Figure 3: P ( Y$_{0}$ > Y$_{1}$ ) = P [ ∪ y ∈$_{R}$ { Y$_{0}$ > y, Y$_{1}$ < y }$^{]}$ and these bounds are sharp. From Lemma 3, sharp bounds on marginal distributions of Y$_{0}$ and Y$_{1}$ are improved based on L mtr 01 ( y, z ) and U mtr 10 ( y | z ) under M. 1 − M. 4, and MTR as follows: F L 0 ( y ) = sup z ∈Z $^{[}$P ( y | 0 , z ) (1 − p ( z )) + L mtr 01 ( y, z ) $^{]}$, F U 0 ( y ) = inf z ∈Z $^{[}$P ( y | 0 , z ) (1 − p ( z )) + U wst 01 ( y, z ) $^{]}$, F L 1 ( y ) = sup z ∈Z $^{[}$P ( y | 1 , z ) p ( z ) + L wst 10 ( y, z ) $^{]}$, F U 1 ( y ) = inf z ∈Z $^{[}$P ( y | 1 , z ) p ( z ) + U mtr 10 ( y, z ) $^{]}$. Now, I show that MTR also has identifying power for the joint distribution. I will use Lemma 4 to bound the joint distribution under MTR. Henceforth, x + denotes max ( x, 0) . Lemma 4 ( ? ) Suppose that marginal distributions F$_{0}$ and F$_{1}$ are known and that F ( a$_{0}$, a$_{1}$ ) = θ where ( a$_{0}$, a$_{1}$ ) ∈ R 2 and θ satisfies max ( F$_{0}$ ( a$_{0}$ ) + F$_{1}$ ( a$_{1}$ ) − 1 , 0) ≤ θ ≤ min ( F$_{0}$ ( a$_{0}$ ) , F$_{1}$ ( a$_{1}$ )) . Then, sharp bounds on the joint distribution F are given as follows: F $^{L}$( y$_{0}$, y$_{1}$ ) ≤ F ( y$_{0}$, y$_{1}$ ) ≤ F U ( y$_{0}$, y$_{1}$ ) , 21