It's not all about what CPU you have. There aren't things within a CPU now to say you have a AMD platform were going to play nicer with you. Point is that around 200 dollars the best option you really have for you is the 7850 now if you were to say $250.00 you have other options you have the 660Ti which is almost in that bracket you have the 7870 but keep this in mind the 7850 is a mid level card you are going to see mid level performance with a 7870 as well its not until the 7950 that you are going to see higher performance.
The OP said he would like to spend 200 or less unless that changes the best option at the price point that he has laid out is the 7850. It's not even debatable there isn't a 200 dollar Nvidia card there is a 220 nvidia card the gtx 660 but its not 200
7850 v 7870: Based off this I don't think you are getting 40-60 dollars more worth of performance. http://www.anandtech.com/bench/Product/549?vs=548
7850 v 7950: This is a more sizable upgradehttp://www.anandtech.com/bench/Product/549?vs=550