Treatment recommendation with distributional targets

We study the problem of a decision maker who must provide the best possible treatment recommendation based on an experiment. The desirability of the outcome distribution resulting from the policy recommendation is measured through a functional capturing the distributional characteristic that the decision maker is interested in optimizing. This could be, e.g., its inherent inequality, welfare, level of poverty or its distance to a desired outcome distribution. If the functional of interest is not quasi-convex or if there are constraints, the optimal recommendation may be a mixture of treatments. This vastly expands the set of recommendations that must be considered. We characterize the difficulty of the problem by obtaining maximal expected regret lower bounds. Furthermore, we propose two regret-optimal policies. The first policy is static and thus applicable irrespectively of the subjects arriving sequentially or not in the course of the experimental phase. The second policy can utilize that subjects arrive sequentially by successively eliminating inferior treatments and thus spends the sampling effort where it is most needed.