Do Rallies Get Longer as Matches Progress?

Italian translation at settesei.it

Yesterday at the New York Open, Paolo Lorenzi battled through three sets to defeat Ryan Harrison. It was a notable result for a number of reasons, starting with the fact that Lorenzi is rarely seen on a hard court when there’s any other option. The 37-year-old Italian is one of the many men defying the aging curve these days, and with the victory, he’ll play at least one tour-level quarter-final for the eighth year in a row, despite not reaching his first until he was 30.

The way in which Lorenzi won the match was almost as unique as his career trajectory. Take a look at the average rally length per set:

Set  Avg Rally  
1          3.2  
2          4.0  
3          4.9

You probably don’t need me to tell you which set Harrison won. The opening frame was serve-dominated, typical of American indoor hard court events. As the match progressed, the points increasingly resembled the clay-court sparring that Lorenzi surely would have preferred.

Theorizing

The Lorenzi-Harrison match was extreme, but it tracks with what I believe to be the conventional wisdom. Throughout a match, players get better at reading their opponents’ games, cutting down on unreturned serves and making it more likely that each point will turn into a more protracted exchange. That’s the theory, anyway. There are some countervailing forces, such as fatigue, which work in the other direction, but in general we expect points to get longer.

Yesterday’s contest didn’t exactly follow that script, though. The rallies might have gotten longer because the two men better predicted each other’s shots, but it doesn’t show up so neatly in aces–Harrison hit aces on between 18% of 21% of his points in each set–or the more inclusive category of unreturned serves:

Set  Points  Unret%  
1        47   42.6%  
2        65   32.3%  
3        73   37.0%

While serve recognition may explain the rally length jump from set 1 to set 2, it goes in the opposite direction from set 2 to set 3. Yes, these are small samples, and yes, unreturned serves don’t tell the whole story. But there are signs that our initial theory is missing something.

More matches

As interesting as Lorenzi is, we’re going to need more players, and more data, to better understand what happens to serve returns and rally length over the course of a match. Let’s start with the main draw singles matches from the 2019 Australian Open. Not only are there are a lot of them, but since they are best of five, we have an opportunity to see how these trends unfold over several sets per match.

For each match, I measured the average rally length and rate of unreturned serves for each set, and then made set-by-set comparisons for the length of the match. For instance, in Lorenzi-Harrison, rally length increased by 25% from set 1 to set 2. Then, for each set, I aggregated all the matches of sufficient length to figure out how much the tour as a whole was changing from one set to the next.

The results are considerably less eye-catching than those of the Lorenzi match. In the following table, the “Avg Rally” and “Unret%” columns show the change in ratio form: If the baseline rate in the first set is 1.0, the rally length in set 2 increases by 0.8% and the number of unreturned serves goes up by 2.4%. I’ve also included example columns, showing realistic rally lengths and unreturned-serve rates for each set based on tournament averages of 3.2 shots by point and 34% of serves unreturned:

Set  Avg Rally  Ex Rally  Unret%  Ex Unret  
1            1      3.20       1     34.0%  
2        1.008      3.23   1.024     34.8%  
3        1.019      3.26   1.033     35.1%  
4        0.987      3.16   1.155     39.3%  
5        1.021      3.27   1.144     38.9% 

The set-to-set differences in rally length are barely enough to qualify for the name. The shift in the rate of unreturned serves, however, is much more striking, all the more so because it moves in the opposite direction that we expected.* Perhaps fatigue–or strategic energy conservation–plays a bigger role than I thought, or servers gain more from familiarity with their opponent than returners do.

* You might wonder if the effect is an artifact of the data, that players who reach 4th and 5th sets are bigger servers. That may be true, but it’s not what we’re seeing here. I’m comparing the stats in each set to the previous set in the match itself, and then averaging the set-to-set changes, weighted by the number of points in the sets. A John Isner 5th set, then, is compared only to an Isner 4th set.

WTA to the rescue

The results are completely different for women. Here is the same data for the 127 main draw women’s singles matches at the Australian Open:

Set  Avg Rally  Ex Rally  Unret%  Ex Unret  
1            1      3.40       1     27.0%  
2        1.035      3.52   0.974     26.3%  
3        1.103      3.75   0.915     24.7%

Still not as dramatic as Harrison-Lorenzi, but the trends are more marked than for the men. The number of unreturned serves drops quite a bit, and rally length increases by an amoun that an attentive spectator might notice. Those two are related–if there are fewer unreturned serves, there are more shots per point, even if we only consider the second shot. Beyond that, there are more opportunities for longer exchanges. In any case, the set-by-set trends for women fit closer to the intial theory than the men’s results did.

As with every aggregate stat, I’m guessing that there is a huge amount of variation among players. Perhaps players who are particularly good in third sets really do return more serves or, as Lorenzi did, shift their tactics in the direction of a more favorable style of play. Looking at these types of numbers for individual competitors is a reasonable next step, but it’s one that will need to wait for another day.

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