Before revision: In this paper we study a treatment allocation problem with multiple treatments, in which the individuals to be treated arrive sequentially. The goal of the policy maker is to treat every individual as well as possible. Which treatment is "best" is allowed to depend on various characteristics (functionals) of the individual-specific outcome distribution of each treatment. For example measures of welfare, inequality, or poverty. We propose the Functional Sequential Allocation policy, and provide upper bounds on the regretit incurs compared to the oracle policy that knows the best treatment for each individual.
After revision: Consider a setting in which a policy maker assigns subjects to treatments, observing each outcome before the next subject arrives. 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. We evaluate different assignment policies according to their regret, that is, the sum of all losses incurred due to assigning subjects to suboptimal treatments. All our results are non-asymptotic, in that we provide finite sample lower and upper bounds on the regretit incurs compared to the oracle policy that knows the best treatment for each individual.