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 senders utility function is state-dependent.
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