It doesn't matter which is cheaper if Linux can only play a very small subset of the games. I certainly wouldn't spend $200+ on a video card and then limit myself in my game selection by refusing to spend an extra $100 on the OS.
Personally, I've never actually been able to get Linux to run properly on arbitrary hardware that I happened to own. I'm sure you could put together a machine with specific hardware that is known to work well with Linux, but if you just pick random parts off the shelf based on performance needs, odds are you'll run into some difficulties trying to get everything working under Linux.
Counter-point: I just spent $500 on a graphics card, and my gaming system is single-boot Linux. Now to be fair, I do play quite a few games through WINE (though fewer than I used to), but the proportion of games which are Windows only and unplayable in WINE isn't as high as it used to be. I did get an Nvidia card (GTX 970) though, mainly because AMD's drivers have such a poor reputation under Linux.
That time spent researching whether or not the parts will actually work with Linux is easily worth the cost of buying a Windows license and just knowing that everything will work as expected.
There's plenty of hardware that doesn't work well under Windows too, either because of driver bugs or because the hardware itself has a design flaw (e.g. the GTX 970 memory architecture). Ultimately you need to do some research no matter what your OS is, if you're going to build the system from scratch. (And if not, the Steam machines are equivalent to a pre-built Windows system.)