笔记

Where $\omega^f \leq \omega, \pi =\alpha-\omega;$ when $\omega^f>\omega, \pi = \alpha\frac{\omega}{\omega^f}-\omega$. The expected gain from the single employee is

$E[\pi]\equiv \bar{\pi}=Prob(\omega^f\leq \omega)(\alpha-\omega)+Prob(\omega^f>\omega)[\alpha E(\frac{\omega}{\omega^f}|\omega^f>\omega)-\omega]$

$=\alpha\bar{e}-\omega$

Where

$E(\frac{\omega}{\omega^f}|\omega^f>\omega)=\int\frac{\omega}{\omega^f}f(\omega^f>\omega)d\omega^f, E(\frac{\omega}{\omega^f},\omega^f>\omega)\equiv \int_{\omega}^\infty\frac{\omega}{\omega^f}f(\omega^f)d\omega^f,$

$\bar{e}\equiv Prob(\omega^f\leq\omega)+E(\frac{\omega}{\omega^f},\omega^f>\omega)$