Sequential price setting SPNE Firms A and B are in an oligopoly. They both face the linear market demand curve $X=A-\alpha P$, where $X$ is total market demand, and $P$ is price. Assume constant marginal costs $C_{A}$ and $C_{B}$, where $C_{B}&lt;C_{A}$.

What will be the SPNE outcome if firm A sets price first? Is the following correct?

So long as firm A sets $P&gt;C_{B}$, then it will get 0 profits (as firm B will profitably undercut and take all market demand). Therefore, it is indifferent between any $P&gt;C_{B}$. If, however, A sets $P \leq C_{B}$, then firm B will not be able to profitably undercut, and so market demand will be split between the two firms, at some $P \leq C_{B}$. But then firm A will be making negative profits, and so can profitably deivate to setting some P>$C_{B}$. 

Is this analysis correct? Or can firm A set some $p\leq C_{B}$ in an SPNE?
Your logic is correct. You need one more step. Using the same logic you can show that, in any SPNE, firm $A$ must set a price greater than the price that maximizes $B$'s profits. Firm $B$ then chooses such price.