Yes, and I wouldn't play one. Reason?
Every FB app/game wants access at any time to your personal info,
and will use/share it with anyone they want.
Your problem isn't sharing the personal information in your account if you give Facebook fake information about yourself.
Does Facebook know my real name? No.
Does Facebook know my actual day, month, or year of birth? No.
Does Facebook know my actual cell phone number? No.
Does Facebook know an email address that I've used for anything other than Facebook? No.
I feel ok using Facebook as much as I do with these simple, obvious precautions.