Multi-dimensional mechanism design problems stand out as some of the most difficult problems in contemporary economic theory. Despite their clear importance in applications and more than four decades of research in economics and computer science, they remain poorly understood.

A principal must decide between two options. Which one she prefers depends on the private information of two agents. One agent always prefers the first option; the other always prefers the second. Transfers are infeasible. One application of this setting is the efficient division of a fixed budget between two competing departments. We first characterize all implementable mechanisms under arbitrary correlation. Second, we study when there exists a mechanism that yields the principal a higher payoff than she could receive by choosing the ex-ante optimal decision without consulting the agents. In the budget example, such a profitable mechanism exists if and only if the information of one department is also relevant for the expected returns of the other department. We generalize this insight to derive necessary and sufficient conditions for the existence of a profitable mechanism in the n-agent allocation problem with independent types.