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.

翻译：信息设计（贝叶斯劝说）中的最优信号方案常涉及随机化或状态空间的不连通划分，这类方案可能因过于复杂而难以审计或传达。本文在线性信息设计框架下，针对连续状态空间提出可解释信息设计的概念。对于一维状态情形，我们限定信息设计者采用由状态空间的确定性单调划分所定义的$K$划分信号方案，其中每个划分部分内的所有状态均发送唯一信号。我们证明可解释性代价——最优可解释信号方案与无约束信号方案性能之比——在最坏情况下精确等于$1/2$，这意味着划分信号方案的性能绝不会低于任意信号方案性能的$2$倍。对于均匀先验分布，该可解释性代价可提升至紧确值$2/3$。随后我们将分析拓展至多维状态空间，研究两种自然的可解释性概念：凸划分策略与轴对齐矩形策略。对于凸划分策略，我们证明其紧确可解释性代价为$1/(m+1)$；对于矩形策略，则在均匀先验下建立了与$K$无关但不可避免随$m$呈指数衰减的可解释性代价保证。在计算层面，我们证明可解释策略的精确优化问题通常具有NP难度，但提出了高效近似方法，包括针对Lipschitz效用函数的完全多项式时间近似方案，以及针对广泛的不连续分段Lipschitz效用函数类、能达到最坏情况$1/2$基准的多项式时间算法。