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

It is often demonstrated that Brouwer’s fixed point theorem can not be constructively proved. Therefore, Kakutani’s fixed point theorem, the Fan-Glicksberg fixed point theorem and the existence of a pure strategy Nash equilibrium in a strategic game with continuous (infinite) strategies and quasi-concave payoff functions also can not be constructively proved. On the other hand, however, Sperner’s lemma which is used to prove Brouwer’s fixed point theorem can be constructively proved. Some authors have presented a constructive (or an approximate) version of Brouwer’s theorem using Sperner’s lemma. Thus, Brouwer’s fixed point theorem can be constructively proved in its constructive version. It seems that constructive versions of Kakutani’s fixed point theorem and the Fan-Glicksberg fixed point theorem can be constructively proved using that of Brouwer’s theorem. Then, can we prove a constructive version of the Fan-Glicksberg fixed point theorem directly by Sperner’s lemma? We present such a proof, and we will show the existence of an approximate pure strategy Nash equilibrium in a strategic game with continuous strategies and quasi-concave payoff functions. We follow the Bishop style constructive mathematics.