Now we just need to figure out how to make people more competitive in the areas they care about...

I'm not sure that replacing intrinsic motivation with the motivation to prove yourself better than others is always a good idea. It will help some but it might also hinder.

If I'm feeling competitive, why would I ever skip the chores and lose?

I could skip the chores and lose because I'm feeling the wrong sort of competitiveness, I tied my feelings of self-worth to the outcome and avoiding thinking of the issue altogether is preferable to facing the prospect of loss.