Analytic Combinatorics of the Mabinogion Urn

The Mabinogion urn is a simple model of the spread of influences amongst versatile populations. It corresponds to a non-standard urn with balls of two colours: each time a ball is drawn, it causes a ball of the other kind to switch its colour. The process stops once unanimity has been reached. This note provides analytic expressions describing the evolution of the Mabinogion urn, based on a time-reversal transformation applied to the classical Ehrenfest urn. Consequences include a precise asymptotic analysis of the stopping-time distribution—it is asymptotically normal in the “unfair” case and akin to an extreme-value (double exponential) distribution in the “fair” case—as well as a characterization of the exponentially small probability of reversing a majority.