Model Selection for Treatment Choice: Penalized Welfare Maximization

This paper studies a new statistical decision rule for the treatment assignment problem. Consider a utilitarian policy maker who must use sample data to allocate one of two treatments to members of a population, based on their observable characteristics. In practice, it is often the case that policy makers do not have full discretion on how these covariates can be used, for legal, ethical or political reasons. Even in cases where policy makers have leeway in how to assign treatment, plausible assumptions may generate useful constraints on treatment assignment. We treat this constrained problem as a statistical decision problem, where we evaluate the performance of decision rules by their maximum regret. We adapt and extend results from statistical learning to develop a decision rule which we call the Penalized Welfare Maximization (PWM) rule. Our study of the PWM rule, which builds on the the Empirical Welfare Maximization (EWM) rule developed in Kitagawa and Tetenov (2015), differs from it in two aspects. First, by imposing additional regularity conditions on the data generating process, we derive bounds on the maximum regret of our rule in classes of treatment allocations of infinite VC dimension, which are not explored in Kitagawa and Tetenov (2015). In particular, we show that our rule is well suited to deal with some allocations of infinite VC dimension that can arise in applications. Second, we argue that our rule can provide a reduction in point-wise regret in situations where sample size is small compared to the complexity of the constraints on assignment. We illustrate our method in a small simulation study where our rule is able to achieve smaller regret than EWM in an empirically relevant setting. We conclude by applying our rule to data from the Job Training Partnership Act (JTPA) study.