I think you have an error in the equation defining V(s). You have component n_c * V(s) for the 'nothing happened' case, but I don't think that's correct. If yo

I think you have an error in the equation defining V(s). You have component n_c * V(s) for the 'nothing happened' case, but I don't think that's correct. If you rolled that nothing happens the turn still passes to your opponent, so I think it should be n_c * V'(s).