Z ⟶ A ⟶ Y, U ⟶ A, U ⟶ Y,

with no arrow

Z \to Y

and no common cause of

Z

with

A

Y

In potential outcome notation, a valid instrument requires:

Instrumental-variable independence: $Z ⊥ {A (0), A (1), Y (0), Y (1)}$ , strengthened to all relevant joint potential outcomes as needed. Coin flipping makes the incentive assignment independent of quitting behavior and blood pressure under either assignment.
Exclusion restriction: $Y (z, a) = Y (a)$ . The incentive can affect blood pressure only by changing whether the subject quits, not through stress, income, or another direct route.
Instrument relevance: $P {A (1) \neq = A (0)} > 0$ , or at least $E [A (1) - A (0)] \neq = 0$ . The monetary incentive must change quitting probability.
Consistency of potential outcomes and no interference: observed quitting and blood pressure equal the potential values under the assigned encouragement and received exposure, and one subject's assignment does not affect another's outcome.