  Generalized P\'olya urns with non-linear feedback are an established
probabilistic model to describe the dynamics of growth processes with
reinforcement, a generic example being competition of agents in evolving
markets. It is well known which conditions on the feedback mechanism lead to
monopoly where a single agent achieves full market share, and various further
results for particular feedback mechanisms have been derived from different
perspectives. In this paper we provide a comprehensive account of the possible
asymptotic behaviour for a large general class of feedback, and describe in
detail how monopolies emerge in a transition from sub-linear to super-linear
feedback via hierarchical states close to linearity. We further distinguish
super- and sub-exponential feedback, which show conceptually interesting
differences to understand the monopoly case, and study robustness of the
asymptotics with respect to initial conditions, heterogeneities and small
changes of the feedback mechanisms. Finally, we derive a scaling limit for the
full time evolution of market shares in the limit of diverging initial market
size, including the description of typical fluctuations and extending previous
results in the context of stochastic approximation.
