Subgame Perfect Nash equilibrium: two stage game
In part a, if $B = D_1 + D_2$, then the SGPE should be

$\left\lbrace D_1 = \frac{W}{3},\ D_2 = \frac{W}{3}, \left\lbrace  P = \alpha (D_1 + D_2), \ S = (1 - \alpha) (D_1 + D_2) \right\rbrace \right\rbrace$

Don't say $P = \alpha \frac{2W}{3}$. That's an action, and the second stage best respond should a strategy (function) to make the equilibrium subgame perfect.