A novel feature of platforms such as marketplaces and social networks is that non-participants may become worse off as others join. We show that in such settings, rational agents can be induced to join the platform despite being better off without it-a phenomenon we call a platform trap. We provide a theory of how platform traps emerge through one or more of the following additional features: (i) the ability of the platform to make dynamic price adjustments; (ii) the interplay between on-platform and off-platform negative externalities; (iii) favorable equilibrium selection where participation decisions admit multiple equilibria.