  Mean Field Games provide a powerful framework to analyze the dynamics of a
large number of controlled objects in interaction. Though these models are much
simpler than the underlying differential games they describe in some limit,
their behavior is still far from being fully understood. When the system is
confined, a notion of "ergodic state" has been introduced that characterizes
most of the dynamics for long optimization times. Here we consider a class of
models without such an ergodic state, and show the existence of a scaling
solution that plays similar role. Its universality and scaling behavior can be
inferred from a mapping to an electrostatic problem.
