  Mean field games are studied by means of the weak formulation of stochastic
optimal control. This approach allows the mean field interactions to enter
through both state and control processes and take a form which is general
enough to include rank and nearest-neighbor effects. Moreover, the data may
depend discontinuously on the state variable, and more generally its entire
history. Existence and uniqueness results are proven, along with a procedure
for identifying and constructing distributed strategies which provide
approximate Nash equlibria for finite-player games. Our results are applied to
a new class of multi-agent price impact models and a class of flocking models
for which we prove existence of equilibria.
