Go to Settings > Game Center, where you can:
Sign out (tap your Apple ID)
Allow invites
Let nearby players find you
Edit your Game Center profile (tap your nickname)
Get friend recommendations from Contacts or Facebook
Specify which notifications you want for Game Center. Go to Settings > Notification Center > Game Center. If Game Center doesn’t appear, turn on Notifications.
Change restrictions for Game Center. Go to Settings > General > Restrictions.