How do I know if Game Center is down or not? I'm not asking to know now, but wondering if Apple has a status page that lists if Game Center is having problems. I'm aware of the iCloud system status web page but it's not clear to me if that is the place where Game Center would be listed if for some reason it was not operating fully or correctly.
No, I have suffered some Game Center outages in the past, but could not find a source directly from Apple which stated so, not even the iCloud status page.

The iCloud System Status page now lists the status for all Apple internet services, including Game Center.