A model problem for first-order mean field games with discrete initial data

In this article, we study a simplified version of a density-dependent first-order mean field game, in which the players face a penalization equal to the population density at their final position. We consider the problem of finding an equilibrium when the initial distribution is a discrete measure. We show that the problem becomes finite-dimensional: the final piecewise smooth density is completely determined by the weights and positions of the initial measure. We establish existence and uniqueness of a solution using classical fixed point theorems. Finally, we show that Newton's method provides an effective way to compute the solution. Our numerical simulations provide an illustration of how density penalization in a mean field game tends to the smoothen the initial distribution.