Feedback-evolving Mean-field Games

A natural assumption in games is to consider static payoffs. Yet, this is not true when the environment changes independently or as a result of players' interactions, e.g., geopolitical decisions in the global financial market, or weather conditions in autonomous driving. Indeed, these environmental aspects have a significant impact on the strategic interactions between players and vice versa. With the growing interest in machine learning approaches, disregarding these environmental changes leads to nonstationarity and instability of the corresponding algorithms. Motivated by this issue, we develop a novel framework for continuous-time finite-state feedback-evolving mean-field games (FEMFG) where the population dynamics are paired with an environmental resource which determines the payoffs and in turn evolves according to the population distribution across the underlying Markov chain. We derive the corresponding initial-terminal value problem and show the conditions for the existence of a feedback-evolving mean-field Nash equilibrium as the solution to the FEMFG, namely, when the population dynamics given by the Kolmogorov equation and the value function obtained via the Hamilton-Jacobi-Bellman equation do not change over time.