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We study a mean field game in continuous time over a finite horizon, T, wherethe state of each agent is binary and where players base their strategicdecisions on two, possibly competing, factors: the willingness to align withthe majority (conformism) and the aspiration of sticking with the own type(stubbornness). We also consider a quadratic cost related to the rate at whicha 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 Nashequilibrium, even though the corresponding N-player game does not. Moreover, itexhibits a very rich phase diagram, where polarized/unpolarized,coherent/incoherent equilibria may coexist, except for T small, where theequilibrium is always unique. We fully describe such phase diagram in closedform and provide a detailed numerical analysis of the N-player counterpart ofthe mean field game. In this finite dimensional setting, the equilibriumselected by the population of players is always coherent (favoring thesubpopulation whose type is aligned with the initial condition), but it doesnot necessarily minimize the cost functional. Rather, it seems that, among thecoherent ones, the equilibrium prevailing is the one that most benefits theunderdog subpopulation forced to change opinion.

Polarization and coherence in mean field games driven by private and social utility