  This paper studies statistical decisions for dynamic treatment assignment
problems. Many policies involve dynamics in their treatment assignments where
treatments are sequentially assigned to individuals across multiple stages and
the effect of treatment at each stage is usually heterogeneous with respect to
the prior treatments, past outcomes, and observed covariates. We consider
estimating an optimal dynamic treatment rule that guides the optimal treatment
assignment for each individual at each stage based on the individual's history.
This paper proposes an empirical welfare maximization approach in a dynamic
framework. The approach estimates the optimal dynamic treatment rule using data
from an experimental or quasi-experimental study. The paper proposes two
estimation methods: one solves the treatment assignment problem at each stage
through backward induction, and the other solves the whole dynamic treatment
assignment problem simultaneously across all stages. We derive finite-sample
upper bounds on worst-case average welfare regrets for the proposed methods and
show $1/\sqrt{n}$-minimax convergence rates. We also modify the simultaneous
estimation method to incorporate intertemporal budget/capacity constraints.
