  This paper studies delegation in a model of discrete choice. In the
delegation problem, an uninformed principal must consult an informed agent to
make a decision. Both the agent and principal have preferences over the
decided-upon action which vary based on the state of the world, and which may
not be aligned. The principal may commit to a mechanism, which maps reports of
the agent to actions. When this mechanism is deterministic, it can take the
form of a menu of actions, from which the agent simply chooses upon observing
the state. In this case, the principal is said to have delegated the choice of
action to the agent.
  We consider a setting where the decision being delegated is a choice of a
utility-maximizing action from a set of several options. We assume the shared
portion of the agent's and principal's utilities is drawn from a distribution
known to the principal, and that utility misalignment takes the form of a known
bias for or against each action. We provide tight approximation analyses for
simple threshold policies under three increasingly general sets of assumptions.
With independently-distributed utilities, we prove a $3$-approximation. When
the agent has an outside option the principal cannot rule out, the constant
approximation fails, but we prove a $\log \rho/\log\log \rho$-approximation,
where $\rho$ is the ratio of the maximum value to the optimal utility. We also
give a weaker but tight bound that holds for correlated values, and complement
our upper bounds with hardness results. One special case of our model is
utility-based assortment optimization, for which our results are new.
