Well, first of all I need to address this: a triumvirate is indeed a kind of pact stipulated among three people, but said parties involved are supposed to be sharing power equally. Now, we know that even in real history, the two Roman triumvirates didn't work out, because eventually everyone involved had their own plans and ended up pursuing them, waging war on the other members.
That being said, the Sith Triumvirate worked a little differently, but to explain this we need a little bit of backstory here.
Since Kreia was blamed for Revan's fall to the Dark Side, she was cast away from the Jedi order (being betrayed for the first time) and started looking for her former apprentice, who had disappeared. Eventually, she reached Malachor and found the Trayus Academy. Under the influence of the Dark Side, she turned into a Sith Lord under the name of Darth Traya, taking control of the whole facility. Some time later, she somehow found two beings, that had been deeply affected by the events that happened at Malachor: Sion and Nihilus, two wounds in the Force. Therefore, Traya took them under her tutelage, taking advance of the situation: she could both observe closely the part those two beings played as dead spots in the Force (something she had been looking for) and she could train them to take her revenge on the Jedi who had turned their back on her. And although Traya understood that both Sion and Nihilus were dangerous to the Sith themselves, as they were both beasts with a fragmented will and excessively dependent on the Force, she decided to teach them the secrets of the Dark Side, and the way to somehow make them complete again, satisfying their hunger and ambition. So the Sith Triumvirate name, at least at first, was just a convention: Traya was a master with two apprentices, and that's it. But after all, she didn't want those two to be subdued to her: she wasn't a leader, and she didn't want to be one. She didn't want to control Sion and Nihilus, and eventually she wasn't even able to. She just accepted to train them, as long as they helped her destroy the Jedi. I think there is a great quote in kotor that sums up what Kreia intended to be for her apprentices: "I am but a mirror whose only purpose is to show you what your eyes cannot yet see." After Sion and Nihilus became Sith Lords, she still maintained her role of master, but once again it was just a convention: the Sith Lords were fully equal to one another, with each one developing different skills. Traya held the secret knowledge of the Dark Side, Sion learned the way to transcend death and the power to exploit pain and anger to sustain himself, and Nihilus understood how to drain the Force from living creatures, annihilating all life to satisfy his hunger. Honestly, I think that Traya was the only one who truly believed in the value of the Sith Triumvirate. But even though she gave her apprentices and fellow Lords knowledge, power and in a way also freedom, she had to be betrayed yet again, mostly because every member of the Triumvirate had their own plans and ideas, and probably no one of them fully understood the others. Quoting Traya, "to be united by hatred is a fragile alliance at best". Sion and Nihilus interpreted Kreia's "whispers in their skulls", as her way to control them, while she just wanted to turn them into perfect Sith Lords who would destroy the Jedi. Sion and Nihilus probably misunderstood or just didn't agree with Traya's purposes and plans("no longer do we suffer beneath teachings that weaken us", says Sion) so they eventually slipped out of what they thought to be her "authority" over them, temporarily allying with each other to defeat her. Kreia was stripped of the Force, cast out from the Trayus academy and betrayed once more. At this point, since Sion and Nihilus weren't able to control and restrain themselves, they became slaves of their needs once again, and parted ways. In this case though, the competition between the two of them became quite strong: but remember that they were not squabbling over who'd control the galaxy (the Sith triumvirate was composed of quite anomalous Sith Lords from this point of view). They just had too different purposes and needs, and so the best thing to do was splitting up. They never cared for each other as long as they didn't interfere in one another's plans. So I think the game is pretty clear on this: the Sith Triumvirate was an alliance with no leaders and no one interested to be one. The power was equally shared but the purposes of its members too various and maybe even constrasting to fully work out. And after all, alliances between Sith are probably never going to succeed, for their own particular mindset, even though the members of the Triumvirate represented exceptional cases among this group.
