  We study a particle approximation for one-dimensional first-order
Mean-Field-Games (MFGs) with local interactions with planning conditions. Our
problem comprises a system of a Hamilton-Jacobi equation coupled with a
transport equation. As we deal with the planning problem, we prescribe initial
and terminal distributions for the transport equation. The particle
approximation builds on a semi-discrete variational problem. First, we address
the existence and uniqueness of a solution to the semi-discrete variational
problem. Next, we show that our discretization preserves some previously
identified conserved quantities. Finally, we prove that the approximation by
particle systems preserves displacement convexity. We use this last property to
establish uniform estimates for the discrete problem. We illustrate our results
for the discrete problem with numerical examples.
