The Power of Choice in a Generalized Pólya Urn Model

Abstract

We establish some basic properties of a “Pólya choice” generalization of the standard Pólya urn process. From a set of k urns, the ith occupied by n _i balls, choose c distinct urns i ₁,...,i _c with probability proportional to $n_{i_1}^\gamma \times \cdots\times n_{i_c}^\gamma$, where γ> 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 < γ< 1, the urn occupancies are asymptotically equal with probability 1. For γ> 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.