# What is the positivity assumption required for matching and ATT estimand?

Does ATT estimand require a less stringent positivity assumption in matching?

For example, if a small treated group is matched to a large control group, most of the control subjects will be discarded after matching. Do we only require positivity assumption to hold in the remaining sample?

More formally, the positivity requirement for the ATT stated in the usual "probability of treatment" terms is $$P(A=1|X=x) < 1$$ for all $$x$$. That is, there are no areas of the covariate distribution where only the treatment is possible; in all areas of the covariate distribution, it is possible to receive control. This is general a much easier criterion to satisfy than the positivity requirement for the ATE.