content_image/313973.jpg content_image/313981.jpg content_image/313979.jpg content_image/313980.jpg
where 

<img src='content_image/313973.jpg'>

<img src='content_image/313981.jpg'>

Suppose that marginal distributions F$_{0}$ and F$_{1}$ are fixed. Lemma 4 shows that sharp bounds on the joint distribution improve when the values of the joint distribution are known at some fixed points. Note that P ( Y$_{1}$ ≥ Y$_{0}$ ) = 1 if and only if F ( y, y ) = F$_{1}$ ( y ) for all y ∈ $_{R}$. As illustrated in Figure 3, 

Therefore, 

<img src='content_image/313979.jpg'>

<img src='content_image/313980.jpg'>

Since for each y ∈ R the value of F ( y, y ) is known from the fixed marginal distribution F$_{1}$ under MTR, sharp bounds on the joint distribution can be derived by taking the intersection of the bounds under the restriction F ( y, y ) = F$_{1}$ ( y ) over all y ∈$_{R}$ . Technical details are presented in Appendix. 

