We show that efficient bargaining is impossible for a wide class of economic settings and property rights. These settings are characterized by (i) the existence of “adverse efficient opt-out types”, whose participation does not change the efficient allocation and who, when they opt out, are the worst type other agents can face, and (ii) non-existence of the “marginal core”, and its multivaluedness with a positive probability. We also examine the optimal allocation of property rights within a given class that satisfies (i), such as simple property rights, liability rules, and dual-chooser rules. We characterize property rights that minimize the expected subsidy required to implement efficiency. With two agents, simple property rights that ar...