  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.
