Equilibrium analysis in game theory using nonlinear functional analysis techniques

Abstract

Game theory provides a mathematical framework for analyzing the tactical choices of rational decision makers. The idea of equilibrium, which states that no player has a motivation to change their existing approach given the strategies of other players, is one of the core ideas of game theory. This study examines a type of undirected graph-based distributed quadratic game. The problem of communication topology constraints is presented and nonlinear dynamics with uncertain time-dependent perturbations are present in the participant's dynamics. Based on a high-gain observer approach, a distributed Nash equilibrium (NE) finding technique is given, and the Lyapunov stability theory is used to study the convergence. It represents that every player approximates the positions of their rival players and that there are differences between the NE and the placements of the minor limitation that finally restricts the players. In addition, chattering problems are eliminated since the offered theory's formulation employs the hyperbolic tangent function to control the perturbation rather than the signum function. In an imitation of the oligopoly match, five enterprises manufacture identical materials in a duopoly market framework; this is done to confirm the effectiveness of the recommended strategy. Our results provide novel perspectives and methods for understanding complicated strategic situations, helping to close the gap between mathematical theory and real-world applications.