= Always-taker
{title2=$(D(0),D(1))=(1,1)$}
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.<Principal Stratification in Causal Inference by Constantine E. Frangakis and Donald B. Rubin>[Section 2, joint intermediate potential values and observed mixtures]{ref}
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