We analyze a fractional mean field game of controls system, showing existence of solutions when the order of the fractional Laplacian is \(s\in \left( \frac{1}{2},1\right) \) . Here the running cost depends nonlocally on the distribution \(\mu \) of not only the states but also optimal strategies. The coupling is assumed to be smoothing and satisfy the Lasry-Lions monotonicity condition. We derive three types of a priori estimates on solutions. First, we use the monotonicity condition to derive moment estimates on \(\mu \) . Second, we derive abstract estimates on fractional parabolic equations and apply them to the mean field game. Third, we derive new estimates on the time regularity of the distribution \(\mu \) by analyzing the associated Lévy process. We apply these estimates and the Leray-Schauder fixed point theorem to establish existence of solutions.
