  We investigate a mean field game model for the production of exhaustible
resources. In this model, firms produce comparable goods, strategically set
their production rate in order to maximise profit, and leave the market as soon
as they deplete their capacities. We examine the related Mean Field Game system
and prove well-posedness for initial measure data by deriving suitable a priori
estimates. Then, we show that feedback strategies which are computed from the
Mean Field Game system provide $\varepsilon$-Nash equilibria to the
corresponding $N$-Player Cournot game, for large values of $N$. This is done by
showing tightness of the empirical process in the Skorokhod ${\rm M 1}$
topology, which is defined for distribution-valued processes.
