  We study the algorithmics of information structure design -- a.k.a.
persuasion or signaling -- in a fundamental special case introduced by Arieli
and Babichenko: multiple agents, binary actions, and no inter-agent
externalities. Unlike prior work on this model, we allow many states of nature.
We assume that the principal's objective is a monotone set function, and study
the problem both in the public signal and private signal models, drawing a
sharp contrast between the two in terms of both efficacy and computational
complexity.
  When private signals are allowed, our results are largely positive and quite
general. First, we show polynomial-time equivalence between optimal signaling
and the problem of maximizing the objective function plus an additive function.
This yields an efficient implementation of the optimal scheme when the
objective is supermodular or anonymous. Second, we exhibit a
(1-1/e)-approximation of the optimal private signaling scheme, modulo an
additive loss of $\epsilon$, when the objective function is submodular. These
two results simplify, unify, and generalize results of [Arieli and Babichenko,
2016] and [Babichenko and Barman, 2016], extending them from a binary state of
nature to many states (modulo the additive loss in the latter result). Third,
we consider the binary-state case with a submodular objective, and simplify and
slightly strengthen the result of [Babichenko and Barman, 2016] to obtain a
(1-1/e)-approximation via an explicitly constructed scheme.
  When only a public signal is allowed, our results are negative. First, we
show that it is NP-hard to approximate the optimal public scheme, within any
constant factor, even when the objective is additive. Second, we show that the
optimal private scheme can outperform the optimal public scheme, in terms of
maximizing the sender's objective, by a polynomial factor.
