Mean field games of controls: Propagation of monotonicities

<p style='text-indent:20px;'>The theory of Mean Field Game Controls considers a class mean field games where the interaction is through joint distribution state and control. It well known that, for standard games, certain monotonicity conditions are crucial to guarantee uniqueness equilibria then global wellposedness master equations. In literature condition could be Lasry–Lions monotonicity, displacement or anti-monotonicity conditions. this paper, we investigate these three types Games show their propagation along solutions equations with common noises. particular, extend semi-monotonicity, whose result new even games. This first step towards Controls.</p>