I am not sure how you can term our education system, western education in general, as competitive. You are taught things, and you try to attain a standard of proficiency. The only real competition is with yourself.
I guess it depends on what you mean by competition or cooperation. As individual methods used on a case by case basis, both can be useful. Education is not a silver bullet scenario. You can't just say, a la Dr Phil or Dr Oz 'this one thing will fix all of your problems, just do this one thing'.
As an example of cooperation, you could have collaborative discussion groups or projects where people learn from each other, and teach each other different aspects of content. Totally valid approach depending on what you are teaching, and the level of your students.
As an example of competition, it is as simple as spelling bees. People strive to perform better than their opponents, and it drives people to better performance, provided it is done correctly, and you are encouraging the good performance, not belittling the bad performance.
Both are valid in their place. What would science be without collaboration and idea sharing? What would political science be without debate? What would english and history be without people simply reading on their own?
Sorry, but this question smacks of an oversimplification of the educational process.