 Always-taker

An always-taker is a unit with potential treatment receipts $D(0)=D(1)=1$: it takes the treatment under either value of binary encouragement $Z$. This $(1,1)$ pair is a principal stratum defined by both potential receipts.

For example, a unit may participate whether or not it receives an invitation. Encouraged participants combine always-takers and compliers; unencouraged participants combine always-takers and defiers. Under instrumental-variable monotonicity, the latter group consists only of always-takers. Even then, comparing observed participants across assignment arms mixes different strata and is not automatically a within-stratum causal effect. The exclusion restriction, when imposed, says assignment cannot affect an always-taker's outcome through any channel other than the treatment whose receipt is unchanged.^[ref]