Perhaps, but in many schools grades are kept secret (at least they are at my university), thereby destroying their ability to foster competition. On the other hand, I know some Japanese schools post grades in public areas, in rank order no less.
I think a good compromise would be to give everyone a histogram with anonymous results of their classmates, and an arrow "you are here". So you know whether you are good or bad, but if you are bad your classmates don't know it's you. (Perhaps as an exception, the best one or the best three names could be made public.)
Here's an internal dialogue I just had.
Q: How do we test rationality skills?
A: We haven't come up with a comprehensive test yet.
Q: Maybe we can test some part of rationality?
A: Sure. For example, you could test resistance to akrasia by making two contestants do some simple chores every day. The one who fails first, loses.
Q: That seems like a pointless competition. If I'm feeling competitive, why would I ever skip the chores and lose?
A: Whoa, wait. If competitiveness can cure akrasia, that's pretty cool!
Now we just need to figure out how to make people more competitive in the areas they care about...