  The celebrated Bayesian persuasion model considers strategic communication
between an informed agent (the sender) and uninformed decision makers (the
receivers). The current rapidly-growing literature mostly assumes a dichotomy:
either the sender is powerful enough to communicate separately with each
receiver (a.k.a. private persuasion), or she cannot communicate separately at
all (a.k.a. public persuasion). We study a model that smoothly interpolates
between the two, by considering a natural multi-channel communication structure
in which each receiver observes a subset of the sender's communication
channels. This captures, e.g., receivers on a network, where information
spillover is almost inevitable.
  We completely characterize when one communication structure is better for the
sender than another, in the sense of yielding higher optimal expected utility
universally over all prior distributions and utility functions. The
characterization is based on a simple pairwise relation among receivers - one
receiver information-dominates another if he observes at least the same
channels. We prove that a communication structure $M_1$ is (weakly) better than
$M_2$ if and only if every information-dominating pair of receivers in $M_1$ is
also such in $M_2$. We also provide an additive FPTAS for the optimal sender's
signaling scheme when the number of states is constant and the graph of
information-dominating pairs is a directed forest. Finally, we prove that
finding an optimal signaling scheme under multi-channel persuasion is,
generally, computationally harder than under both public and private
persuasion.
