Teorema de punto fijo de Kakutani en topología y su aplicación en la existencia de equilibrio de Nash en la teoría de juegos

Abstract

The Escuela Superior Politécnica de Chimborazo (ESPOCH) does not have enough bibliography about the fixed-point theorem in topology, therefore, the objective is to generate a reference document, in which Kakutani’s fixed point theorem in topology and its application in the existence of Nash equilibrium in game theory is studied. For this purpose, we have used a non-interactive qualitative methodological design with descriptive scope, with the purpose of to produce a guiding document for mathematics students of the Escuela Superior Politécnica de Chimborazo (ESPOCH). As a result, a bibliographic document was generated, which consists of 3 chapters. In the first chapter there are topological preliminaries such as: open and closed sets, continuous functions, homeomorphisms, connected spaces, metric spaces, compact spaces and the intermediate value theorem. In the second chapter, it is observed that Brouwer’s fixed point theorem which consists of fixed-point properties, homotropies, circular functions of degree and the non-retraction theorem with which the equivalence between Brouwer’s fixed point theorem in two dimensions is demonstrated. Finally, the third chapter introduces Kakutani’s fixed point theorem and its application in game theory, which is composed of multivalued functions, correspondences, game theory and the existence of Nash equilibrium. In conclusion, from the topological results mentioned in the final guidance paper, Brouwer’s fixed point theorem was proved in two dimensions. Subsequently, through the tools of game theory and Kakutani’s fixed point theorem in two dimensions, the proof of the existence theorem of Nash equilibrium was achieved.