Universal scaling in sports ranking

TL;DR

This paper reveals that sports ranking systems, such as tennis, exhibit universal power-law distributions in scores and prize money, and introduces a simple model explaining this scaling based on rank-dependent win probabilities.

Contribution

It identifies universal power-law behavior in sports rankings and proposes a sigmoidal-based toy model to explain the origin of this universal scaling.

Findings

Scores and prize money follow universal power laws.

The probability of higher-ranked players winning depends on rank difference.

The toy model reproduces empirical power-law distributions.

Abstract

Ranking is a ubiquitous phenomenon in the human society. By clicking the web pages of Forbes, you may find all kinds of rankings, such as world's most powerful people, world's richest people, top-paid tennis stars, and so on and so forth. Herewith, we study a specific kind, sports ranking systems in which players' scores and prize money are calculated based on their performances in attending various tournaments. A typical example is tennis. It is found that the distributions of both scores and prize money follow universal power laws, with exponents nearly identical for most sports fields. 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 will top 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 tournaments. The simulations yield results consistent with the empirical findings. Extensive studies indicate the model is robust with respect to the modifications of the minor parts.