My basic intuition is that consciousness arises as a byproduct of any sufficiently interconnected (i.e. Complex) network which processes input from the outside world. The problem with AI right now is bottlenecking of both kinds — we don't have computers powerful enough to support the level of complexity of the human brain (100bn neurons with an average of 7k synapses each), and we don't have good enough sensors for capturing the real world (hint: touch is a BIG one).
What sort of game do you have in mind? I definitely think a game in which humans interact with an AI in an artificially-supported information-rich environment (say, the VR of 5-10 years from now) could help provide a suitable environment for "growing" such an AI. Consider the insane amount of Stimulation and calibration human babies require for development! It takes years for the visual system to stablilize, for example.