Explainable Information Design

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 $K$-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 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 natural explainability notions: convex-partitional policies and axis-aligned rectangular policies. For convex-partitional policies, we prove a tight PoE of $1/(m+1)$, while for rectangular policies we establish a PoE guarantee under uniform prior that is independent of $K$ but unavoidably exponential in $m$. On the computational side, we prove that the exact optimization of explainable policy is NP-hard in general, but provide efficient approximation methods, including an FPTAS for Lipschitz utility functions and a polynomial-time algorithm that achieves the worst-case $1/2$ benchmark for the broad class of discontinuous, piecewise Lipschitz, utility functions.