Efficient propensity score regression estimators of multi-valued treatment effects for the treated

We study the role of the propensity scores in estimating treatment effects for the treated with a multi-valued treatment.  Assume assignment to one of the multiple treatments is random given observed characteristics.  Valid causal comparisons for the subpopulation who has been treated a particular treatment level are based on two propensity scores - one for the treatment level and one for the counterfactual level.  In contrast to the binary treatment case, these two propensity scores do not add up to one.  This is the key feature that allows us to distinguish different roles of the propensity scores and to provide new insight in well-known paradoxes in the binary treatment effect and missing data literature.  We formally show that knowledge of the propensity score for the treated level decreases the semiparametric efficiency bound, regardless of knowledge of the propensity score for the counterfactual level.  We propose efficient kernel regression estimators that project on a nonparametrically estimated propensity score for the counterfactual level and the true propensity score for the treated level.  A surprising result is implied for the binary treatment effect for the treated: when the propensity scores are known, using one estimated propensity score is not efficient.  Our efficient estimator regresses on a normalized propensity score that utilizes the information contained in the nonparametrically estimated and the true propensity scores.