Normalized matching property for competing urn models

We study the competing urn model in which m balls are placed independently into n urns according to (possibly distinct) ball distributions. Kahn and Neiman (2010) showed that, under identical ball distributions, the induced urn measure has conditional negative association property and asked whether this remains true without assuming identical distributions. We answer this in the affirmative by showing that the competing urn model satisfies the normalized matching property. This, in turn, implies conditional negative association for the induced urn measure with non-identical ball distributions, resolving the question of Kahn and Neiman.