  For two classes of Mean Field Game systems we study the convergence of
solutions as the interest rate in the cost functional becomes very large,
modeling agents caring only about a very short time-horizon, and the cost of
the control becomes very cheap. The limit in both cases is a single first order
integro-partial differential equation for the evolution of the mass density.
The first model is a 2nd order MFG system with vanishing viscosity, and the
limit is an aggregation equation. The result has an interpretation for models
of collective animal behaviour and of crowd dynamics. The second class of
problems are 1st order MFGs of acceleration and the limit is the kinetic
equation associated to the Cucker-Smale model. The first problem is analyzed by
PDE methods, whereas the second is studied by variational methods in the space
of probability measures on trajectories.
