  We consider a class of deterministic mean field games, where the state
associated with each player evolves according to an ODE which is linear w.r.t.
the control. Existence, uniqueness, and stability of solutions are studied from
the point of view of generic theory. Within a suitable topological space of
dynamics and cost functionals, we prove that, for nearly all mean field
games(in the Baire category sense) the best reply map is single valued for a.e.
player. As a consequence, the mean field game admits a strong (not randomized)
solution. Examples are given of open sets of games admitting a single solution,
and other open sets admitting multiple solutions. Further examples show the
existence of an open set of MFG having a unique solution which is
asymptotically stable w.r.t. the best reply map, and another open set of MFG
having a unique solution which is unstable. We conclude with an example of a
MFG with terminal constraints which does not have any solution, not even in the
mild sense with randomized strategies.
