I can't find much NON-BIASED info as far any comparison between the 3 projectors, other than ART's reviews somewhat, but I have compared the 8500ub to the hc3800.
The problem is the hc4000 now has DC3 and the Epson 8350 now has d7 fine (or whatever its called). So they've changed out enough in both these PJ's that we don't know for sure. Art also claims the Epson 8350 was sharper than the 8100 even when his 8350 had worse convergence? (wow now that could indicate a huge up in Epson sharpness, but I have my doubts).
Art seemed to say the 8350 still could not quite match the ANSI contrast of the 8500. According to ART, in bright scenes the 8500 still had somewhat better blacks then the 8350, and in dark scenes the 8500 left the 8350 in the dust (dynamic contrast). Judging from what ART has said, the dynamic and ANSI contrast of the 8350 is still only a tad higher than the 8100 and not near the 8500 (At least not in dynamic contrast).
Problem with this is, the 8350 is a much brighter projector than the 8500 and he was using a new LAMP, so hopefully ART took the time to calibrate them to the same brightness when doing his eyeball contrast comparisons (he usually does, but sometimes he forgets or is lazy - if any of us had to compare so many PJ's, I'm sure we'd do the same thing).
This is all very confusing, so we have to wait and see for more reviews and info.
I have compared the hc3800 to the Epson 8500.
For gaming, sharpness is fairly important (depends on what type of gaming I suppose), so the hc4000 might get my vote, but then again some are claiming the Epsons are even sharper this year (has yet to been proven though). For some gaming, black levels are important as well so that gives the Epson a +1.
It's a tough call, but I'd say if gaming is the main thing and you are willing to return your PJ several times if the convergence is off, I'd probably at least give the Epsons a try.
If you want something cheaper, you might just grab the hc4000 and the Epson 8350 at the same time from a place you know will allow a return, and then compare them yourself and ship the one back that you don't want.
If you look in the Epson threads, they will basically kill you if you try to compare anything and claim that you are trashing Epson.