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can be written as follows: for each z ∈ Z and any y ∈$_{R}$ , 

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While the probability P ( y, 1 | z ) is observed, the counterfactual probability P$_{1}$ ( y, 0 | z ) is never ob- served. Let p = sup z ∈Z p ( z ) , p = inf z $_{∈Z}$p ( z ). Note that p and p are well defined under M. 4. 

For z ∈ Z such that p ( z ) < p, the counterfactual probability P$_{1}$ ( y, 0 | z ) can be decomposed as follows: 

The second equality follows from M. 3. 

Note that P ( Y$_{1}$ ≤ y, p ( z ) < U ≤ p ) is point-identified as follows: 

However, P ( Y$_{1}$ ≤ y, p < U ) is never observed. Note that for 

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lim p ( z ) → p P$_{1}$ ( y | 0 , z ) can be any value between 0 and 1. Therefore, I can derive bounds on P ( Y$_{1}$ ≤ y, p < U ) by plugging 0 and 1 into the counterfactual distribution P ( y | 0 , z ). Similarly, the other counterfac- tual probability P$_{0}$ ( y, 1 | z ) can be partially identified. 

