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In many markets the interaction between individuals and competing firms is
mediated by a strategic platform. A defining feature of modern platforms is their
extensive ability to control interactions. To understand such markets, I incorporate
within-side externalities into a general mechanism-design model of many-to-many
matching in two-sided markets. I characterize platform-optimal mechanisms and
establish comparative statics. I analyze the implications and welfare effects of
changes in market structure, including vertical integration by the platform and
changes in the platform's information about firms. I then extend the canonical
Dixit-Stiglitz model of monopolistic competition by giving the platform control
over consideration sets.
