Constructive proof of the existence of Nash Equilibrium in a finite strategic game with sequentially locally non-constant payoff functions by Sperners lemma

arXiv:1103.1980v3 [math.LO] 23 Aug 2011 CONSTRUCTIVE PROOF OF THE EXISTENCE OF NASH EQUILIBRIUM IN A FINITE STRATEGIC GAME WITH SEQUENTIALLY LOCALLY NON-CONSTANT PAYOFF FUNCTIONS BY SPERNER’S LEMMA YASUHITO TANAKA Abstract. Using Sperner’s lemma for modified partition of a simplex we will constructively prove the existence of a Nash equilibrium in a finite strategic game with sequentially locally non-constant payofffunctions. We follow the Bishop style constructive mathematics. 1. Introduction It is often said that Brouwer’s fixed point theorem can not be constructively proved1. Thus, the existence of a Nash equilibrium in a finite strategic game also can not be constructively proved. Sperner’s lemma which is used to prove Brouwer’s theorem, however, can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer’s theorem using Sperner’s lemma. See [?] and [?]. Thus, Brouwer’s fixed point theorem can be constructively proved in its constructive version. Also Dalen in [?] states a conjecture that a uniformly continuous function f from a simplex to itself, with property that each open set contains a point x such that x ̸= f(x), which means |x −f(x)| > 0, and also at every point x on the boundaries of the simplex x ̸= f(x), has an exact fixed point. We call such a property of functions local non-constancy. Further we define a stronger property sequential local non-constancy. In another paper [?] we have constructively proved Dalen’s conjecture with sequential local non-constancy. In this paper we present a proof of the existence of a Nash equilibrium in a finite strategic game with sequentially locally non-constant payofffunctions by Sperner’s lemma. We consider Sperner’s lemma for modified partition of a simplex, and utilizing it prove the existence of such a Nash equilibrium. 2000 Mathematics Subject Classification. Primary 26E40, Secondary 91A10. Key words and phrases. Nash equilibrium, Sperner’s lemma, sequentially locally non-constant payofffunctions. 1[?] provided a constructive proof of Brouwer’s fixed point theorem. But it is not constructive from the view point of constructive mathematics ´a la Bishop. It is sufficient to say that one dimensional case of Brouwer’s fixed point theorem, that is, the intermediate value theorem is non-constructive. See [?] or [?]. On the other hand, in [?] Orevkov constructed a computably coded continuous function f from the unit square to itself, which is defined at each computable point of the square, such that f has no computable fixed point. His map consists of a retract of the computable elements of the square to its boundary followed by a rotation of the boundary of the square. As pointed out by Hirst in [?], since there is no retract of the square to its boundary, Orevkov’s map does not have a total extension. Brouwer’s fixed point theorem can be constructively, in the sense of constructive mathematics ´a la Bishop, proved only approximately. But the existence of an exact fixed point of a function which satisfies some property of local non-constancy may be constructively proved. 1 2 YASUHITO TANAKA A B C D E F G H Figure 1. Example of graph In the next section we prove a modified version of Sperner’s lemma. In Section 3 we present a proof of the existence of a Nash equilibrium in a finite strategic game with sequentially locally non-constant payofffunctions by the modified version of Sperner’s lemma. We follow the Bishop style constructive mathematics according to [?], [?] and [?]. 2. Sperner’s lemma To prove Sperner’s lemma we use the following simple result of graph theory, Handshaking lemma2. A graph refers to a collection of vertices and a collection of edges that connect pairs of vertices. Each graph may be undirected or directed. Figure 1 is an example of an undirected graph. Degree of a vertex of a graph is defined to be the number of edges incident to the vertex, with loops counted twice. Each vertex has odd degree or even degree. Let v denote a vertex and V denote the set of all vertices. 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 edges must be even. This is a simple lemma. But for completeness of arguments we provide a proof. Proof. Prove this lemma by double counting. Let d(v) be the degree of vertex v. The number of vertex-edge incidences in the graph may be counted in two different ways; by summing the degrees of the vertices, or by counting two incidences for every edge. Therefore, X v∈V d(v) = 2e, where e is the number of edges in the graph. The sum of the degrees of the vertices is therefore an even number. It could happen if and only if an even number of the vertices had odd degree. □ Let ∆denote an n-dimensional simplex. n is a finite natural number. For exam- ple, a 2-dimensional simplex is a triangle. Let partition or triangulate a simplex. Figure 2 is an example of partition (triangulation) of a 2

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