How well can fine balance work for covariate balancing

Fine balance is a matching technique to improve covariate balance in observational studies. It constrains a match to have identical distributions for some covariates without restricting who is matched to whom. However, despite its wide application and excellent performance in practice, there is very little theory indicating when the method is likely to succeed or fail and to what extent it can remove covariate imbalance. In order to answer these questions, this paper studies the limits of what is possible for covariate balancing using fine balance and near-fine balance. The investigations suggest that given the distributions of the treated and control groups, in large samples, the maximum achievable balance by using fine balance only depends on the matching ratio (ie, the ratio of the sample size of the control group to that of the treated group). In addition, the results indicate how to estimate this matching ratio threshold without knowledge of the true distributions in finite samples. The findings are also illustrated by numerical studies in this paper.