  In persuasion problems where the receiver's action is one-dimensional and his
utility is single-peaked, optimal signals are characterized by duality, based
on a first-order approach to the receiver's problem. A signal is optimal iff
the induced joint distribution over states and actions is supported on a
compact set (the contact set) where the dual constraint binds. A signal that
pools at most two states in each realization is always optimal, and such
pairwise signals are the only solutions under a non-singularity condition on
utilities (the twist condition). We provide conditions under which higher
actions are induced at more or less extreme pairs of states. Finally, we
provide conditions for the optimality of either full disclosure or negative
assortative disclosure, where signal realizations can be ordered from least to
most extreme. Optimal negative assortative disclosure is characterized as the
solution to a pair of ordinary differential equations.
