  Mean field game theory studies the behavior of a large number of interacting
individuals in a game theoretic setting and has received a lot of attention in
the past decade (Lasry and Lions, Japanese journal of mathematics, 2007). In
this work, we derive mean field game partial differential equation systems from
deterministic microscopic agent dynamics. The dynamics are given by a
particular class of ordinary differential equations, for which an optimal
strategy can be computed (Bressan, Milan Journal of Mathematics, 2011). We use
the concept of Nash equilibria and apply the dynamic programming principle to
derive the mean field limit equations and we study the scaling behavior of the
system as the number of agents tends to infinity and find several mean field
game limits. Especially we avoid in our derivation the notion of measure
derivatives. Novel scales are motivated by an example of an agent-based
financial market model.
