equilibria can be formulated as an optimal transportation problem using the { 0 , 1 } -valued cost function. Establishing sharp bounds on the DTE is also an optimal transportation problem with an indicator function as the cost function. The DTE can be written as the integration of an indicator function with respect to the joint distribution π as follows: Since marginal distributions of potential outcomes are given as µ$_{0}$ and µ$_{1}$, establishing sharp bounds re- duces to picking a particular joint distribution maximizing or minimizing the DTE from all possible joint distributions having µ$_{0}$ and µ$_{1}$ as their marginal distributions. Then the DTE is bounded as follows: where Π ( µ$_{0}$, µ$_{1}$ ) is the set of joint distributions that have µ$_{0}$ and µ$_{1}$ as marginal distributions. For the indicator function, the Kantorovich duality lemma for { 0 , 1 }− valued costs in Villani (2003) can be applied as follows: Lemma 3 (Kantorovich duality for { 0 , 1 } -valued costs) The sharp lower bound on the DTE has the following dual representation: Similarly, the sharp upper bound on the DTE can be written as follows: where where 14 A E = { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ ∈ A s.t. y$_{1}$ − y$_{0}$ ≤ δ } . sup π ∈ Π( µ$_{0}$,µ$_{1}$ ) ∫ 1 { y$_{1}$ − y$_{0}$ ≤ δ } dπ = 1 − inf F ∈ Π( F$_{0}$,F$_{1}$ ) ∫ 1 { y$_{1}$ − y$_{0}$ > δ } dπ = 1 − sup A ⊂$_{R}$ $^{{}$µ$_{0}$ ( A ) − µ$_{1}$ $^{(}$A E $^{)}$; A is closed } A D = { y$_{1}$ ∈$_{R}$|∃ y$_{0}$ ∈ A s.t. y$_{1}$ − y$_{0}$ ≥ δ } . inf π ∈ Π( µ$_{0}$,µ$_{1}$ ) ∫ 1 { y$_{1}$ − y$_{0}$ < δ } dπ (6) = sup A ⊂$_{R}$ $^{{}$µ$_{0}$ ( A ) − µ$_{1}$ $^{(}$A D $^{)}$; A is closed } inf π ∈ Π( µ$_{0}$,µ$_{1}$ ) ∫ 1 { y$_{1}$ − y$_{0}$ < δ } dπ ≤ F$_{Δ}$ ( δ ) ≤ sup π ∈ Π( µ$_{0}$,µ$_{1}$ ) ∫ 1 { y$_{1}$ − y$_{0}$ ≤ δ } dπ, F$_{Δ}$ ( δ ) = Pr ( Y$_{1}$ − Y$_{0}$ < δ ) = ∫ 1 { y$_{1}$ − y$_{0}$ < δ } dπ.