Generally, in any game, I'll just kill everything that moves.
I do, however, get a little annoyed when someone does better than me, and just for a sort of "ha, I'm better than you", I'll hunt them down until they're dead.
I don't ignore everyone else completely, though. I won't put much effort into killing them, but I'll do it if they become a problem, shooting me, etc. This does become disadvantageous, as I tend to disgregard my score in my blind rage that must end up with my prey dead. Probably several bullet holes to the face, or at least rings, concerning SRB2.