Bounding causal effects analytically, rather than numerically, is appealing for its interpretability and conceptual clarity. Existing sharp methods rely on optimization-based approaches such as the Balke-Pearl framework, whose computational complexity grows rapidly. An alternative line of work derives bounds heuristically using probability laws and generic inequalities, and some recent papers have claimed or conjectured that this approach can yield sharp analytical bounds with substantially lower complexity. In this paper, we show that this perceived advantage is illusory. In particular, in a discrete instrumental variable setting, we show that any sharp analytical bound for the average treatment effect must be expressible as a maximum (minimum) over a collection of linear terms whose cardinality grows exponentially in the number of values taken by the outcome. In parallel, we show that the number of instrumental variable inequalities itself also grows exponentially. Consequently, bounds and inequalities expressed using only polynomially many such terms cannot be sharp. As a constructive complement, the paper is accompanied by codes implemented in python and R to derive sharp analytical bounds and sharp inequalities with optimal efficiency, matching the lower bounds proven in this paper. These codes are available online.

翻译：以解析而非数值方式界定因果效应，因其可解释性与概念清晰性而备受青睐。现有尖锐方法依赖基于优化的框架（如Balke-Pearl框架），其计算复杂度随规模快速增长。另一类替代方法通过概率定律与通用不等式启发式推导界限，近期部分论文声称或猜想该方法能以显著更低的复杂度得到尖锐解析界。本文证明这一感知优势实为错觉。具体而言，在离散工具变量设定下，我们证明：平均处理效应的任意尖锐解析不等式必然可表示为多个线性项的最大（最小）值，且这些线性项的基数随结果变量取值数量呈指数增长。同时，我们证明工具变量不等式本身的数目亦呈指数增长。因此，仅使用多项式量级项表达的不等式与界限无法达到尖锐性。作为建设性补充，本文附有Python与R语言实现的代码，能以本文所证下界匹配的最优效率导出尖锐解析界限与尖锐不等式。相关代码已公开发布于网络。