Explainable Information Design

The optimal signaling schemes in information design (Bayesian persuasion) problems often involve non-explainable randomization or disconnected partitions of state space, which are too intricate to be audited or communicated. We propose explainable information design in the context of information design with a continuous state space, restricting the information designer to use $K$-partitional signaling schemes defined by deterministic and monotone partitions of the state space, where a unique signal is sent for all states in each part. We first prove that the price of explainability (PoE) -- the ratio between the performances of the optimal explainable signaling scheme and unrestricted signaling scheme -- is exactly $1/2$ in the worst case, meaning that partitional signaling schemes are never worse than arbitrary signaling schemes by a factor of 2. We then study the complexity of computing optimal explainable signaling schemes. We show that the exact optimization problem is NP-hard in general. But for Lipschitz utility functions, an $\varepsilon$-approximately optimal explainable signaling scheme can be computed in polynomial time. And for piecewise constant utility functions, we provide an efficient algorithm to find an explainable signaling scheme that provides a $1/2$ approximation to the optimal unrestricted signaling scheme, which matches the worst-case PoE bound. A technical tool we develop is a conversion from any optimal signaling scheme (which satisfies a bi-pooling property) to a partitional signaling scheme that achieves $1/2$ fraction of the expected utility of the former. We use this tool in the proofs of both our PoE result and algorithmic result.