Insight: Serial correlation is defined as patterns in the sequence of serve directions in a point game. Servers and returners can use such patterns to their advantage.
As I write in Part 1, if there are unequal win probabilities across serve directions, the server and returner may be able to take advantage of them. Another potential inefficiency, which the returner in particular can capitalize on, is serial correlation in the sequence of serve directions in a point game. (In theory, there can also be serial correlation in the sequence of the returner’s court positioning across serves, but if there is, I can’t observe it without player tracking data.)
What is serial correlation? We can see in Table 1 that both players use mixed strategies in each point game; that is, they serve a certain percentage of the time to the returner’s left and a certain percentage to his right. In turn, serial correlation consists of any patterns in the sequence of those serve directions across serves in those point games. For instance, does serving to the returner’s left on the current serve in a point game make him any more or less likely to serve to the returner’s left on the next serve in that point game? More likely means there is positive serial correlation, whereas less likely means there is negative serial correlation. Ultimately, if there is any serial correlation, the returner may be able to take advantage of it because he’ll have an idea where the next serve is going!
To test for serial correlation, Walker and Wooders use a “runs” test, where a run is a sequence of serves in a point game that are all hit to the same direction. Given a certain number of serves hit to each direction in a point game, they want to see how many runs there are. The more runs relative to the total number of serves, the more negative the serial correlation, and vice versa. As a simple case, look at Murray’s second serves to the deuce court. He hit 18 of them to Raonic’s left (backhand) but only one to Raonic’s right. Given that, there can only be two runs (if the serve to Raonic’s right was the first or last serve of the point game), or three runs (if the serve to Raonic’s right was not the first or last serve). Ultimately, there were three runs, so the serial correlation is slightly negative, though not statistically significant.
The details of the runs test are in WW, but F(ri-1) in Table 1 is the probability of having no more than one fewer than the number of runs in the point game, and F(ri) is the probability of having no more than the number of runs in the point game. We can therefore reject the null hypothesis of no serial correlation at the 5% level if F(ri-1) is greater than 0.975 (in which case the serial correlation is negative) or if F(ri) is less than 0.025 (in which case it’s positive). Ultimately, we can only reject one null hypothesis at the 10% level, specifically for negative serial correlation in Raonic’s second serves to the deuce court. Essentially, that result is about what we would expect even if the data were random. (A joint Kolmogorov-Smirnov test for all the point games is also insignificant, but as in Part 1, such tests are low-powered for only eight point games.) Then again, it’s always possible it isn’t a false positive, considering WW find a bias toward negative serial correlation among tennis pros (see Gauriot, Page, and Wooders (2022) for more details).
In the end, the main point of this article is to explain what serial correlation is and show how to test for it. Moreover, even though it’s always possible for serial correlation not to be suboptimal (for instance, if serving to one direction on the current point affects your win probability when going back there on the next point; my co-authored paper “Disequilibrium Play in Tennis” delves more into this concept), it is still useful to check for serial correlation in your matches because you can use that information to your advantage and prevent your opponent from doing the same to you.
Table 1: Runs Tests for Serial Correlation
