  Optimal treatment regimes are personalized policies for making a treatment
decision based on subject characteristics, with the policy chosen to maximize
some value. It is common to aim to maximize the mean outcome in the population,
via a regime assigning treatment only to those whose mean outcome is higher
under treatment versus control. However, the mean can be an unstable measure of
centrality, resulting in imprecise statistical procedures, as well as unfair
decisions that can be overly influenced by a small fraction of subjects. In
this work, we propose a new median optimal treatment regime that instead treats
individuals whose conditional median is higher under treatment. This ensures
that optimal decisions for individuals from the same group are not overly
influenced either by (i) a small fraction of the group (unlike the mean
criterion), or (ii) unrelated subjects from different groups (unlike marginal
median/quantile criteria). We introduce a new measure of value, the Average
Conditional Median Effect (ACME), which summarizes across-group median
treatment outcomes of a policy, and which the optimal median treatment regime
maximizes. After developing key motivating examples that distinguish median
optimal treatment regimes from mean and marginal median optimal treatment
regimes, we give a nonparametric efficiency bound for estimating the ACME of a
policy, and propose a new doubly robust-style estimator that achieves the
efficiency bound under weak conditions. Finite-sample properties of the
estimator are explored via numerical simulations and the proposed algorithm is
illustrated using data from a randomized clinical trial in patients with HIV.
