The implication seems to be that by creating a set textbook for personal conduct (the Pact) and set guidelines for the operation of a society (The Order), the creators of the fifty silos will be able to build a society of sharing and caring humans who don't make the same mistakes they themselves made.
Why were there 50 silos?
Because the creators anticipated a high failure rate
Are the silos competing with each other?
Not directly, and in some cases they seem to be linked to common resources such as power supplies
What's the end-game here?
“They didn’t give us a chance. That’s not what this is.” He [Bernard] waved at the room around him. “These are prisons. Cages, not homes. Not meant to protect us, but meant to force us, by pain of death, to bring about their vision.”
“Their vision for what?” [said Luke]
“For a world where we’re too much the same, where we’re too tightly invested in each other to waste our time fighting, to waste our resources guarding those same limited resources.” He lifted his mug and took a noisy sip. “That’s my theory, at least. From decades of reading. The people who did this, they were in charge of a powerful country that was beginning to crumble. They could see the end, their end, and it scared them suicidal. As the time began to run out—over decades keep in mind—they figured they had one chance to preserve themselves, to preserve what they saw as their way of life. And so, before they lost the only opportunity they might ever have, they put a plan into motion.”
Wool Omnibus: Chapter 18
The principle seems to be that once sufficient time has passed, the silo dwellers (however many are left, any however many silos are left) will eventually emerge into a new world, where their psychology has been completely reshaped by their experiences in the silo, in a world where those who're selfish, aggressive and uncooperative have been weeded out of the general population and where their racial, religious and social homogeneity is sufficient to prevent the old religion-based, nation-based, society-based wars from flaring up.