  We study a mean field game in continuous time over a finite horizon, T, where
the state of each agent is binary and where players base their strategic
decisions on two, possibly competing, factors: the willingness to align with
the majority (conformism) and the aspiration of sticking with the own type
(stubbornness). We also consider a quadratic cost related to the rate at which
a change in the state happens: changing opinion may be a costly operation.
Depending on the parameters of the model, the game may have more than one Nash
equilibrium, even though the corresponding N-player game does not. Moreover, it
exhibits a very rich phase diagram, where polarized/unpolarized,
coherent/incoherent equilibria may coexist, except for T small, where the
equilibrium is always unique. We fully describe such phase diagram in closed
form and provide a detailed numerical analysis of the N-player counterpart of
the mean field game. In this finite dimensional setting, the equilibrium
selected by the population of players is always coherent (favoring the
subpopulation whose type is aligned with the initial condition), but it does
not necessarily minimize the cost functional. Rather, it seems that, among the
coherent ones, the equilibrium prevailing is the one that most benefits the
underdog subpopulation forced to change opinion.
