Why three medals at the Olympics?

It is interesting to follow Olympics in various countries. While everywhere there is focus on domestic athletes, big countries, say the US, only consider gold medals to be worth mentioning, while in small ones, say Denmark, even Olympic diplomas (ranks 4-8) make major news. This brings up the question: why reward more than the winner?

Pavlo Blavatskyy demonstrates that giving a single prize elicits a lot of effort from a few athletes. Giving several prizes reduces the effort of those few, but this may increase the overall effort as more athletes participate.

Given, as I suggested above, only gold medals are worthwhile in big countries, I wonder whether we can see more effort from the top athletes from big countries, while we see more effort from small country citizens among diploma-worthy athletes. An empirical test of this would be difficult, though. Indeed, a prediction of this theory would to find proportionally more athletes from small countries in ranks 2-8. But larger country athletes could simply be missing from because a fellow citizen won.