We were utilising Turn Based Games via GameCenter since iOS 6, once it was introduced (running absolutely same code for years).
However, since iOS 10 invites do not work any more. The person being invited does receive a message in Messages saying "Invitation to play %game_name$".
No invitation push notification delivered to game itself (new turn notifications are delivered successfully).
No new game appears in current game list to accept invitation (we use standard GK*ViewControllers) as it was displayed before.
If the user tap the message in Messages an alert view with "Retrieving..." text is displayed for a short period of time and nothing happens.
Apple states (https://developer.apple.com/videos/play/wwdc2016/611/) no code changes are needed. But it's definitely broken (or "no changes" statement is false).
Any ideas, colleagues?
Invitations via iMessage don't work when the two devices use the same iCloud login. I was experiencing the problems you described until I changed the iCloud login to match the iMessage login on each device.