On the Tractability of Public Persuasion with No Externalities

Persuasion studies how a principal can influence agents' decisions via strategic information revelation --- often described as a signaling scheme --- in order to yield the most desirable equilibrium outcome. Recently, there has been a large body of algorithmic study of designing optimal public signaling schemes, a.k.a., public persuasion, which however is rifle with computational intractability results. In this paper, we design efficient and tight algorithms for public persuasion, and focus on a fundamental multi-agent persuasion model with no inter-agent externalities and binary actions introduced by Arieli and Babichenko. En route, we develop new algorithmic techniques which may be of independent interests. First, we prove that optimal public persuasion is fixed parameter tractable. Our main result here relies on an interesting connection to a basic question in combinatorial geometry: how many cells can $n$ hyperplanes divide $R^d$ into? We use this connection to show a new characterization of public persuasion, which then enables efficient algorithm design. Second, we relax agent incentives and show that optimal public persuasion admits a bi-criteria PTAS for monotone submodular objectives and this approximation is tight. To prove this result, we establish an intriguing "noise stability" property of submodular functions which strictly generalizes the key result of Cheraghchi et al., originally motivated by applications of learning submodular functions and differential privacy. Finally, motivated by automated persuasion implemented as software, we consider relaxing the equilibrium concept of the model to coarse correlated equilibrium. Here we use a sophisticated primal-dual analysis to establish the polynomial-time equivalence between optimal public persuasion and the combinatorial problem of directly maximizing the sender's objective minus any linear function.