>But if we could see the sequence of all the moves after the game (and we knew the value of pi), we could work out whether this was a game that likely used pi to determine its moves. The game that used pi is, in a sense, “watermarked”.

Wouldn't pi contain any such sequence of numbers? Therefore you'd have to allow only certain regions of pi, and therefore, its not random anymore and we could just shortcut the whole game?