  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.
