Using Elo to re-rank the ATP’s top 20 in a preview of the French Open.

In this ongoing series of articles on statistical modeling of the ATP, we’ve encountered mathematical representations of tennis that rely on a few basic statistics (e.g. percentage of points won on serve, on return, on different surfaces). So far, none of these metrics have taken account of wins and losses against other players.

The closest we’ve come to incorporating opponents into the model was in the previous iteration (see Parts II and III) with opponents’ historical percentage of points won on serve and on return. This article will explore the Elo rating system and use an adjusted version of it to re-rank the top 20 players in the ATP. Adjustments will be made to tailor the Elo rankings to the upcoming French Open.

The Elo rating system, named after physicist Aprad Elo, originally served as improved ranking protocol for chess. Fortunately, the math underlying Elo fits any head-to-head competition, including tennis. The basic mechanics of Elo are fairly straightforward. FiveThirtyEight, which probably deserves the credit for popularizing Elo as a means of ranking sports competitors, has a solid overview of it here.

The essential features of Elo are as follows: (1) For tennis, Elo ratings change based on the binary outcome of win or loss; how close the match was does not matter. (2) Elo is zero-sum, so the winner gains points from the loser. (3) Upset victories award more points to the winner than victories that occurred in line with the rankings. (4) Elo is self-correcting, insofar as an underrated player will perform better in the long-run than the system originally predicted, thereby increasing his Elo rating to the appropriate level. (5) 1500 is the average Elo score in most systems.

It should be noted that Elo is by no means a perfect system. In fact, you can quibble with some of the underlying assumptions quite easily. For example, can you assume the performance of a player in a match to be a normally distributed random variable? How severely should Elo ratings change after an upset? This FiveThirtyEight article explores some of these issues in an entertaining format.

Now, let’s establish what the Elo ratings actually are for the top players on tour. Importantly, Elo is a historical rating system. Unless the system uses a very high K factor (meaning it is very sensitive to upsets), the “best” players will remain at the top of the heap, even if they have not played well lately. You will see this aspect of the system reflected in the Elo ratings below (source):

There are obviously some major differences between these rankings and the standard ATP rankings (which are based on points awarded from the slams, Masters 1000 tournaments, and several others). Though Elo does provide a means for probabilistic modeling of head-to-head matches, the draw for the French Open is not out at the time of writing. Therefore, I will make adjustments to the Elo ratings to devise a new ranking of the top 20 players that, hopefully, will provide some more predictive capacity over the standard ATP rankings.

I make two primary adjustments to the Elo ratings for the French Open. The first adjustment incorporates a player’s winning percentage on clay, a unique surface that makes some players much better and others much worse. The second adjustment tries to build in a measure of momentum in the season thus far via year-to-date winning percentage. I consider the clay adjustment more important than momentum, so I weighted it more heavily. The table below compares the three relevant rankings: ATP vs Elo vs Adjusted Elo.

Some interesting results emerge from a comparison of these three ranking systems. Notice that, on face, Adjusted Elo seems to provide a more sensible seeding of the players for the French than the normal ATP rankings. Federer will not play at Roland Garros, so Adjusted Elo leaves us with Nadal as the favorite, then Djokovic, Murray, Nishikori, Thiem, Wawrinka, and Zverev. Given the way things shook out in Rome, Adjusted Elo seems to be on point. We’ll find out for sure once play is over at Roland Garros. I will conclude this analysis with a post-French evaluation in the next installment.

Edited by Emily Berman, Coleman Gray.

- Roger Federer
- Pete Sampras
- Rafael Nadal
- Serena Williams

## What do you think?

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