Bayesian Persuasion in Networks: Divisibility and Network Irrelevance

Abstract: We study a multiple-receiver Bayesian persuasion model in which the sender wants to persuade a critical mass of receivers. Receivers are connected in a network and can perfectly observe their immediate neighbors' signals, which complicates the problem of the sender. We simplify the problem by considering signaling schemes (``experiments'') in which certain receivers are never targeted, effectively dividing the network and preventing some of the information spillovers. Using this divided network approach, we derive lower bounds on the value of the sender, find experiments that achieve them, and provide sufficient conditions for the lower bounds to coincide with the optimal value. Our approach is robust to adding and severing some connections, sometimes even when adding more connections than existed in the original network. Finally, we demonstrate how our approach can be implemented if the network is non-divisible (e.g., a star) and compute the lower bound for such cases. This allows us to better interpret cultural phenomena such as echo chambers and opinion polarization.