Yes, the AI will "fight" each other for a city. However, the AI will not wage holy-motherfucking-war against itself like they will assuredly do against you on Deity. On Deity, as soon as they feel like it, the AI will demand something from you. If you have a great person actually in a city, your only choice is to give him up, or face never ending swarms. If you don't have one, they will just ask for tech, then tech, then tech, then decide they've taken everything they can from you and swarm.
Not only do they swarm you, but they dominate battles they had no business winning. 48 defense (me) vs. 22 offense (AI) TWICE in a row. That's the worst I've seen, but they routinely win battles they shouldn't when they are determined to take a city from you. They develop large numbers of troops, yet somehow continue building other things in the cities.
I'm all for a challenge, but it should never be a challenge based on ridiculous shit. Show the AI how to place cities better, show them how to improve culture, show them how to improve science, but do NOT just have them pull bullshit.