This study focuses on the development of an online inertial mirror descent algorithm for characterizing a novel class of averaged \(\varepsilon \) -Nash equilibrium in a class of non-cooperative multiplayer games with dynamic strategies in continuous time. Each player’s dynamic strategy is subject to constraints defined on a compact and convex set. Using the Tanaka-Yokohama formula, which characterizes the Nash \(\varepsilon \) -equilibrium, the strategy is determined for each player in the dynamic game. The Min–Max property of this function confirms the existence of the Nash \(\varepsilon \) -equilibrium. The algorithm design employs the Legendre–Fenchel transform and a selected proxy function to facilitate the inertial mirror descent approach for the averaged trajectories of the game dynamics. This transformation is instrumental in proving the convergence to the Nash \(\varepsilon \) -equilibrium with a rate of \(O(t^{-1})\) . In addition, various proxy functions are proposed and analyzed for their effectiveness in constructing the online inertial mirror descent algorithm. A numerical example contributes to evidence of the application of the mirror descent algorithm presented in this study, considering two players with a state of three components each. A particular selection of a proxy function characterizes the existence of the Nash \(\epsilon \) -equilibrium.
