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.
Keywords
mean field games, optimal control, Nash equilibrium,
congestion games
