Differentially Private Bayesian Persuasion

The tension between persuasion and privacy preservation is common in real-world settings. Online platforms should protect the privacy of web users whose data they collect, even as they seek to disclose information about these data (e.g., to advertisers). Similarly, hospitals may share patient data to attract research investments with the obligation to preserve patients' privacy. To address these issues, we study Bayesian persuasion under differential privacy constraints, where the sender must design an optimal signaling scheme for persuasion while guaranteeing the privacy of each agent's private information in the database. To understand how privacy constraints affect information disclosure, we explore two perspectives within Bayesian persuasion: one views the mechanism as releasing a posterior about the private data, while the other views it as sending an action recommendation. The posterior-based formulation leads to privacy-utility tradeoffs, quantifying how the tightness of privacy constraints impacts the sender's optimal utility. For any instance in a common utility function family and a wide range of privacy levels, a significant constant gap in the sender's optimal utility can be found between any two of the three conditions: epsilon-differential privacy constraint, relaxation (epsilon, delta)-differential privacy constraint, and no privacy constraint. We further geometrically characterize optimal signaling schemes under popular privacy constraints (epsilon-differential privacy, (epsilon, delta)-differential privacy and Renyi differential privacy), which turns out to be equivalent to finding concave hulls in constrained posterior regions. Finally, we develop polynomial-time algorithms for computing optimal differentially private signaling schemes.