What sort of devices? Will people really sit there logged into multiple devices, and expect them all to rise in realtime?
I mean, wouldn't a user play on one device. Finish, and maybe pick up another device - and when they do, you refresh all the stats when the app loads?
I would suggest you use RTT for real-time notifications - but we actually assume that that a user would only have one logged in RTT session - so that won't work.
You could send push notifications to all of the devices. Push notifications can have data payloads - so your client app(s) can catch the updated data, or maybe the notification is just a signal to the client to request updated data... (likely more flexible).
Alternatively, I guess your option best option would be to have the devices poll every <x> minutes or so...
Honestly, the use case still seems a bit weird though - I'm probably just not understanding something