It's going to end in bloodshed if all 3 of them are on equal footing. Every time. So the answer is then, as someone said, one senior Augustus, and 2 junior Caesar's. To provide an example of what I am talking about, let me use Diocletian's tetrarchy system:
Diocletian retires as per OTL. Rather than have Maximian retire with him, he stipulates Maximian is now the senior Augustus, and is to retire in 10 years. Galerius is raised to junior Augustus. All 3 emperors + Diocletian agree on Galerius's successor as junior Caesar. When Maximian retires, Galerius moves his base to Italy and becomes the senior Augustus, with Constantius Chlorus being appointed junior Augustus, moving his base to Nicomedia. Galerius's successor becomes senior Caesar and moves to Gaul (to replace Constantius), and everyone agrees on a successor to Constantius as junior Caesar.
So it works like this: Every 10 years, the emperors alternate power bases. So the junior most Caesar starts in the Balkans. In 10 years, he becomes the senior Caesar, and moves to Gaul. 10 years from then, he comes the junior Augustus and moves to the east, and 10 years from then, he comes the senior Augustus and moves to Italy.
This, I feel, is the best way to do it. You can of course customize it for a 3 way split, but the basic principles apply. The idea is to always have a senior Augustus who's been Augustus for awhile to always be in place-not having 2 Augustus's retire at the same time and thus opening up a power vacuum like Diocletian did. This allows for there to be a clear top dog. The alternating of bases every 10 years as well, helps to prevent any one person from developing a strong power base and base of support.
This isn't perfect by any means. But if you can keep it in place for, say, 2 generations at least before a civil war, then I think it can be institutionalized to the point where it becomes the de facto state.
This is again a well thought model. I also see 4 emperors not 3, because you got 4 major fronts (Germania, Pannonia, Moesia, Syria) and some minor ones (Britannia, Africa, Egypt, ...). The minor commands could be perhaps a training ground for candidates.
I also agree, that it was a big mistake, that Diocletian enforced Maximian to retire with him. However, it is a miracle, that the tetrarchy worked at all. Perhaps because Maximian and Diocletian have been friends, or at least good guys or rational guys, like Marcus Aurelius and Lucius Verus. Imagine Septimus Severus introduces this model with Caracalla and Geta behind him. You know how this story ended.
And I see some more issues:
1. a roman officer in the 2nd/3rd century was usually 30 years old, when he became an officer of equestrian rank (prefect, tribune, ...). Regardless, if he started as centurio ex equite romano (about 10 years to become primipulus) or if he started as prefect after a civil career (duumvir). Afterwards you need at least about another 10 years to call this guy an experienced officer able to lead an army. Well, 45 to 50 years old would be even better, if it comes to experience.
That means, if the junior Caesar is choosen from the most experienced commanders, he is at least 40 years old. I am not mentioning guys from senatorial rank, because they had near to null military experience, when they became a commander as legatus legionis; also about the age of 30. Finally, we should not forget officers ex caliga, which are usually even 15 years older.
So your junior Caesar is about 40 years old and needs another 40 years (4 x 10) until he goes to retirement at the age of 80. That means, most emperors will die during their career, which of course accelerates the career of all. But it would perhaps better to change positions all 5 years. If not some guys might become dangerously impatient, if one Augustus really gets that old.
2. What happens, if the senior Augustus does not want to go to retirement? You know, he commands the central field army, which should be the strongest.
3, What happens if one emperor shows incompetence? Some guys are not as good as they looked like, when they have been appointed. Others change after illness, or suffering from dementia or other diseases, which reduces their performance. Or they become that greedy and corrupt, that they have to be replaced. But how to replace or accuse somebody, who commands a regional field army?
3. What happens, if some emperor starts a conspiration to murder a higher one, in order to progress faster? As long as this does not become obvious, it is no problem. But if. Or if just enough rumours are spread.
4. What happens, if an Augustus wants his son as junior Caesar and the 2 others disagree? If the name of this guy is Constantine, I see trouble ahead.
5. How do you avoid, that a rather unexperienced son becomes junior Caesar? If he fails, he usually gets murdered by the legions and the legions appoint someone by themselves.
6. Looking to the travelling times in the roman world, do you really believe, that the 3 emperors can meet in time, whenever a new junior Caesar has to be appointed?
My questions don't mean, that I disagree to your model. Actually a rotation model, where the high military commanders become always promoted automatically, until they reach the top with a high age sounds pretty good. This model sounds more stable than any other.
That was exactly one major problem of the republic. You first became consul, and then military commander. And afterwards just senator again. Waiting another 10 years to get a chance to become consul again. That was exactly the wrong order. At least for guys like Julius Caesar and others.
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