FIXED POINT THEORETIC ANALYSIS OF GAME THEORY

Abstract

The existence of Nash equilibrium is one of the most fundamental results in game theory with wide applications across economics, optimization, and computer science. This study examines the existence of Nash equilibrium in games with finite strategies using fixed point theorems. The paper partly focuses on Brouwer’s fixed-point theorem, Kakutani’s fixed-point theorem, and Sperner’s lemma, which pave the core foundation in proving the existence of the Nash equilibrium. This paper focuses on the importance of the Nash equilibrium being a fixed point of the best response correspondence. This paper also contains the statement and proof of the important lemma of Sperner in combinatorics and applies it in the proof of Brouwer’s fixed-point theorem, which demonstrates an intriguing interrelation between combinatorics, topology, and game theory. Finally, the paper discusses the existence of the Nash equilibrium in infinite strategic games using Schauder’s fixed-point theorem.