Referring to the OUYA developer documentation, it shows how to get a list of item receipts the player has purchased. You can check if the "full game" item is in the receipts and if so the player has indeed purchased the game. If you want to make sure the game works even without an internet connection, when the user purchases the "full game" item you can then store something akin to a license file, perhaps store the user's UUID in a file somewhere and then you can compare the contents of the file to the user's UUID (if they match the user has indeed purchased the game). The UUID is unique to that player, and is also unique from one game to the next, which makes it a bit more difficult to easily spoof, for example by a hacker creating a fake "game" that reports the UUID (since their "game" is different it will report a different ID than your game).
You didn't remember the plot of the Doctor Who movie because there was none; Just a bunch of plot holes strung together.
@liuyang - Killa is suggesting that you cache the purchases locally and encrypt it by user id. That way a "clone" of the license file would only successfully decrypt for the user logged into the console that made the purchase.
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If you want to make sure the game works even without an internet connection, when the user purchases the "full game" item you can then store something akin to a license file, perhaps store the user's UUID in a file somewhere and then you can compare the contents of the file to the user's UUID (if they match the user has indeed purchased the game). The UUID is unique to that player, and is also unique from one game to the next, which makes it a bit more difficult to easily spoof, for example by a hacker creating a fake "game" that reports the UUID (since their "game" is different it will report a different ID than your game).