We consider the problem of learning how to optimally allocate treatments whose cost is uncertain and can vary with pre-treatment covariates. This setting may arise in medicine if we need to prioritize access to a scarce resource that different patients would use for different amounts of time, or in marketing if we want to target discounts whose cost to the company depends on how much the discounts are used. Here, we show that the optimal treatment allocation rule under budget constraints is a thresholding rule based on priority scores, and we propose a number of practical methods for learning these priority scores using data from a randomized trial. Our formal results leverage a statistical connection between our problem and that of learning heterogeneous treatment effects under endogeneity using an instrumental variable. We find our method to perform well in a number of empirical evaluations.

翻译：我们研究如何学习在成本不确定且可能随治疗前协变量变化的情况下，实现最优治疗分配的问题。这一设定在医学中可能源于需要优先分配稀缺资源，而不同患者使用该资源的时间不同；在市场营销中则可能源于需要针对折扣进行定向投放，而折扣对公司造成的成本取决于其使用程度。本文表明，在预算约束下的最优治疗分配规则是基于优先得分的阈值规则，并提出了多种实用方法，利用随机试验数据学习这些优先得分。我们的理论结果借助了工具变量法，建立了该问题与内生性条件下异质性治疗效果学习之间的统计联系。多项实证评估表明，我们的方法性能表现良好。