What will happen in reality
The first emperor is ailing. The five families, the Alphaei, Betii, Gammae, Deltae and Epsilones, make ready to cope with his impending death. Outwardly, each family is preparing a designated heir to compete in the imperial trials; the Betii are the only family foolish enough to put forward a heir of their own blood risking his demise or bloody sacrifice. The other four families have seen to it that an adopted commoner, remarkably good with a sword, is designed for this position, with the understanding that should he win he will promptly abdicate in favor of one of his adoptive brothers.
In parallel, the families put out feelers to the other familes, looking for alliances. After all, the priesthood is a purely ideological force, made up of corruptible men; the priesthood doesn't count. Each family seeks to establish an alliance with two other families, thus establishing an overwhelming force and putting the minority in the position to accept a settlement.
Eventually, the scheming rich Epsilones arrive at an understanding with the charismatic Alphaei and the rural and frankly semi-barbaric Betii; they will rule the empire jointly, with the Alphaei having paramountcy over the Gammae and the Betii over the Deltae. The Epsilones, themselves, require nothing else but the right to free trade over the entire empire, with a monopoly on maritime transport.
When the emperor dies, the priests try to initiate the deadly imperial contest; but the Alphaei, Betii and Epsilones make it known that such bloody games belong to an obsolete, dark and violent era, and the one living God looks unfavorably upon those ready to shed innocent blood. The Gammae resign themselves to their fate, but the foolish Deltae try to resist invoking an idealized tradition. However, the people and the soldiers, from the ranks of whom true power raises, are sick and tired of blood and destruction, and hail the new peaceful order.
And now the five dominant families are three.
The rich Epsilones become richer from trade and from their monopoly on maritime transport. Soon the time will come when they will persuade the newly elevated Betii to restore the unity of the empire by taking on the haughty and ancient Alphaei...
We are supposed to learn from history
This system won't work. It will break down when the first emperor dies or retires.
I will tell you a true story; oversimplified, of course, but basically true. More or less.
Once upon a time, there was this large and diverse empire, called the Roman empire. It was a wonderful empire, except for one flaw. It did not have any fixed rule for succession; once an emperor died, or retired, it was unclear who will become emperor next. This flaw led to a long period of instability and civil war, generally called the Crisis of the Third Century. This dark and bloody period lasted for half a century, and it ended only when a wise man, who had been born in a poor family in a remote province, rose through the ranks of the army and eventually succeeded in putting an end to the civil wars by defeating all warring factions.
His name was Diocles, meaning Glory of God; one single name, for he came from a Greek-speaking family. When acceded to the throne he adoped the regnal name Imperator Caesar Caius Aurelius Valerius Diocletianus Augustus. In history books he is usually called Diocletian.
To ensure the stability of the empire, and to avoid future civil wars, he invented a great system for regulating the succession to the imperial throne. His system is called the Tetrarchy, or Rule of Four.
An alegorical propaganda statue representing the Roman tetrarchy. The tetrarchs are representing embracing, in sign of harmony, in a porphyry sculpture dating from the 4th century, produced in Asia Minor; today on a corner of Saint Mark's in Venice, next to the Porta della Carta. Photograph by Nino Barbieri, available on Wikimedia under the GNU Free Documentation License, Version 1.2 or later.
The system was supposed to work as follows:
There were two principal emperors, called Augusti; each of them ruled one half of the empire, co-ordinating between them any major political or military matters. Diocletian himself took the eastern half, and raised Maximian to the rank of Augustus of the West.
Each of the two principal emperors appointed a subordinate emperor, called a Caesar, who ruled over a part of the principal emperor's portion of the empire, and served as successor designate. Diocletian appointed Galerius as his Caesar, and Maximian appointed Constantius Chlorus.
When a principal emperor died or retired, his place was to be taken by his Caesar, who thus became an Augustus and appointed a new Caesar.
The system sort of worked for 29 years, from 284 (when Diocletian and Maximian inaugurated it) to 313. Then the empire fell into civil wars again; the complicated and messy wars ended only in 324, when Constatine emerged as sole emperor. Upon the death of Constantine, the imperial rule devolved to his three sons who were supposed to rule jointly, each controlling directly a portion of the empire.
More general strife and civil wars followed, until 395, when Arcadius and Honorius, the two sons of Theodosius I, divided the empire between them. From that point forward the notional unity of the empire was never again materialized. The western half fell into anarchy and descended into the long night of the Middle Ages; the eastern half met with varying fortunes, and remained a great power for many centuries.
Why did the tetrarchy fail?
It failed because it was a complicated system, vulnerable to human failings; ambition, tribalism, misplaced loyalties, all contributed to its failure. The great diversity of the empire was not helpful. In retrospect, we realize that the empire was too large, way beyond what the technical, economic and political levels of the time could sustain. But the essential failure was the mismatch between the lofty ideology and the dismal reality.