W_i(z, x) = \int_{-\infty}^z\left(\mathbf E\left[\vphantom{1^2}V_1\mid X_1=x, Y_1=y\right] - \beta(y)\right)g(y|x)dy - \ - \left(\vphantom{1^2}1 - G(z|x)\right)\beta(z) = \ = \int_{-\infty}^z\left(\vphantom{1^2}v(x,y) - \beta(y)\right)g(y|x)dy - \left(\vphantom{1^2}1 - G(z|x)\right)\beta(z).
{}