Drawing from an Urn is Isometric

Drawing (a multiset of) coloured balls from an urn is one of the most basic models in discrete probability theory. Three modes of drawing are commonly distinguished: multinomial (draw-replace), hypergeometric (draw-delete), and Pólya (draw-add). These drawing operations are represented as maps from urns to distributions over multisets of draws. The set of urns is a metric space via the Wasserstein distance. The set of distributions over draws is also a metric space, using Wasserstein-over-Wasserstein. The main result of this paper is that the three draw operations are all isometries, that is, they preserve the Wasserstein distances.