What kind of specs the power supply has? What is it's maker and model? And also, there is usually a sticker on it, stating the specs, specially the amount of amperes on the V12 line or lines is important, as "500W" really doesn't mean much. It will read something like "12V 20A", and it may have several 12V lines. But if it has negative "-12V" line, that's not important.
If the power supply is not powerful enough, it will shut down when overloaded if it has the proper overload protection. So, for that part, the behaviour you desribe would perfectly fit into weak power supply. Weak by desing, or by age and wear.
Specially if the games that do this, are very demaning.
But your CPU is nt really using a lot of power, so that most power supplies should handle it and any graphics card.
Also what is your graphics card, specific model? If you can't find it anywhere, then download GPU-Z. The high end models consume many times the amount of power than the low end 6800 series model. Just the 4 number model number is enough, manufacturer probably not important.
If the power supply could be ruled out as less probably reason, then other reason could be motherboard. It's really in practise the only other component besides PSU that can completely cut power.(unless there is some very unlikely physical damage in a bad place, making a shortcircuit causing the PSU to shutdown)
A CPU can shutdown the system too if going above a thermal thresshold, but it would throttle it's own executing far before that, making it really inpractical to ever reach unless the CPU would not have a cooler.
Graphics card, CPU, memory and others may crash the computer, but then it should remain on, or restart.