I would ask if the shutdown is instant when playing the games in question or is it an overtime thing.
It could be very much an inrush issue which I would say a lot of people don't know or even understand.
For most motors, inrush is normally assumed to be 4 times the normal current (large inductive motors) for reference.
I run EMC for a living and do power quality testing. I have seen items that pull mA of power and have 6A inrushes.
I would say that I would go with nothing short of a 850w PSU, this would give you the most overhead since you have to remember inrush is not exclusive to one item on your board but ALL items.
Also it would account for slight power dips due to dirty power.