Ideally what you want to do is contact a game development studio and pitch your idea. Unfortunately, every game development studio I've worked at does not contract work out to individuals; we only contract out to companies larger than ourselves, and usually only for porting an existing game to a different platform (i.e. make Wii ports from 360 games, etc).
If such a company exists [...]
You can stop there. No company will do this. Think about it from their perspective. What might happen as a consequence of learning about this idea for a game? On the plus side, maybe the idea is so wonderful that they can make good money with it. But the prior is not good, because the idea is coming from someone that knows nothing about the industry. On the down side, if they ever want to publish a game that has any similarities to the idea, they can get sued. I cannot imagine any development studio would be stupid enough to be interested in learning about this idea.