On two-color monotonic self-equilibrium urn models

In this study, we focus on a class of two-color balanced urns with multiple drawings that has the property of monotonic self-equilibrium. We give the definition of a monotonic self-equilibrium urn model by specifying the form of its replacement matrix. At each step, a sample of size m ≥ 1 is drawn from the urn, and the replacement rule prespecified by a matrix is applied. The idea is to support whichever color that has fewer counts in the sample. Intuitively, for any urn scheme within this class, the proportions of white and blue balls in the urn tend to be equal asymptotically. We observe by simulation that, when n is large, the number of white balls in the urn within this class is around half of the total number of balls in the urn on average and is normally distributed. Within the class of affine urn schemes, we specify subclasses that have the property of monotonic self-equilibrium, and derive limiting behavior of the number of white balls using existing results. The class of non-affine urn schemes is not yet well developed in the literature. We work on a subclass of non-affine urn models that has the property of monotonic self-equilibrium. For the special case that one ball is added into the urn at each step, we derive limiting behavior of the expectation and the variance and prove convergence in probability for the proportion of white balls in the urn. An optimal strategy on urn balancing and application of monotonic self-equilibrium urn models are also briefly discussed