Unified continuous-time q-learning for mean-field game and mean-field control problems

This paper studies the continuous-time q-learning in mean-field jump-diffusion models in a setting where the environment simulator does not provide direct access to the population distribution. We propose the integrated q-function in decoupled form (decoupled Iq-function) and establish its martingale characterization, which provides a unified policy evaluation rule for both mean-field game (MFG) and mean-field control (MFC) problems. Moreover, we consider the learning procedure where population distribution is updated based on the representative agent's state values. Depending on the task to solve the MFG or MFC problem, we can employ the decoupled Iq-function differently to characterize the mean-field equilibrium policy or the mean-field optimal policy respectively. Based on these theoretical findings, we devise a unified parametric q-learning algorithm for both MFG and MFC problems by utilizing test policies and the averaged martingale orthogonality condition. In two applications within and beyond LQ framework, we illustrate the effectiveness and efficiency of our unified parametric q-learning algorithm for both MFG and MFC learning tasks.