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 (those with a higher score are treated first), 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：我们考虑如何学习在治疗成本不确定且随治疗前协变量变化时，如何最优地分配治疗的问题。这一设定可能出现在医学领域——当需要优先分配一种稀缺资源，而不同患者使用该资源的时间长短不同时；也可能出现在市场营销中——针对折扣进行定向投放，而折扣给公司带来的成本取决于其使用程度。在此，我们证明预算约束下最优的治疗分配规则是基于优先级得分的阈值规则（得分更高者优先接受治疗），并提出若干实用方法，用于利用随机试验数据学习这些优先级得分。我们的形式化结果揭示了该问题与使用工具变量学习内生性条件下异质性治疗效应问题之间的统计联系。实证评估表明，我们的方法在多项应用中表现优异。