Dominance is build into the bell curve

In sports we often have incredibly dominant athletes, that for a while make it very clear who is on top. In chess that is particularly noticeable with Carlsen topping the rating list for almost ten years now, Kasparov before him being head and shoulders above everyone else and many other champions enjoying long undisputed reigns (Karpov, Casablanca) or at least short extremely dominant stretches (Alekhine, Fischer).

But the same holds for many other sports as well. Federer, Klitschko, Tiger Woods, Carl Lewis, etc. etc. I recently realized that this is at least partly a direct consequence of abilities being normally distributed. In a normal distribution the number of athletes within an ability bucket of a certain size drops exponentially the farther out from the mean you look. Intuitively this means that the top athlete might get his very own bucket, while the preceding bucket is already filled with, let’s say, ten rivals. Consequently the average distance to the next best athlete must be exponentially smaller the closer you get to the mean.

This exponentially bigger distance to the next best rival is what we call dominance. Counterintuitively this entails that the stronger the competition in a given sport the more dominant the top athlete is likely to be. Simply because the top in a very competitive field is going to be farther out from the mean. Of course this mathematical relationship doesn’t hold as strongly in teams sports. In teams sports dominance is more likely to result from winner-takes-all dynamics.

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