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Functional Sequential Treatment Allocation

Consider a setting in which a policy maker assigns subjects to treatments, observing each outcome before the next subject arrives. Initially, it is unknown which treatment is best. However, the sequential nature of the problem permits learning about the effectiveness of the treatments, which we measure by a functional of the associated outcome distributions, for example an inequality, welfare or poverty measure. In the present article, we evaluate assignment policies according to their regret, that is, the sum of all losses incurred due to assigning subjects to suboptimal treatments. We first study explore-then-commit (ETC) policies. These are policies, where one initially explores which treatment is of the highest quality, typically through a randomized controlled trial, and subsequently fully commits to the "inferred superior" (but potentially suboptimal) treatment. Then, we introduce and study the Functional Upper Confidence Bound (F-UCB) policy, which interweaves exploration and exploitation and is thus not of the ETC type. Our results show, in particular, that the F-UCB policy (i) performs much better than any ETC policy, and (ii) is near minimax optimal. We also show that a suitably adapted F-UCB policy is near minimax optimal under minimal assumptions when covariate information is available.

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