What Does it Mean to be Number 1 in Tennis?

The Economist has an interesting commentary on debate between professional tennis players Rafael Nadal and Roger Federer over how to rank players. Nadal has just resigned from his role on the Association of Tennis Player's Council, presumably over the issue.

Rankings are important not simple because of public interest, but because they dictate seedings in tournaments. which has an impact on the quality of competition faced in then earlier rounds of a tournament.  

Here is an excerpt:

Players receive a certain share of points, used to determine rankings, depending on how well they do in tournaments. Under the current system, they must defend those points the following year. Mr Nadal failed in this task in 2011, when he lost several titles to Serbia’s Novak Djokovic along with his number one position. Under a two-year system, he would not have had to defend the points he won in 2010 until this year, and would still have been the top seed entering competitions in early 2012.

Besides Mr Nadal’s status, other rankings would be less changeable under a two-year system. That justifies Mr Federer’s description of this approach as “boring”. Although self-interest also motivates the Swiss star, who wants to be able swiftly to recapture the number one spot he last held in 2009 (he is now ranked third, behind Mr Djokovic and Mr Nadal), introducing a less dynamic scoring method is unlikely to help popularise tennis. A two-year rankings calculation would also sit incongruously with the season, which stretches over 12 months, and even puzzle or annoy spectators. Many would surely think it unfair if Mr Djokovic had remained the world number two after winning both Wimbledon and the US Open, two of the four prestigious “grand slams”.

The ATP methodology for ranking players takes up 7 pages of fine print in the ATP's rulebook (here in PDF), and is convoluted and unintuitive. Like any quantitative metric the ranking system raises a number of important questions, such as, what it the point of ranking in the first place? Like any process that turns subjective judgments into quantitative values, there are a wide range of legitimate points of debate.

If the point is to identify the world's best tennis player, then over what time period? Obviously big tournaments identify the best player over the period of that tournament, which is typically a few weeks. A ranking system that spans multiple tournaments has to assign a value to the tournament and the player's performance.

For example, winning one of the 4 Grand Slams is judged under the ATP ranking system to be twice as valuable as winning one of the 9 ATP World Tour Master's 1000 events, and losing in the finals of a Grand Slam is 20% better than winning one of the 1000 events.

How does this compare to prize money? The winner of the Western & Southern Open (an ATP 1000 event) to be held in Cincinnati in August will get $535,600 whereas the winner of last year's US Open received $1.8 million and second place got $900,000. From an economic standpoint, winning the US Open is about 3 times more important than winning the Western & Southern Open. Clearly, the economic incentives are not aligned with the points awarded.

The method of ranking might can also have other consequences. A recent analysis recently found that fewer players were taking home a greater share of prize money, during a time period when the ranking system was changed. When changes were introduced in 2009 one observer warned that it would lead to "The rich would get richer while the poor will either stay poor or get poorer."

Lo and behold, via USA Today, that is exactly what has happened:

A USA TODAY analysis of the Association of Tennis Professionals prize money from 1990-2011 shows the wealth disparity between players ranked in the top 100 has never been greater. The study, which uses a commonly accepted method of measuring income distribution called the Gini co-efficient, also demonstrates that the gap has been greater over the course of the past three years than ever since the ATP's inception in 1990.

The prize money figures also include money earned at the four majors, which are not governed by the ATP.

Novak Djokovic, Rafael Nadal and Roger Federer, who have shared the top three ranking spots since 2007, have been more dominate in terms of prize money accumulation than any trio since the men's tour was formed more than two decades ago. They've raked in between 20% and 26% of available prize money the past five years. The only other trio ever to break 20% was Federer-Nadal-Andy Roddick in 2006.

The 20% mark had never been crossed before — not in the heydays of the Boris Becker-Stefan Edberg-Ivan Lendl or Pete Sampras-Andre Agassi-Jim Courier rivalries.

So here is my solution to the rankings issue for tennis. The convoluted and impractical points system should be retired. In its place the ATP should simply implement a ranking system based on prize money earned in ATP-sanctioned events over the past 12 month period (such that no tournament is counted twice). The ATP could seek to standardize prize money by event (impossible however for the Grand Slams as they are not run under the ATP).

Implementing such a change would lead to big differences in rankings. For women, here is how the points rankings look as of 19 March:

Azarenka
Sharapova
Kvitova
Radwanska
Stosur
Wozniacki
Bartoli
Li
Zvonareva
Petkovic

And here is how the money rankings look:

Azarenka
Sharapova
Radwanska
Kvitova
Clijsters
Goerges
Wozniacki
Kerber
Ivanovic
Bartoli

The top 4 are the same, but then only 2 of the next 6.

Using prize winnings as the metric of ranking clearly aligns the player's economic incentives with a measure of their professional performance. Such a metric would also allows players to weigh their participation and effort in various tournaments according to a simple, intuitive metric. When trying to turn complex and multidimensional characteristics into a single measure, sometimes the most simple and straightforward option is the best.