Leakage-Robust Bayesian Persuasion

We introduce the concept of leakage-robust Bayesian persuasion. Situated between public persuasion [KG11, CCG23, Xu20] and private persuasion [AB19], leakage-robust persuasion considers a setting where one or more signals privately sent by a sender to the receivers may be leaked. We study the design of leakage-robust persuasion schemes and quantify the price of robustness using two formalisms: - The first notion, $k$-worst-case persuasiveness, requires a scheme to remain persuasive as long as each receiver observes at most $k$ leaked signals. We quantify the Price of Worst-case Robustness (PoWR$_k$) -- i.e., the gap in sender's utility as compared to the optimal private scheme -- as $\Theta(\min\{2^k,n\})$ for supermodular sender utilities and $\Theta(k)$ for submodular or XOS utilities, where $n$ is the number of receivers. This result also establishes that in some instances, $\Theta(\log k)$ leakages are sufficient for the utility of the optimal leakage-robust persuasion to degenerate to that of public persuasion. - The second notion, expected downstream utility robustness, relaxes the persuasiveness and considers the impact on sender's utility when receivers best respond to their observations. By quantifying the Price of Downstream Robustness (PoDR) as the gap between the sender's expected utility over random leakage patterns as compared to private persuasion, we show that over several natural and structured distributions of leakage patterns, PoDR improves PoWR to $\Theta(k)$ or even $\Theta(1)$, where $k$ is the maximum number of leaked signals observable to each receiver across leakage patterns in the distribution. En route to these results, we show that subsampling and masking are general-purpose algorithmic paradigms for transforming private persuasion signaling schemes to leakage-robust ones, with minmax optimal loss in the sender's utility.