# Sport: Universal Law Discovered for Player Rankings

## The same universal laws seems to govern player rankings in sports as diverse as tennis, fencing, snooker and many others. How come?

Nov 16, 2011

About 100 years ago, the Italian economist Vilfredo Pareto noticed that about 80 per cent of the land in Italy was owned by about 20 per cent of the people. Puzzled by this observation, he went on to check it elsewhere. To his surprise, the same principle applied in other countries too.

Since then, the so-called Pareto Principle has cropped up in everything from the distribution of wealth within a population to the distribution of crime within the criminal community.

Today, Weibing Deng at Hua-Zhong Normal University in China and a few pals, study the distribution of wealth and success in 12 different sports, including tennis, hockey, volleyball, baseball, basketball and so on (all sports in which rankings of individual players are available).

Having analysed the rankings these sports, they say that wealth and success follow power laws. That’s as expected.

But the surprise is that almost all sports follow exactly the same law–the Pareto Principle. In other words, regardless of the sport, 20 percent of the players enjoy 80 per cent of the success and prize money.

Exactly how this rule emerges in sports with different rules, governing bodies and tournament structures is something of a puzzle.

However, it means there is certain predictability in the outcome of events in which two players are pitted against each other. To test the nature of this predictability, Deng and co have found a model that exactly reproduces the statistics of the real sport.

They make a few assumptions about the players involved, the most interesting being that the probability of one player beating another depends only on their difference in ranking. So the number 1 ranked player is just as likely to beat the number 10 player as the number 75 is of beating the number 85. In fact, the same probability applies to any two players separated by ten places in the rankings.

Deng and co fit this model to the ATP and WTA tennis rankings for men and women and say that it reproduces more or less exactly the real results.

The model allows for one other variable. They say that in the example above of players separated by ten places, their model shows there is more uncertainty in men’s tennis than in women’s tennis over which one will win. This extra variable is therefore an indicator of the competitiveness of a sport.

There is one interesting applications that Deng and co suggest: gambling. With this model, it should be a straightforward to see whether a bookmaker is offering odds that are incompatible with the Pareto Principle and to bet accordingly.

At least that’s the principle. In practice their can be all kinds of confounding factors, such as the structure of specific tournaments, which may not allow all players to compete or divide them into additional categories.

Of course, if this model can produce some new insight, there’ll only be a short window in which to profit from it until the bookies themselves begin to use it too. And with the publication of this paper, that window may already be closing.

Ref: arxiv.org/abs/1111.2919: Universal Scaling In Sports Ranking