  This paper studies a penalized statistical decision rule for the treatment
assignment problem. Consider the setting of a utilitarian policy maker who must
use sample data to allocate a binary treatment to members of a population,
based on their observable characteristics. We model this problem as a
statistical decision problem where the policy maker must choose a subset of the
covariate space to assign to treatment, out of a class of potential subsets. We
focus on settings in which the policy maker may want to select amongst a
collection of constrained subset classes: examples include choosing the number
of covariates over which to perform best-subset selection, and model selection
when approximating a complicated class via a sieve. We adapt and extend results
from statistical learning to develop the Penalized Welfare Maximization (PWM)
rule. We establish an oracle inequality for the regret of the PWM rule which
shows that it is able to perform model selection over the collection of
available classes. We then use this oracle inequality to derive relevant bounds
on maximum regret for PWM. An important consequence of our results is that we
are able to formalize model-selection using a "hold-out" procedure, where the
policy maker would first estimate various policies using half of the data, and
then select the policy which performs the best when evaluated on the other half
of the data.
