It would almost certainly be proxy wars between the U.S. and China. With MAD in effect, that's pretty much what you've got. If the two countries ever did fight directly, however, it would be because of resource wars: a mad scramble to secure and gobble up what's left of the world's remaining limited resources, mostly fossil fuels. That of course brings us back to MAD, and there's really only one way this would ultimately play out if the two leviathan countries went to war 'face to face'. If you've ever played the Fallout games, basically that.