Calculus of the ex ante expected social surplus (perfect information)
The ex ante expected social surplus is
\begin{equation} S^{*}\left(b,k\right)=\int\nolimits_{0}^{1}\left[\pi\left(X^{\ast}\left(\delta\right),Y^{*}\left(\delta\right)\right)\left(b-\delta\right)-X^{\ast}\left(\delta\right)-Y^{*}\left(\delta\right)\right]d\delta.\nonumber \\ \end{equation}Using (LABEL:eq:X*) and (LABEL:eq:Y*) and integrating, we get
\begin{align} S^{*}\left(b,k\right) & =\int\nolimits_{0}^{k}\left(b-\delta\right)d\delta-\left(b-k\right)\int_{k}^{1}\left(\frac{b-2\delta+k}{b+\delta-2k}\right)d\delta\notag \\ & =-2b+2k+3bk-\frac{5}{2}k^{2}+3\left(b-k\right)^{2}\ln\left(1+\frac{1-k}{b-k}\right).\notag \\ \end{align}observing that \(\left(b-2\delta+k\right)/\left(b+\delta-2k\right)\) admits \(3\left(b-k\right)\ln\left(b+\delta-2k\right)-2\delta\) as an antiderivative. \(\blacksquare\)