IAENG International Journal of Applied Mathematics, 41:2, IJAM_41_2_08 ______________________________________________________________________________________ Proof of Constructive Version of the Fan-Glicksberg Fixed Point Theorem Directly by Sperner’s Lemma and Approximate Nash Equilibrium with Continuous Strategies: A Constructive Analysis Yasuhito Tanaka, Member, IAENG Abstract—It is often demonstrated that Brouwer’s fixed fixed point theorem can be constructively proved using that point theorem can not be constructively proved. Therefore, of Brouwer’s theorem. Kakutani’s fixed point theorem, the Fan-Glicksberg fixed point Then, can we prove a constructive version of the Fan- theorem and the existence of a pure strategy Nash equilibrium in a strategic game with continuous (infinite) strategies and Glicksberg fixed point theorem directly by Sperner’s lemma? quasi-concave payoff functions also can not be constructively We present such a proof, and we will show the existence of proved. On the other hand, however, Sperner’s lemma which an approximate pure strategy Nash equilibrium in a strategic is used to prove Brouwer’s fixed point theorem can be con- game with continuous strategies and quasi-concave payoff structively proved. Some authors have presented a construc- functions. tive (or an approximate) version of Brouwer’s theorem using Sperner’s lemma. Thus, Brouwer’s fixed point theorem can be Let (pi)i2I be a family of semi-norms on a locally convex constructively proved in its constructive version. It seems that space where I is an index set, for example, the set of constructive versions of Kakutani’s fixed point theorem and positive integers, and F be a finitely enumerable subset the Fan-Glicksberg fixed point theorem can be constructively of I. A constructive version of the Fan-Glicksberg fixed proved using that of Brouwer’s theorem. Then, can we prove a point theorem states that for any compact and convex val- constructive version of the Fan-Glicksberg fixed point theorem directly by Sperner’s lemma? We present such a proof, and ued multi-function (multi-valued function or correspondence) we will show the existence of an approximate pure strategy with closed graph from a compact and convex set in a Nash equilibrium in a strategic game with continuous strategies locally convex space to the set of its inhabited (nonempty) and quasi-concave payoff functions. We follow the Bishop style subsets, there exists an approximate fixed point. We consider constructive mathematics. a uniform version of the property of closed graph for multi- Index Terms—constructive version of the Fan-Glicksberg functions, and call such a multi-function a multi-function fixed point theorem, approximate pure strategy Nash equi- with uniformly closed graph, or say that a multi-function librium, continuous strategy, quasi-concave payoff function, uniformly has a closed graph. An approximate fixed point x Sperner’s lemma. Pof a multi-function Φ for each " > 0 is a point such that − ⊂ i2F pi(x Φ(x)) < " is satisfied for each F I, where I. INTRODUCTION pi(x − Φ(x)) = infy2Φ(x) pi(x − y). An approximate pure T is often demonstrated that Brouwer’s fixed point strategy Nash equilibrium is a state where strategies chosen I theorem can not be constructively proved. Therefore, by all players are best responses each other within the range Kakutani’s fixed point theorem, the Fan-Glicksberg fixed of ". We call such strategies approximate best responses. point theorem and the existence of a pure strategy Nash In the next section we prove Sperner’s lemma. Our proof equilibrium in a strategic game with continuous (infinite) is a standard constructive proof. In Section 3 we prove strategies and quasi-concave payoff functions also can not be a constructive version of the Fan-Glicksberg fixed point constructively proved. On the other hand, however, Sperner’s theorem by Sperner’s lemma. In Section 4 we will show lemma which is used to prove Brouwer’s fixed point theorem the existence of an approximate pure strategy Nash equi- can be constructively proved. Some authors have presented librium in a strategic game with continuous strategies and a constructive (or an approximate) version of Brouwer’s quasi-concave payoff functions. We follow the Bishop style theorem using Sperner’s lemma. Thus, Brouwer’s fixed point constructive mathematics according to [1], [2] and [3]. theorem can be constructively proved in its constructive version. See [4] and [9]. It seems that constructive versions II. SPERNER’S LEMMA of Kakutani’s fixed point theorem and the Fan-Glicksberg To prove Sperner’s lemma we use the following simple result in graph theory, Handshaking lemma1.A graph refers Manuscript received December 29, 2010; revised April 20, 2011. This work was supported in part by the Ministry of Education, Science, Sports to a collection of vertices and a collection of edges that and Culture of Japan, Grant-in-Aid for Scientific Research (C), 20530165, connect pairs of vertices. Each graph may be undirected or and the Special Costs for Graduate Schools of the Special Expenses for directed. Fig. 1 is an example of an undirected graph. The Hitech Promotion by the Ministry of Education, Science, Sports and Culture of Japan in 2010. degree of a vertex of a graph is defined to be the number of Yasuhito Tanaka is with the Faculty of Economics, Doshisha University, Kyoto, Japan. e-mail: [email protected]. 