Two-sided market models in which platforms compete via two-part tariffs, i.e. a subscription and a per-transaction fee, are often plagued by a continuum of equilibria. This paper augments existing models by allowing for heterogeneous rading behavior of agents on both sides. We show that this simple method yields a unique equilibrium even in the limit as the heterogeneity vanishes. In case of competitive bottlenecks we find that in this equilibrium platforms benefit from the possibility to price discriminate if per-transaction costs are relatively large. This is the case because two-part tariffs allow platforms to better distribute these costs among the two sides. Under two-sided single-homing price discrimination hurts platforms if per-transaction fees can be negative.