This is intended to provoke you, I get like that sometimes:
For me healthy competition is an exploration of possibility, an investigation of trials to see how good one can be, and that requires a team. A group that supports by pushing and pulling, experimenting and failing together.
If healthy competition requires cooperation in teams, imagine what we humans could do in full cooperation instead of competition. Linux, anyone? This is also why countries that implement a war economy throw competition out and expect all businesses to cooperate.
A war economy or indeed the bringing together of action via a common enemy has proven a very efficient way to get things done. The target might be wrong and the forcing nature of the authority too but, one based on betterment of opportunity through voluntary action would achieve the same. The problem is of course that people only volunteer for the things that benefit them directly, even if they will have just as large or larger indirect benefits from pulling their weight for others.
I don't know enough about Linux :P