1For another constructive proof of Sperner’s lemma, see [7]. (Advance online publication: 24 May 2011) IAENG International Journal of Applied Mathematics, 41:2, IJAM_41_2_08 ______________________________________________________________________________________ 2 H C D F 2 1 E 1 2 2 G 0 2 2 1 A B 0 0 1 0 1 Fig. 1. Example of graph 0 1 edges incident to the vertex, with loops counted twice. Each 1 1 0 0 vertex has odd degree or even degree. Let v denote a vertex and V denote the set of all vertices. Fig. 2. Partition and labeling of 2-dimensional simplex Lemma 1 (Handshaking lemma): Every undirected graph contains an even number of vertices of odd degree. That is, the number of vertices that have an odd number of incident whose k-th coordinate (k = 0; 1; : : : ; n) is 1 and all edges must be even. other coordinates are 0 is labeled with k. − It is a simple lemma. But for completeness of arguments 2) If a vertex of simplices of K is contained in an n 1- we provide a proof. dimensional face of ∆, then that vertex is labeled with some number which is the same as the number of a Proof: Prove this lemma by double counting. Let d(v) vertex of that face. be the degree of vertex v. The number of vertex-edge 3) If a vertex of simplices of K is contained in an n − incidences in the graph may be counted in two different 2-dimensional face of ∆, then that vertex is labeled ways: by summing the degrees of the vertices, or by counting with some number which is the same as the number two incidences for every edge. Therefore, X of a vertex of that face. And so on for cases of lower d(v) = 2e; dimension. v2V 4) A vertex contained in inside of ∆ is labeled with arbitrary number among 0, 1, ::: , n. where e is the number of edges in the graph. The sum of A small simplex of K which is labeled with the numbers the degrees of the vertices is therefore an even number. It 0, 1, ::: , n is called a fully labeled simplex. Now let us prove could happen if and only if an even number of the vertices Sperner’s lemma. had odd degree. Lemma 2 (Sperner’s lemma): If we label the vertices of Let ∆ denote an n-dimensional simplex. n is a positive K following above rules 1) ∼ 4), then there are an odd integer at least 2. For example, a 2-dimensional simplex is number of fully labeled simplices. Thus, there exists at least a triangle. Let partition or triangulate the simplex. Fig. 2 is one fully labeled simplex. an example of partition (triangulation) of a 2-dimensional Proof: Our proof is standard. But again for completeness simplex. In a 2-dimensional case we divide each side of ∆ of arguments we provide a proof in Appendix A. in m equal segments, and draw the lines parallel to the sides n ∆ of ∆. Then, the 2-dimensional simplex is partitioned into m2 Since and partition of are finite, the number of small triangles. We consider partition of ∆ inductively for cases of simplices constructed by partition is also finite. Thus, we can n higher dimension. In a 3 dimensional case each face of ∆ constructively find a fully labeled -dimensional simplex of K is an 2-dimensional simplex, and so it is partitioned into m2 through finite steps. triangles in the way above mentioned, and draw the planes parallel to the faces of ∆. Then, the 3-dimensional simplex III. CONSTRUCTIVE VERSION OF THE FAN-GLICKSBERG is partitioned into m3 trigonal pyramids. And so on for cases FIXED POINT THEOREM of higher dimension. In this section we will prove a constructive version of the Let K denote the set of small n-dimensional simplices Fan-Glicksberg fixed point theorem using Sperner’s lemma. of ∆ constructed by partition. The vertices of these small The classical Fan-Glicksberg theorem ([5] and [6]) is stated simplices of K are labeled with the numbers 0, 1, 2, ::: , n as follows. subject to the following rule. Theorem 1 (The Fan-Glicksberg fixed point theorem): 1) The vertices of ∆ are respectively labeled with 0 Let X be a compact and convex subset of a locally to n. We label a point (1; 0;:::; 0) with 0, a point convex space E, and Φ be a compact and convex valued (0; 1; 0;:::; 0) with 1, a point (0; 0; 1 :::; 0) with 2, multi-function with closed graph from X to the set of its ::: , a point (0;:::; 0; 1) with n.