I think there are some very good ideas in this thread, but I must play devil's advocate and point several things out.
First, the point about not being able to log into another arena for X period of time? No good. I'm the type of person that will see his own team has WAY too many pilots, and log out, go to another arena. Or I'm the type of person that logs in, finds out from .SR later that his squaddies are in another arena and switch. This only works if everybody looks for the arena where their team has all the players -- and not all players do this.
Secondly, the idea of not seeing enemy teams? On the surface I find it a cool idea. However, I think you're over-analyzing the end result. Players don't log in, check the roster, and say "Hey, check it out, the knits are way outnumbered again, let's hit them while the other team is hitting them too." It's a matter of the path of least resistance. Even if you can't SEE how many players team X has, it will have less players, be less capable of putting up a strong defense, and both other teams will gravitate toward this path of least resistance. It's not a concious process, I'd say. So you wouldn't be able to prevent it.