  Covariate balance is crucial for unconfounded descriptive or causal
comparisons. However, lack of balance is common in observational studies. This
article considers weighting strategies for balancing covariates. We define a
general class of weights---the balancing weights---that balance the weighted
distributions of the covariates between treatment groups. These weights
incorporate the propensity score to weight each group to an analyst-selected
target population. This class unifies existing weighting methods, including
commonly used weights such as inverse-probability weights as special cases.
General large-sample results on nonparametric estimation based on these weights
are derived. We further propose a new weighting scheme, the overlap weights, in
which each unit's weight is proportional to the probability of that unit being
assigned to the opposite group. The overlap weights are bounded, and minimize
the asymptotic variance of the weighted average treatment effect among the
class of balancing weights. The overlap weights also possess a desirable
small-sample exact balance property, based on which we propose a new method
that achieves exact balance for means of any selected set of covariates. Two
applications illustrate these methods and compare them with other approaches.
