This paper models two-sided market platforms, which connect third-party suppliers (developers) of many different products and services to users who demand a variety of these products. From a positive perspective, our model provides a simple explanation for the stark differences in platform pricing structures observed across a range of industries, including software for computers and an increasing number of electronic devices, videogames, digital media, etc. We show that the optimal platform pricing structure shifts towards making a larger share of profits on developers when users have a stronger preference for variety and also when there is uncertainty with respect to the availability, or a limited supply, of third-party (high-quality) products. From a normative perspective, we show that the increasingly popular public policy presumption that open platforms are inherently more efficient than proprietary ones -in terms of induced product diversity, user adoption and total social welfare- is not justified in our framework. The key welfare tradeoff is between the extent to which a proprietary platform internalizes business-stealing, product diversity and indirect network effects and the two-sided deadweight loss it creates through monopoly pricing.