Trolldor said:
Uhh... I think they'd figure out to climb on top of one another before they decided to slaughter one another.
That's more optimistic than you think it is. Have you ever had a puzzle that you couldn't figure out, that you looked at in a million different ways and still couldn't figure it out? The "obvious" answer isn't always obvious.
Throw in the survival-instinct-makes-us-violent factor. People going into the Vaults are doing so to escape what is viewed at their time to be an inevitable nuclear holocaust. They end up in this Vault, not knowing anyone, not knowing if anyone is trustworthy, and then have to work together to figure this out, while they know the world outside is blowing up? Not as easy as it first appears at all.
There's also the isolation factor. This room is small for ten people. Tall but small and with no way of finding privacy. There are constant irritations that will settle in after the first hour as people realise they have to use toilets in the full view of everyone else, or ask everyone to look away. With the women, imagine the irritation at the men having the full ability to look, and perhaps vice versa. Fear settles in through this, alongside mistrust.
Out of twelve randomly selected people, there is also the probability that someone has an unknown or unexpressed mental issue. They might even try to manipulate others if, in their panic after getting into the Vault and assuming the bombs have dropped, they immediately think of the "kill others to break the screen" option, assuming the Vault itself is a sick joke designed to turn them into murderers. Despair can do that to people.
Not to mention, have you ever tried to stand on top of a group of twelve people without falling down? It's actually a demanding acrobatic feat a lot of the time. It takes co-ordination and often friendship. And we know from reality TV that groups of randomly selected people do NOT all immediately become friends with each other if they end up in a tightly packed situation with no way out. If they work together it is not through sturdy ties but a mutual goal, and this one might not seem so inspiring when they find themselves locked in the Sphere in the first place.