It always seemed to me that the Yonkous don't want to actively take out each other because they have a fragile balance with the Shishibukais and the Marines, so if someone is down and the others are weakened, then a vacuum of power happens and everything goes to hell, like it seemed to happen with Blackbeard, with the difference that the marines took a very heavy blow too in that case so they couldn't stop him, and the other 3 kept their part of the balance.
If there is an all out fight between the yonkous, whatever remains will be attacked by marines/shishis and that is a big problem.
So there is a constant struggle, but not an all out war.