  We study the selection of covariate adjustment sets for estimating the value
of point exposure dynamic policies, also known as dynamic treatment regimes,
assuming a non-parametric causal graphical model with hidden variables, in
which at least one adjustment set is fully observable. We show that recently
developed criteria, for graphs without hidden variables, to compare the
asymptotic variance of non-parametric estimators of static policy values that
control for certain adjustment sets, are also valid under dynamic policies and
graphs with hidden variables. We show that there exist adjustment sets that are
optimal minimal (minimum), in the sense of yielding estimators with the
smallest variance among those that control for adjustment sets that are minimal
(of minimum cardinality). Moreover, we show that if either no variables are
hidden or if all the observable variables are ancestors of either treatment,
outcome, or the variables that are used to decide treatment, a globally optimal
adjustment set exists. We provide polynomial time algorithms to compute the
globally optimal (when it exists), optimal minimal, and optimal minimum
adjustment sets. Our results are based on the construction of an undirected
graph in which vertex cuts between the treatment and outcome variables
correspond to adjustment sets. In this undirected graph, a partial order
between minimal vertex cuts can be defined that makes the set of minimal cuts a
lattice. This partial order corresponds directly to the ordering of the
asymptotic variances of the corresponding non-parametrically adjusted
estimators.
