I don't know the exact ratio that is needed if at all any. But I have noticed within all the different games I've played as a conquerer, that I use one strategy to make them give me capping.
I take over (or better yet if they are in horrible locations, raze) two to three non-important cities to the civ. Then I take a huge force and pile it out side the capital city. I kill all but one weak unit, and allow one turn to pass. So far, no matter how big the civ (4 cities or 15) on the next turn the civ offers the capping.
I do this, because I want the vassal to still be a useful resource. Once you accept them as a vassal you cannot trade back cities to them. Which I think is stupid. Some cities are nothing more than a distance cost for my ussual empires I build. If the civ has useful cities and you always retain a warlike status, they become a production whore for extra military units.
But back to the AI offerring capping. I ussually cannot get to pull up the diplomacy screen myself and have the option available. It will still say "no, we are doing fine on our own." But on the next turn they will offer it to me.
Large size ones too.
On a certain game (WOTM1) I'm playing right now, and can only talk about upto 500AD, I was able to have the Korians offer me capping and they still had more than 4 cities left. It was like 7, I believe.
And on a past game, Frederick had the largest civ at 20~ something cities. And I only razed 2 useless cities on their tundra coastline. With literally no damage really done to his empire I piled artillery and infantry outside Berlin, Hamburg and Frankfurt (Berlin and Hamburg were on the top 5 cities of the world) knocked all but one unit out in each. And next turn, he offered capping. It was a huge shock to me, I just wanted to see if he would. Big pay off. In that game I could really afford my own empire, let alone his too.
I wonder if anyone really knows what the AI takes into account before it becomes an option.
