The Power of Choice in a Generalized Polya Urn Model

HTML clipboard /*-->*/ /*-->*/ We introduce a "Polya choice" urn model combining elements of the well known "power of two choices" model and the "rich get richer" model. From a set of $k$ urns, randomly choose $c$ distinct urns with probability proportional to the product of a power $\gamma>0$ of their occupancies, and increment one with the smallest occupancy. The model has an interesting phase transition. If $\gamma \leq 1$, the urn occupancies are asymptotically equal with probability 1. For $\gamma>1$, this still occurs with positive probability, but there is also positive probability that some urns get only finitely many balls while others get infinitely many.