Major-minor mean field games: common noise helps

Abstract: The objective of this work is to study the existence, uniqueness, and stability of equilibria in mean field games involving a major player and a continuum of minor players over finite intervals of arbitrary length. Following earlier articles addressing similar questions in the context of classical mean field games, the cost functions for the minor players are assumed to satisfy the Lasry-Lions monotonicity condition. In this contribution, we demonstrate that if, in addition to the monotonicity condition, the intensity of the (Brownian) noise driving the major player is sufficiently high, then -- under further mild regularity assumptions on the coefficients -- existence, uniqueness, and stability of equilibria are guaranteed. A key challenge is to show that the threshold (beyond which the noise intensity must be taken) can be chosen independently of the length of the time interval over which the game is defined. Building on the stability properties thus established, we further show that the associated system of master equations admits a unique classical solution. To the best of our knowledge, this is the first result of its kind for major-minor mean field games defined over intervals of arbitrary length.