  In this paper, we address an instance of uniquely solvable mean-field game
with a common noise whose corresponding counterpart without common noise has
several equilibria. We study the selection problem for this mean-field game
without common noise via three approaches.
  A common approach is to select, amongst all the equilibria, those yielding
the minimal cost for the representative player. Another one is to select
equilibria that are included in the support of the zero noise limit of the
mean-field game with common noise. A last one is to select equilibria supported
by the limit of the mean-field component of the corresponding $N$-player game
as the number of players goes to infinity. The contribution of this paper is to
show that, for the class under study, the last two approaches select the same
equilibria, but the first approach selects another one.
