If you are real concerned about 3D, then the Sharp xvz-30,000 provides the best deal but it is usually $200 over that price if you get it when it is on sale. Another one is the Mitsubishi hc5 (same as hc9000d), but it is also $200 over that budget. The Sharp is much better at 3D than the Mits.
Yet another, which is only $100 over your budget is the Epson 5010 refurb from VisualApex.com. However, still $100 over
The truth is, $1,600 is a bad cut-off point, $1,800 or less would be more realistic for a projector that also does good 3D.
If you want to forget about 3D and just focus on 2D in that price range, then the B-Stock JVC HD-250 (if you can still get them) is the best 2D projector under $1,500.
All said, it depends what you want, best 2D, best 3D, and if you are willing to buy from Ebay, etc...
If you want a good mix of dark blacks and good 3D, I'd go with the refurb Epson 5010 as it is the best of all worlds (though does have some gaming lag).
Mits hc5 = Best 2D via Ebay, not sure about gaming lag but I think it's low
Sharp xvz30,0000 = Best 3D and good 2D, I think gaming lag is too high for FPS
Epson 5010 = Best Blacks in 2D (a bit rougher due to pixel fill in bright scenes), also very good 3D but not as good as the Sharp, but lots of gaming lag
Beyond that, I would probably look at a Mitsubishi hc7800 or hc7900, but the lens isn't as sharp as some other DLP projectors.
Under $1,000, the w1070 is the king if you can fit it in your mounting range with its very short throw.