  We present a new combined \textit{mean field control game} (MFCG) problem
which can be interpreted as a competitive game between collaborating groups and
its solution as a Nash equilibrium between groups. Players coordinate their
strategies within each group. An example is a modification of the classical
trader's problem. Groups of traders maximize their wealth. They face cost for
their transactions, for their own terminal positions, and for the average
holding within their group. The asset price is impacted by the trades of all
agents. We propose a three-timescale reinforcement learning algorithm to
approximate the solution of such MFCG problems. We test the algorithm on
benchmark linear-quadratic specifications for which we provide analytic
solutions.
