This paper considers the estimation of treatment assignment rules when the
policy maker faces a general budget or resource constraint. Utilizing the
PAC-Bayesian framework, we propose new treatment assignment rules that allow
for flexible notions of treatment outcome, treatment cost, and a budget
constraint. For example, the constraint setting allows for cost-savings, when
the costs of non-treatment exceed those of treatment for a subpopulation, to be
factored into the budget. It also accommodates simpler settings, such as
quantity constraints, and doesn't require outcome responses and costs to have
the same unit of measurement. Importantly, the approach accounts for settings
where budget or resource limitations may preclude treating all that can
benefit, where costs may vary with individual characteristics, and where there
may be uncertainty regarding the cost of treatment rules of interest. Despite
the nomenclature, our theoretical analysis examines frequentist properties of
the proposed rules. For stochastic rules that typically approach
budget-penalized empirical welfare maximizing policies in larger samples, we
derive non-asymptotic generalization bounds for the target population costs and
sharp oracle-type inequalities that compare the rules' welfare regret to that
of optimal policies in relevant budget categories. A closely related,
non-stochastic, model aggregation treatment assignment rule is shown to inherit
desirable attributes.Comment: 70 pages, 7 figure