How to Sample and When to Stop Sampling: The Generalized Wald Problem and Minimax Policies
Abstract: "Acquiring information is expensive. Experimenters need to care-fully choose how many units of each treatment to sample and when to stop sampling. The aim of this paper is to develop techniques for incorporating the cost of information into experimental design. In particular, we study sequential experiments where sampling is costly and a decision-maker aims to determine the best treatment for full scale implementation by (1) adaptively allocating units to two possible treatments, and (2) stopping the experiment when the expected welfare (inclusive of sampling costs) from implementing the chosen treatment is maximized. Working under the diﬀusion limit, we describe the optimal policies under the minimax regret criterion. Under small cost asymptotics, the same policies are also optimal under parametric and non-parametric distributions of outcomes. The minimax optimal sampling rule is just the Neyman allocation; it is independent of sampling costs and does not adapt to previous outcomes. The decision-maker stops sampling when the average diﬀerence between the treat-ment outcomes, multiplied by the number of observations collected until that point, exceeds a speciﬁc threshold. We also suggest methods for inference on the treatment eﬀects using stopping times and discuss their optimality."