Proof of Corollary 1 First, I prove that in the dual representation The proof consists of two parts: (i) deriving the lower bound and (ii) deriving the upper bound. Part 1. The sharp lower bound the function ϕ is nondecreasing. Recall that Pick ( y ′ $_{0}$, y ′ $_{1}$) and ( y ′′ $_{0}$, y ′′ $_{1}$) with y ′′ 0 > y ′ 0 in the support of the optimal joint distribution. Then, The inequality in the second line of (A.10) is satisfied because y ′′ 1 ≥ y ′′ 0 > y ′ $_{0}$. The inequality in the third line of (A.10) holds because 1 { y$_{1}$ − y$_{0}$ < δ } is nondecreasing in y$_{0}$ . 63 ϕ ( y ′ $_{0}$) = inf y$_{1}$ ≥ y$_{0}$ { 1 { y$_{1}$ − y ′ 0 < δ } − ψ ( y$_{1}$ ) } (A.10) ≤ 1 { y ′′ 1 − y ′ 0 < δ } − ψ ( y ′′ $_{1}$) ≤ 1 { y ′′ 1 − y ′′ 0 < δ } − ψ ( y ′′ $_{1}$) = ϕ ( y ′′ $_{0}$) . ϕ ( y$_{0}$ ) = inf y$_{1}$ ≥ y$_{0}$ { 1 { y$_{1}$ − y$_{0}$ < δ } − ψ ( y$_{1}$ ) } . inf F ∈ Π( F$_{0}$,F$_{1}$ ) ∫ { 1 { y$_{1}$ − y$_{0}$ < δ } + λ ( 1 ( y$_{1}$ < y$_{0}$ )) } dF = sup ( ϕ,ψ ) ∈ Φ$_{c}$ ∫ ϕ ( y$_{0}$ ) dµ$_{0}$ +$^{∫}$ ψ ( y$_{1}$ ) dµ$_{1}$,