What laptop can run games on medium-high settings without lag for under 800-850$?

I think you would be very happy with the Lenovo IdeaPad Y510p with its dual GT755 graphics cards. This one is just a little above your stated budget but I believe it to be well worth it.

http://www.newegg.com/...6834313742

Some other good possibilities:

http://www.newegg.com/...geSize=100

This one has the Intel HD graphics 4600 which is just a little shy of the performance of the GTX 560 Ti and 650 Ti as far as performance. Note, the HD Graphics 4600 is a huge jump ahead of the 4400 graphics.
http://www.newegg.com/...3Z91AN4583
http://www.newegg.com/...6834258520

Other thoughts:
http://www.newegg.com/...%20%241000