AI summaryⓘ
The authors study how well simple and easy-to-understand ways of sharing information (called explainable signaling schemes) work compared to the best possible but more complicated methods. They focus on situations where the information can be represented along a single line and signals correspond to dividing this line into consecutive parts. They find exact measures of how much performance is lost when restricting to explainable signals under a uniform assumption about the unknown state. Specifically, for two signals, the best explainable scheme achieves half the value of the best unrestricted scheme, and for three or more signals, it achieves two-thirds of that value. These results complete an open question about utility types and signal counts from earlier work.
information designsignaling schemeexplainable policystate spaceuniform priorprice of explainabilitybinary utilitiesone-dimensional linear settingsignal partitionoptimal signaling
Authors
Francesco Bacchiocchi, Tommaso Cesari, Roberto Colomboni
Abstract
In information design, an informed sender aims to influence a receiver's decision by committing to a signaling scheme. However, optimal signaling schemes often rely on randomization or assign the same signal to disconnected regions of the state space, making them difficult to interpret or communicate. Motivated by these limitations, we focus on explainable information design in the one-dimensional linear setting, where an explainable policy partitions the state space into at most $K$ consecutive intervals and deterministically sends a distinct signal for each interval. We study the price of explainability, defined as the worst-case ratio between the optimal value achieved by an explainable signaling scheme and that achieved by an unrestricted signaling scheme using the same number of signals. Under a uniform prior, Chen et al. [2026] established a tight $2/3$ guarantee when the explainable signaling scheme was allowed to use additional signals. They also showed that the same $2/3$ guarantee holds when both the explainable and unrestricted signaling schemes use at most $K$ signals, provided that utilities are binary-valued and $K \geq 4$, leaving the case of arbitrary bounded utilities open. We resolve this question completely. Under a uniform prior, the price of explainability is exactly $1/2$ for $K=2$ and exactly $2/3$ for every $K \geq 3$. For both regimes, we also show that the corresponding ratios are tight.