Compute Nash Equilibrium in median voter game
The calculations look correct. 

For the PSNE, I would just say since there is no mutual BRs in pure strategy, there is no PSNE, and leave it at that.

In the MSNE, A would play $-1$ with probability $\frac{1-s}{2-s}$, $0$ with $\frac{s}{2-s}$ and $1$ with $\frac{1-s}{2-s}$; B would play $-1$ with $\frac{s}{2-s}$, $0$ with $\frac{2-3s}{2-s}$ and $1$ with $\frac{s}{2-s}$.