Does someone invite you to a game and nothing comes up on your screen? Or do you invite someone and the game just sits there waiting and waiting?
If this happens, then chances are you’ve turned off Alerts for Game Center.
To fix this, go into Settings -> Notifications -> Game Center and enable Alerts. Game Center needs Alerts enabled for it function properly.
Yes this does leave you open to getting trumpeted at 3 oclock in the morning, but it is what it is.