If you use Little Snitch and are annoyed by constant alerts like this...
1. Add rules to Little Snitch – you can tell Little Snitch to allow or deny this process forever.
2. You can disable the gamed process so that it won't try and connect.
If you have no interest in Game Center, particularly if you are running OS X server, it may be more efficient to disable the gamed process, as well as others you may not be using.