The Power of Choice in a Generalized Pólya Urn Model

We establish some basic properties of a "Polya choice" generalization of the standard Polya urn process. From a set of kurns, the ith occupied by n i balls, choose cdistinct urns i 1 ,...,i c with probability proportional to $n_{i_1}^\gamma \times \cdots\times n_{i_c}^\gamma$, where i¾?> 0 is a constant parameter, and increment one with the smallest occupancy (breaking ties arbitrarily). We show that this model has a phase transition. If 0 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.