  We focus on the scenario in which an agent can exploit his information
advantage to manipulate the outcome of an election. In particular, we study
district-based elections with two candidates, in which the winner of the
election is the candidate that wins in the majority of the districts.
District-based elections are adopted worldwide (e.g., UK and USA) and are a
natural extension of widely studied voting mechanisms (e.g., k-voting and
plurality voting). We resort to the Bayesian persuasion framework, where the
manipulator (sender) strategically discloses information to the voters
(receivers) that update their beliefs rationally. We study both private
signaling, in which the sender can use a private communication channel per
receiver, and public signaling, in which the sender can use a single
communication channel for all the receivers. Furthermore, for the first time,
we introduce semi-public signaling in which the sender can use a single
communication channel per district. We show that there is a sharp distinction
between private and (semi-)public signaling. In particular, optimal private
signaling schemes can provide an arbitrarily better probability of victory than
(semi-)public ones and can be computed efficiently, while optimal (semi-)public
signaling schemes cannot be approximated to within any factor in polynomial
time unless P=NP. However, we show that reasonable relaxations allow the design
of multi-criteria PTASs for optimal (semi-)public signaling schemes. In doing
so, we introduce a novel property, namely comparative stability, and we design
a bi-criteria PTAS for public signaling in general Bayesian persuasion problems
beyond elections when the sender's utility function is state-dependent.
