  We propose a new mean-field game model with two states to study
synchronization phenomena, and we provide a comprehensive characterization of
stationary and dynamic equilibria along with their stability properties. The
game undergoes a phase transition with increasing interaction strength. In the
subcritical regime, the uniform distribution, representing incoherence, is the
unique and stable stationary equilibrium. Above the critical interaction
threshold, the uniform equilibrium becomes unstable and there is a multiplicity
of stationary equilibria that are self-organizing. Under a discounted cost,
dynamic equilibria spiral around the uniform distribution before converging to
the self-organizing equilibria. With an ergodic cost, however, unexpected
periodic equilibria around the uniform distribution emerge.
