At the beginning of each Hunger Games, the contestants are lifted up through a tube into the arena that the games will be held. The tributes have a lot of time interacting with other tributes while training for the games. What would have happened if they had met up and decided to stay on the starting platforms, rather than playing the game.
The starting points of the Hunger Games, the pedestals the elevator tubes are tied to, have explosives in them. Any attempt to ruin the game by not participating would be solved by remotely detonating a pedestal or two to motivate participation.
Remember in the first movie, when Katniss walked too far away from the action, the game maker motivated her back towards the other tributes, with a well placed fire. Then they also release genetically modified pit bulls to corral the last 3 tributes to the cornucopia for the finale.
And we know that the Capital has no issue with extortion and blackmail. They may not kill the Tributes, instead threatening their family, their district, or their love ones. Joanne had her family killed for her defiance, Haymitch had no one to threaten so he got away, but Finnick was basically pimped and sold to be raped because he didn't want his lover to be hurt, and Snow threatened Peeta's life if Katniss didn't fall in line. Most people may not care about their lives, but rarely would let their loved ones get hurt.
The Capital has the means and experience in getting the tributes to do what they want most of the time.
To quote Snow:
'If the Head Gamemaker, Seneca Crane, had had any brains, he would have blown you to dust right there and then. But he had an unfortunate sentimental streak. So here you are. Can you guess where he is?' I nod, because by the way he says it, it's clear that Seneca Crane has been executed.
If nobody tries to kill each other, which districts 1 & 2 would, the gamemakers would send out traps and mutaitions to kick start it.
The Gamemakers would probably kill them instantly but try to leave one tribute so they still have a victor. They could just set off all mines around the tributes or send in some bombs.