Stationary and initial-terminal value problem for collective decision making via mean-field games

Given a large number of homogeneous players that are distributed across three possible states, we consider the problem in which these players have to control their transition rates, following some optimality criteria. The optimal transition rates are based on the players' knowledge of their current state and of the distribution of all the other players, thus introducing mean-field terms in the running and the terminal cost. The first contribution is a mean-field model that takes into account the macroscopic and the microscopic dynamics. The second contribution is the study of the mean-field equilibrium resulting from solving the initial-terminal value problem, involving the Kolmogorov equations and the Hamilton-Jacobi ODEs. The third contribution is the analysis of a stationary equilibrium for the system, which can be obtained in the asymptotic limit from the nonstationary equilibrium. We reframe our analysis within the context of Lyapunov's linearisation method and stability theory of nonlinear systems.