Designing Discontinuities

Abstract Discontinuities can be fairly arbitrary but also cause significant impacts on outcomes in larger systems. Indeed, their arbitrariness is why they have been used to infer causal relationships among variables in numerous settings. Regression discontinuity from econometrics assumes the existence of a discontinuous variable, a threshold, that splits the population into distinct partitions to estimate the causal effects of a given phenomenon. Here we consider the design of partitions for a given discontinuous variable to optimize a certain effect previously studied using regression discontinuity. To do so, we propose a quantization-theoretic approach to optimize the effect of interest, first learning the causal effect size of a given discontinuous variable and then applying dynamic programming for optimal quantization design of discontinuities to balance the gain and loss in that effect size. We also develop a computationally-efficient reinforcement learning algorithm for a dynamic programming formulation of optimal quantization. We demonstrate our approach by designing optimal time zone borders for counterfactuals of social capital, social mobility, and health, including novel examples that may be of independent empirical interest. We further demonstrate our approach by designing letter grade breakpoints for college courses to improve motivation for students.