  Lagrangian duality underlies both classical and modern mechanism design. In
particular, the dual perspective often permits simple and detail-free
characterizations of optimal and approximately optimal mechanisms. This paper
applies this same methodology to a close cousin of traditional mechanism
design, one which shares conceptual and technical elements with its more mature
relative: the burgeoning field of persuasion. The dual perspective permits us
to analyze optimal persuasion schemes both in settings which have been analyzed
in prior work, as well as for natural generalizations which we are the first to
explore in depth. Most notably, we permit combining persuasion policies with
payments, which serve to augment the persuasion power of the scheme. In both
single and multi-receiver settings, as well as under a variety of constraints
on payments, we employ duality to obtain structural insights, as well as
tractable and simple characterizations of optimal policies.
