strategies, averaging out all of the possible distributions of the opponents $x_j$ with their mixed strategies $\sigma_j$. In particular, $J(x_i,u,x_j,u')$ is the payoff of the game between player $x_j$ with strategy $u$ against player $x_j$ with strategy $u'$. For the heuristic interpretation and the modeling issues related to we refer the interested reader to [@BorgersSarin; @CH2017]. The paper [@AFMS2018] introduced suitable notions (namely, Lagrangian and Eulerian) of solutions to the mean-field limit of showing convergence of the particle model to the limit description by means of stability estimates in the Wasserstein metric arising from the well-posedness theory of ODE’s in Banach spaces [@Brezis; @Cartan; @Dieudonne].

While sharing some common features, the dynamics described by and show a relevant difference in the structures of the velocity fields in the right-hand sides. Indeed, in our setting, it is natural to postulate that the velocity of the agents is also depending on the behavior of the ones around them, at least in a small neighborhood. Such a feature is not present in the first equation in , whereas it is encoded in our model by considering a velocity field $v_\Psi$ depending on the global state of the system. Beside this main distinction, also 