Reworded: There is no performance benefit by using ATI & AMD. FYI, Intel offers better performance these days. NVIDIA & ATI cards will perform differently depending on the game....therefore there no definite company to go with.
If comparing ATI + AMD and NVIDIA + AMD.....the game performance will be strictly defendant on the video card and how a particular video card performs in a certain game.
In short, don't worry about pairing. Look at benchmarks for games and decide on which manufacturer is best to go with.