  Some units from a population receive the same treatment that is different
from treatments available for other reservoir populations. The minimal
sufficient statistic $s$ for the pre-treatment $x$-covariates's distributions
in the populations is the coarsest balancing score. $s$ is used to select
matching units for simultaneous causal comparisons of multiple
treatments.Necessary and sufficient conditions on the posterior distribution of
the treatment variable (given $x$) determine whether a statistic is either
sufficient or minimal sufficient for the x-covariates' distributions. Results
in the literature are thus extended. Strong ignorability of treatment
assignment given $s(x)$ is also established. Consequently, the expected
treatments' differences given $s(x)$ are shown to be simultaneously unbiased
for the average causal effects of all treatments' differences. The existing
statistical theory for $s$ and its estimates support their use in causal
inference.
