  Mean field games are concerned with the limit of large-population stochastic
differential games where the agents interact through their empirical
distribution. In the classical setting, the number of players is large but
fixed throughout the game. However, in various applications, such as population
dynamics or economic growth, the number of players can vary across time which
may lead to different Nash equilibria. For this reason, we introduce a
branching mechanism in the population of agents and obtain a variation on the
mean field game problem. As a first step, we study a simple model using a PDE
approach to illustrate the main differences with the classical setting. We
prove existence of a solution and show that it provides an approximate
Nash-equilibrium for large population games. We also present a numerical
example for a linear--quadratic model. Then we study the problem in a general
setting by a probabilistic approach. It is based upon the relaxed formulation
of stochastic control problems which allows us to obtain a general existence
result.
