Explainable Information Design

Abstract

Optimal signaling schemes in information design (Bayesian persuasion) often involve randomization or disconnected partitions of state space, which might be too intricate to be audited or communicated. We propose explainable information design in the context of linear information design with a continuous state space. In the case of single-dimensional state, we restrict the information designer to use interval-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 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 $2$. For a uniform prior, this PoE can be improved to a tight $2/3$. We then extend the analysis to multi-dimensional state spaces by studying two notions of explainability: convex-partitional policies and axis-aligned rectangular policies. We prove a tight PoE of $1/(m+1)$ for convex-partitional policies, while for rectangular policies we establish a PoE guarantee under uniform prior that is independent of the number of signals but unavoidably exponential in the dimension $m$. We also study the computational complexity of explainable information design, proving that the exactly optimal explainable policy is NP-hard to compute, but an explainable policy with $1/2$ approximation guarantee can be computed in polynomial time for piecewise Lipschitz utility functions.