Lecture on the first SFB/TR 15 meeting, Gummersbach, July, 18 - 20, 2004: We develop a general theoretical framework of trade on a platform on which buyers and sellers interact. The platform may be owned by a single large, or many small independent or vertically integrated intermediaries. We provide a positive and normative analysis of the impact of platform ownership structure on platform size. The strength of network effects is important in the ranking of ownership structures by induced platform size and welfare. While vertical integration may be welfare-enhancing if network effects are weak, monopoly platform ownership is socially preferred if they are strong. These are also the ownership structures likely to emerge.