Well, on a touchscreen would be a start....perhaps with user adjustable directional pad and buttons? Who knows! Just throwing ideas around...
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Perhaps you misunderstood...of course you can continue to play games on your idevice when out and about, but with this solution...designed for the living room, the user downloads a (in this case) game to their Apple TV and uses the device to play it, connected to Apple TV via Bluetooth. Nothing to do with AirPlay! The app runs natively on the Apple TV. A controller app (or modified Remote App) can be configured by the user to whatever makes playing easier. Or, there is no reason the game can't send control button info to the controller app, specifically for that game.
Edit: plus, with iCloud support for gaming progress, one can just play on the big screen, then pick up on the device only later (on a train or something)
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I like this idea! If some of the members here had their way we would all still be wired to everything. Plus it's amazing how posts stating, " oh, where would you know where to touch the screen to control it', are so prehistoric! It can be tailored for each game individually. Game devs would be required to submit a controller app layout which is sent to the device upon startup....the user familiarises themselves with this and off they go to play!