Open Access

Thomas Michael Gottfried

• Generalized Pólya urns with non-linear feedback are an established probabilistic model to describe the dynamics of growth processes with reinforcement. Depending on the feedback function, this process exhibits monopoly, where a single, random agent (or colour) achieves full market share, or it converges to a deterministic limit point. This work provides a comprehensive account of the properties of this process for fairly general feedback. In the monopoly case, we derive results on the prediction of the monopolist for large initial market size and in the deterministic case, we give an asymptotic description of the evolution of market shares. We describe in detail how monopolies emerge in a transition from sub-linear to super-linear feedback via hierarchical states close to linearity. Moreover, we derive a scaling limit for the dynamics and characterize the fluctuations in a functional central limit theorem. By choosing a different approach, we generalize known results on the number ofGeneralized Pólya urns with non-linear feedback are an established probabilistic model to describe the dynamics of growth processes with reinforcement. Depending on the feedback function, this process exhibits monopoly, where a single, random agent (or colour) achieves full market share, or it converges to a deterministic limit point. This work provides a comprehensive account of the properties of this process for fairly general feedback. In the monopoly case, we derive results on the prediction of the monopolist for large initial market size and in the deterministic case, we give an asymptotic description of the evolution of market shares. We describe in detail how monopolies emerge in a transition from sub-linear to super-linear feedback via hierarchical states close to linearity. Moreover, we derive a scaling limit for the dynamics and characterize the fluctuations in a functional central limit theorem. By choosing a different approach, we generalize known results on the number of steps won by the losing agents in the case of strong monopoly, which even provides information on dependencies between several losers and on the tail distribution of further related quantities like the time of monopoly. As a seeming paradox, losers with feedback close to the identity are most likely to win in many steps. Finally, we suggest an extended version of the non-linear Pólya urn as a model for the dynamics of wealth distribution within an economy, where we particularly highlight the empirical observation that wealth is distributed significantly more unequal than wages. This allows some interesting predictions for future developments.…