Я полагаю, что предоставление invitation handler
является ответом:
https://developer.apple.com/documentation/gamekit/gkmatchmaker
To receive invitations from other players, your game must provide an
invitation handler. After your game successfully authenticates the local
player, it should immediately set the inviteHandler property. The invite
handler is called immediately if your game was launched in response to a
push notification.
РЕДАКТИРОВАТЬ: ОТСУТСТВУЕТ, это свойство явно УСТАРЕЛО, так как iOS 7