Universal scaling in sports ranking
Ranking is a ubiquitous phenomenon in human society. On the web pages of Forbes, one may find all kinds of rankings, such as the world's most powerful people, the world's richest people, the highest earning tennis players, and so on and so forth. Herewith, we study a specific kind sports ranking systems in which players' scores and/or prize money are accrued based on their performances in different matches. By investigating 40 data samples which span 12 different sports, we find that the distributions of scores and/or prize money follow universal power laws, with exponents nearly identical for most sports. In order to understand the origin of this universal scaling we focus on the tennis ranking systems. By checking the data we find that, for any pair of players, the probability that the higher ranked player tops the lower ranked opponent is proportional to the rank difference between the pair. Such a dependence can be well fitted to a sigmoidal function. By using this feature, we propose a simple toy model which can simulate the competition of players in different matches. The simulations yield results consistent with the empirical findings. Extensive simulation studies indicate that the model is quite robust with respect to the modifications of some parameters.