This paper addresses the issue of limited overlap in the estimation of average treatment effects (ATE) by proposing a systematic approach to define optimal estimands. Limited overlap in covariate distributions can lead to imprecise estimates and sensitivity to model specification. The authors introduce two new estimands: the Optimal Subpopulation Average Treatment Effect (OSATE) and the Optimally Weighted Average Treatment Effect (OWATE). OSATE focuses on subpopulations where the average treatment effect can be estimated more precisely, while OWATE uses weights based on the propensity score to improve precision. The paper demonstrates that under certain conditions, the optimal selection rules for OSATE depend solely on the propensity score. A simple rule is to drop units with estimated propensity scores outside the range [0.1, 0.9]. This approach is consistent with previous informal methods but provides a systematic way to choose the cutoff point. The OSATE is defined as the average treatment effect within a selected subpopulation, and the OWATE is a weighted average treatment effect where the weights depend on the propensity score. The authors argue that focusing on these estimands acknowledges the difficulties in making inferences about the population of interest and provides more precise estimates. They also note that these methods can be useful in situations where the primary interest is to determine whether a treatment may harm or benefit some group in a broader population. Additionally, these methods can provide useful information when making inferences about treatment effects for fixed populations. The paper also discusses the properties of estimators for OSATE and OWATE, showing that the optimal weights for OWATE are proportional to the product of the propensity score and one minus the propensity score. The authors illustrate these methods using data from the LaLonde dataset, showing that the OSATE method can significantly reduce variance by focusing on a subpopulation with high overlap in covariate distributions. The paper concludes that the proposed methods provide a systematic approach to dealing with limited overlap in the estimation of average treatment effects, leading to more precise estimates and better inferences. The results are applicable to a wide range of distributions and can be implemented on real data. The authors emphasize that these methods are not meant to replace traditional estimands but to provide additional insights and improve the precision of treatment effect estimates in the presence of limited overlap.This paper addresses the issue of limited overlap in the estimation of average treatment effects (ATE) by proposing a systematic approach to define optimal estimands. Limited overlap in covariate distributions can lead to imprecise estimates and sensitivity to model specification. The authors introduce two new estimands: the Optimal Subpopulation Average Treatment Effect (OSATE) and the Optimally Weighted Average Treatment Effect (OWATE). OSATE focuses on subpopulations where the average treatment effect can be estimated more precisely, while OWATE uses weights based on the propensity score to improve precision. The paper demonstrates that under certain conditions, the optimal selection rules for OSATE depend solely on the propensity score. A simple rule is to drop units with estimated propensity scores outside the range [0.1, 0.9]. This approach is consistent with previous informal methods but provides a systematic way to choose the cutoff point. The OSATE is defined as the average treatment effect within a selected subpopulation, and the OWATE is a weighted average treatment effect where the weights depend on the propensity score. The authors argue that focusing on these estimands acknowledges the difficulties in making inferences about the population of interest and provides more precise estimates. They also note that these methods can be useful in situations where the primary interest is to determine whether a treatment may harm or benefit some group in a broader population. Additionally, these methods can provide useful information when making inferences about treatment effects for fixed populations. The paper also discusses the properties of estimators for OSATE and OWATE, showing that the optimal weights for OWATE are proportional to the product of the propensity score and one minus the propensity score. The authors illustrate these methods using data from the LaLonde dataset, showing that the OSATE method can significantly reduce variance by focusing on a subpopulation with high overlap in covariate distributions. The paper concludes that the proposed methods provide a systematic approach to dealing with limited overlap in the estimation of average treatment effects, leading to more precise estimates and better inferences. The results are applicable to a wide range of distributions and can be implemented on real data. The authors emphasize that these methods are not meant to replace traditional estimands but to provide additional insights and improve the precision of treatment effect estimates in the presence of limited overlap.