will having a moderate or strict NAT effect the lag, multiplayer and host migration issues on console?

I ask this as i have noticed a few players have mentioned they have moderate NAT.

If it does inter fear with MP in any way why don't Crytek restrict players with anything other than an open NAT. I mean unless NAT is open you simply cannot join a session online.

What do you guys think?

Any one have any ideas on this?