Regret Analysis in Threshold Policy Design

Threshold policies are targeting mechanisms that assign treatments based on whether an observable characteristic exceeds a certain threshold. They are widespread across multiple domains, such as welfare programs, taxation, and clinical medicine. This paper addresses the problem of designing threshold policies using experimental data, when the goal is to maximize the population welfare. First, I characterize the regret (a measure of policy optimality) of the Empirical Welfare Maximizer (EWM) policy, popular in the literature. Next, I introduce the Smoothed Welfare Maximizer (SWM) policy, which improves the EWM's regret convergence rate under an additional smoothness condition. The two policies are compared studying how differently their regrets depend on the population distribution, and investigating their finite sample performances through Monte Carlo simulations. In many contexts, the welfare guaranteed by the novel SWM policy is larger than with the EWM. An empirical illustration demonstrates how the treatment recommendation of the two policies may in practice notably differ.