Optimal Dynamic Treatment Regimes and Partial Welfare Ordering

Dynamic treatment regimes are treatment allocations tailored to heterogeneous individuals. The optimal dynamic treatment regime is a regime that maximizes counterfactual welfare. We introduce a framework in which we can partially learn the optimal dynamic regime from observational data, relaxing the sequential randomization assumption commonly employed in the literature but instead using (binary) instrumental variables. We propose the notion of sharp partial ordering of counterfactual welfares with respect to dynamic regimes and establish mapping from data to partial ordering via a set of linear programs. We then characterize the identified set of the optimal regime as the set of maximal elements associated with the partial ordering. We relate the notion of partial ordering with a more conventional notion of partial identification using topological sorts. Practically, topological sorts can be served as a policy benchmark for a policymaker. We apply our method to understand returns to schooling and post-school training as a sequence of treatments by combining data from multiple sources. The framework of this paper can be used beyond the current context, e.g., in establishing rankings of multiple treatments or policies across different counterfactual scenarios.