Abstract: Nash systems are strongly coupled systems of semilinar parabolic equations that describe closed-loop Nash equilibria in stochastic differential games. Despite existence, uniqueness and regularity for this class of systems is now classical, new questions regarding the regularity of solutions as the number of players increases have been posed within the study of large population games. It is indeed desirable to justify limiting models for these games as the number of players goes to infinity; from a PDE perspective, this amounts to obtain Lipschitz estimates (and further rather precise information on higher order derivatives) as the number of equations goes to infinity. The talk will be devoted to the presentation of some results in this direction, obtained in collaboration with D. F. Redaelli (University of Rome Tor Vergata). This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.