  Mean field games were introduced by J-M.Lasry and P-L. Lions in the
mathematical community, and independently by M. Huang and co-workers in the
engineering community, to deal with optimization problems when the number of
agents becomes very large. In this article we study in detail a particular
example called the 'seminar problem' introduced by O.Gu\'eant, J-M Lasry, and
P-L. Lions in 2010. This model contains the main ingredients of any mean field
game but has the particular feature that all agent are coupled only through a
simple random event (the seminar starting time) that they all contribute to
form. In the mean field limit, this event becomes deterministic and its value
can be fixed through a self consistent procedure. This allows for a rather
thorough understanding of the solutions of the problem, through both exact
results and a detailed analysis of various limiting regimes. For a sensible
class of initial configurations, distinct behaviors can be associated to
different domains in the parameter space . For this reason, the 'seminar
problem' appears to be an interesting toy model on which both intuition and
technical approaches can be tested as a preliminary study toward more complex
mean field game models.
