  We study a model of delegation in which a principal takes a multidimensional
action and an agent has private information about a multidimensional state of
the world. The principal can design any direct mechanism, including stochastic
ones. We provide necessary and sufficient conditions for an arbitrary mechanism
to maximize the principal's expected payoff. We also discuss simple conditions
which ensure that some convex delegation set is optimal. A key step of our
analysis shows that a mechanism is incentive compatible if and only if its
induced indirect utility is convex and lies below the agent's first-best
payoff.
