  There are many settings in which a principal performs a task by delegating it
to an agent, who searches over possible solutions and proposes one to the
principal. This describes many aspects of the workflow within organizations, as
well as many of the activities undertaken by regulatory bodies, who often
obtain relevant information from the parties being regulated through a process
of delegation. A fundamental tension underlying delegation is the fact that the
agent's interests will typically differ -- potentially significantly -- from
the interests of the principal, and as a result the agent may propose solutions
based on their own incentives that are inefficient for the principal. A basic
problem, therefore, is to design mechanisms by which the principal can
constrain the set of proposals they are willing to accept from the agent, to
ensure a certain level of quality for the principal from the proposed solution.
  In this work, we investigate how much the principal loses -- quantitatively,
in terms of the objective they are trying to optimize -- when they delegate to
an agent. We develop a methodology for bounding this loss of efficiency, and
show that in a very general model of delegation, there is a family of
mechanisms achieving a universal bound on the ratio between the quality of the
solution obtained through delegation and the quality the principal could obtain
in an idealized benchmark where they searched for a solution themself.
Moreover, it is possible to achieve such bounds through mechanisms with a
natural threshold structure, which are thus structurally simpler than the
optimal mechanisms typically considered in the literature on delegation. At the
heart of our framework is an unexpected connection between delegation and the
analysis of prophet inequalities, which we leverage to provide bounds on the
behavior of our delegation mechanisms.
