AI 中文总结
研究一维线性设置下可解释信息设计的可解释性价格，通过将状态空间划分为至多\(K\)个连续区间并确定性发送信号，解决了均匀先验下不同\(K\)值时可解释性价格的问题，给出了\(K = 2\)时为\(1/2\)、\(K \geq 3\)时为\(2/3\)且比率严格的结果。
AI 中文摘要
在信息设计中，有信息的发送者旨在通过采用信号方案来影响接收者的决策。然而，最优信号方案往往依赖随机化或为状态空间的不相连区域分配相同信号，难以解释或传达。受这些限制启发，我们聚焦于一维线性设置中的可解释信息设计，其中可解释策略将状态空间划分为至多\(K\)个连续区间，并为每个区间确定性地发送不同信号。我们研究可解释性价格，即可解释信号方案与无限制信号方案使用相同数量信号时所达到的最优值的最坏情况比率。在均匀先验下，Chen等人在可解释信号方案可使用额外信号时建立了\(2/3\)的严格保证。他们还表明，当可解释和无限制信号方案都至多使用\(K\)个信号时，在效用为二值且\(K \geq 4\)时同样的\(2/3\)保证成立，任意有界效用的情况仍未解决。我们完全解决了这个问题。在均匀先验下，\(K = 2\)时可解释性价格恰好为\(1/2\)，\(K \geq 3\)时恰好为\(2/3\)。对于这两种情况，我们还表明相应比率是严格的。
英文摘要
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.
