There are a lot of variables. Color gamut and luminance are the two main ones as far as PC rendering and displaying that on a monitor, and even the cameras broadcasting a game into a digital format that come through our televisions. For example, NTSC or SMPTE is the old standard definition television and dvd player color gamut, while ITU-R 709 or Rec. 709/BT.709 is the HDTV standard color gamut.
Wide color gamuts are used in photography editing programs for a larger color range as long as the monitor can display them properly. sRGB is the windows standard and what you want to shoot for when calibrating most monitors for the most accurate results when displaying color and luminance.
PC's generally use a luminance of 0-255, while TV's use a luminance of 16-235. I believe this is the main issue, because it was the issue for me. Seeing the Chicago Bulls jersey that looks red on television in real life, and kind of pink in games is really a turn off. My old tv allowed me to change the luminance levels from 0-255 to 16-235 and the jersey looked red again. Not only that, but the cheap seats in the arena's were no longer pitch black and just a dark gray. While it lowered contrast, it brought out some extra details and made the scene look more natural to me.
What's the difference? At a black level of 16, anything under that is clipped and detail is lost. The same goes for the white clipping point of 235 and anything above that. Colors in between get changed as well.
So we have real life jerseys illuminated by lights in an arena that are different for each one, shot by a camera/cameras that have it's own settings and standards for capturing light and color, converted to a format for broadcasting, delivered to us and displayed on our televisions that have different ways of reproducing color and in different formats. Then we have a game with color information from schematics, converted to an image format, rendered in a certain format with different shaders and lighting applied, displayed through our console/pc's to our tv or monitor.
Check these examples and tell me which one you think looks closer to the real thing. I edited one of them.

