Optimal Decision Rules Under Partial Identification

I consider a class of statistical decision problems in which the policy maker must decide between two alternative policies to maximize social welfare (e.g., the population mean of an outcome) based on a finite sample. The central assumption is that the underlying, possibly infinite-dimensional parameter, lies in a known convex set, potentially leading to partial identification of the welfare effect. An example of such restrictions is the smoothness of counterfactual outcome functions. As the main theoretical result, I obtain a finite-sample decision rule (i.e., a function that maps data to a decision) that is optimal under the minimax regret criterion. This rule is easy to compute, yet achieves optimality among all decision rules; no ad hoc restrictions are imposed on the class of decision rules. I apply my results to the problem of whether to change a policy eligibility cutoff in a regression discontinuity setup. I illustrate my approach in an empirical application to the BRIGHT school construction program in Burkina Faso (Kazianga, Levy, Linden and Sloan, 2013), where villages were selected to receive schools based on scores computed from their characteristics. Under reasonable restrictions on the smoothness of the counterfactual outcome function, the optimal decision rule implies that it is not cost-effective to expand the program. I empirically compare the performance of the optimal decision rule with alternative decision rules.