  'Rich get richer' rule comforts previously often chosen actions. What is
happening to the evolution of individual inclinations to choose an action when
agents do interact ? Interaction tends to homogenize while each individual
dynamics tends to reinforce its own position. Interacting stochastic systems of
reinforced processes were recently considered in many papers, where the
asymptotic behavior was proven to exhibit a.s. synchronization. We consider in
this paper models where, even if interaction among agents is present, absence
of synchronization may happen due to the choice of an individual non-linear
reinforcement. We show how these systems can naturally be considered as models
for coordination games, technological or opinion dynamics.
