I think this is an interesting premise for a story world, but like others have mentioned, what is the conflict? What is going to happen when the survivors meet these tribes - do all the tribes want to kill them? Will some of the survivors join these tribes, and vice versa? How will the survivors manage to live in this new world?

Like Lorenzo pointed out, there's many genre options that this premise alone could take the game; if you told us more about what the main conflicts were or even gave ideas as to how a player would play them out, then it'd be easier to answer your original question.

Like Lorenzo pointed out, there's many genre options that this premise alone could take the game; if you told us more about what the main conflicts were or even gave ideas as to how a player would play them out, then it'd be easier to answer your original question.

The OP doesn't have a partly designed game to "take" one way or the other, he has only an intention to make a game that leverages the cool setting and backstory he invented; I think he partly realizes that starting from a "premise" doesn't provide much direction to game design, unless (not his case) there are so many constraints that design becomes finding a reasonable game that includes certain mandatory assets and events.

The more common and natural approach is deciding on a type of game and then choosing or remodeling a setting for it, finding a coherent and interesting world and story that supports the game. In which setting are the entities and behaviours required by the game realistic and/or relevant? What story themes are a good match for what the player does?
For example, suppose you want to make a FPS in the OP's setting. you can figure out interesting and varied environments to fight in (e.g. a large fallen skyscraper, turned sideways; depopulated farmland on the surface; strange "lost world" jungle; the streets of a city or village; tunnels or caves; narrow footings in the middle of enormous cliffs created by the cave-ins) leaving other places undeveloped; interesting weapons, defining more or less explicitly what technology is available; interesting enemies, again developing specific details at the expense of others and adapting them to the game's needs; and a somewhat interesting hero.
Then you can choose a story in which your protagonist goes through your environments, acquires your weapons, has fun fights with your enemy types, and accomplishes something meaningful.

If it had more details I think it could make a good story. Like explaining exactly why the earth began to break (maybe some natural disaster or supernatural disaster, and what caused it), and how that world underneath the earth's surface came to be, why the tribes want to kill the people on the surface, who is in charge of the tribes, why are they from different time periods, and what happened to the people that survived when they fell underneath the surface?

The story may not seem original at first glance but you could take it and spin so many ideas with it that it could become original.

@ OP:

Are you asking "is this a good game idea"? If so, it belongs in Game Design, not Writing.
But you didn't say anything about what the game itself would be - what genre, what platform, what play mechanic.

I agree with Tom. You're asking the wrong question, essentially. Instead of asking whether this would be a good idea or not, you should instead ask yourself (or us) how this can become a good idea. It's all about implementation.

As for my own answer, I'd say that you could make it like the Fallout universe - dark satirical humor and you can just about sell any wild idea that you can think of. Only problem is that you'd have to be good at that type of humor.