On mean field games models for exhaustible commodities trade

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.