Suppose $x$ is a random variable while $y$ is a number. Let \begin{equation*} D(y)=\mathbb{E} \max\left\{x,y\right\}-\max\left\{\mathbb{E}x, y\right\}, \end{equation*} which can be interpreted as the "benefit" of unfolding $x$. The result is \begin{equation*} argmax_{y} D(y)=Ex \end{equation*} My proof is as follows.
Let $C(y)=\mathbb{E} \max\left\{x,y\right\}=yF(y)+\int_y^{+\infty} xf(x)dx$. $\frac{\partial C}{\partial y}=F(y)+yf(y)-yf(y)=F(y)$. So if $y<Ex$, then $\frac{\partial D}{\partial y}=F(y)\geqslant 0$. If $y>Ex$, then $\frac{\partial D}{\partial y}=F(y)-1\leqslant 0$.
I was wondering whether this is a well-known result and what the related results/insights are.