v ' 11 11 xeX L 11 11J xeX 11 11

g(x*,y*) = y*d(x*, Fi) = max^yd(x*,F^ = max d(x*, Fi).

1=0 yeA i=0 i=0,...,m

Therefore, x* = y* and

yjx*d(x*,F) = Yjy*d(x*,F) = max d(x*,F).

i=0 i=0

Let/(x*) = {i:x* > 0}. Then/(x*) = 0 and

x*d(x*, F) = ^x*d(x*, F) = max d(x*, F).

ie/(x*) i=0 i_ ,...,m

It must be

d(x*, F) = max d(x*, F), Vi e /(x*). Additionally, since

x* e co{ei: i e /(x*)} c (J Fi,

ie/(x*)

there exists i0 e /(x*) such that x* e Fi0, which implies max d(x*, F) = d(x*, Fir) = 0,

i=0,...,m V 7 V 0/

that is, d(x*, Fi) = 0 for all i = 0,..., m. Since Fi is a closed set, it follows that x* e Fi. Therefore,

x*ef| Fi.

This completes the proof. □

3.4 Variational inequality

The variational inequality is an important tool in the study of optimization theory and game theory [6]; we also refer to early celebrated works [14] and [15]. Here, we deduced the variational inequality by Nash equilibrium theorem directly.

Theorem 3.4 (Variational inequality) LetX be a nonempty, convex, bounded, and closed subset of Rn, and p : X —^ Rn be a continuous function. Then, there exists x* e X such that

(<p(x*),y -x*) > 0, Vy e X.

Proof Define two mappings f,g: X x X —► R by

f (x, y) = —l|x - y||,

g(x, y) = (p(x), x - y).

Obviously, f, g are continuous on X x X, andf (•,y) and g(x, •) are concave for any x e X and any y e X.

For the game T = (X,X;f,g), by Theorem l.l there exists (x*,y*) e X x X such that

f (x*,y*) = —1|x* -y* | = max[-||x -y* ||] = -min||x -y* || = 0, g(x*,y*) = (p(x*),x* - y*) = max(^(x^,x* -y).

Therefore, x* = y* and

0 = (p(x*),x* -x*) = (p(x*),x* -y*) = max(p(x*),x* -y),

which implies

(p(x*),x* -y> < 0, Vy e X, that is,

(p(x*),y -x*> > 0, Vy e X. This completes the proof. □

4 Concluding remarks

Nash equilibrium is a very important notion in the game theory. In general, the Nash equilibrium theorem can be derived from the Brouwer and Kakutani fixed point theorems. However, there is no proof for the Kakutani and Brouwer fixed point theorems via the Nash equilibrium theorem. In this paper, we fill these gaps. We show that the Kakutani fixed point theorem, Walras equilibrium theorem (set-valued excess demand function), and generalized variational inequality can be derived from the Nash equilibrium theorem with the aid of an inverse of the Berge maximum theorem. For the single-valued situation, we derive the Brouwer fixed point theorem, Walras equilibrium theorem (single-valued excess demand function), KKM lemma, and variational inequality from the Nash equilibrium theorem directly, without any recourse.

Moreover, it is known that the Nash equilibrium theorem has been extended by Ky Fan to Hausdorff topological vector spaces (see Theorem 4 in [l6]). We next apply the Fan extension of the Nash equilibrium theorem to give an infinite-dimensional extension of the Brouwer fixed point theorem (i.e., the Tychonoff fixed point theorem).

Theorem 4.1 (see Theorem 4 in [l6]) Let N = {l,..., n} be a finite set of players. Suppose that, for each i e N, Xi is a nonempty, convex, and compact set in a locally convex Hausdorff topological vector space Ei, f : X := WieNXi —^ R is continuous, andfi(xi,x-i) is quasi-concave in xifor any x-i, where -i = N\{i}. Then, there exists x* e X such that

fi(x**, x-) = maxfi(ui, x-), Vi e N.

Theorem 4.2 (Tychonoff fixed point theorem)a LetX be a compact convex subset of a locally convex Hausdorfftopological vector space E, and p : X ^ X be a continuous function. Then, there exists x* e X such that x* = p (x*).

Proof Let X be a compact convex subset of a locally convex Hausdorff topological vector space E, p : X ^ X be a continuous function, and P be a separating family of semi-norms that generates the topology of E. For every p e P,set Fp = {x e X: p(x - p(x)) = 0}. We have to prove that Hpep Fp = 0. Since X is compact and the sets Fp are closed, it suffices to show that, for any finite set {pi,...,pn} ^ P, p|n=i F(pi) = 0. To this end, apply the Nash equilibrium theorem (Ky Fan's version) to the functions f (x, y) = - ^ni=l pi(x - y) an<d g(x,y) = - J]n=lpi(p(x) - y). The following proof is similar to that given in Theorem 3.l.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details

1 Department of Mathematics, Guizhou University, Guiyang, 550025, P.R. China. 2School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, P.R. China. 3School of Economics, Shanghai University of Finance and Economics, Shanghai, 200433, P.R. China. 4Key Laboratory of Mathematical Economics (SUFE), Ministry of Education, Shanghai, 200433, P.R. China.

Acknowledgements

This research is supported by National Natural Science Foundation of China (Nos. 11501349,61472093,11361012), the Chen Guang Project sponsored by the Shanghai Municipal Education Commission and Shanghai Education Development Foundation (No. 13CG35) and the Youth Project for Natural Science Foundation of Guizhou Educational Committee (No. [2015]421).The authors wish to thank the anonymous referees for their constructive comments and suggestions that significantly improved the exposition of the paper.

Endnote

a This result and its proof has been suggested by an anonymous referee. Received: 2 November 2015 Accepted: 17 June 2016 Published online: 24 June 2016 References

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