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

Convergence of some Mean Field Games systems to aggregation and flocking models