This article provides some of the analytical detail behind this post on Mark Lebedew's blog, discussing whether it is better to serve first or receive first in a volleyball match. Note that the numbers in the first table differ slightly from those presented by Mark, because this was a different simulation run.
Indicators of set win
Ben Raymond (ben@untan.gl), Mark Lebedew (@homeonthecourt)
How strongly is the outcome of a set determined by various markers along the way? For example, how often does the first team to reach 8 points win the set? Is serving first an advantage or a disadvantage? Serving first is expected to be a slight disadvantage, because the serving team must win at least 2 breakpoints to win the set.
1 Simulations
We first run simulations to check what numbers we might expect from two closely matched teams. We simulated 10000 sets, using a constant sideout rate of 0.667 (the average overall sideout rate) for both teams. The team serving first in each simulated set was chosen at random, and the outcome of each point was randomly assigned according to the sideout rate.
| Indicator | Number of sets | Simulated set win rate | Confidence interval |
|---|---|---|---|
| Served first | 10000 | 0.478 | 0.468–0.488 |
| First to 1 point | 10000 | 0.551 | 0.541–0.561 |
| First to 8 points | 10000 | 0.689 | 0.680–0.698 |
| First to 8 (with 3 point margin) | 4218 | 0.784 | 0.772–0.797 |
| First to 16 points | 10000 | 0.790 | 0.782–0.798 |
| First to 16 (with 3 point margin) | 5926 | 0.885 | 0.877–0.893 |
| First to 20 points | 10000 | 0.850 | 0.843–0.857 |
| First to 20 (with 3 point margin) | 6370 | 0.937 | 0.930–0.942 |
These results agree with the notion that serving first should be a slight disadvantage between closely-matched teams.
2 Real data
Now let’s see what the data look like. From the PlusLiga:
| Indicator | Number of sets | Set win rate | Confidence interval | Set win rate from simulations |
|---|---|---|---|---|
| Served first | 543 | 0.499 | 0.456–0.542 | 0.478 |
| First to 1 point | 543 | 0.567 | 0.524–0.609 | 0.551 |
| First to 8 points | 543 | 0.703 | 0.663–0.742 | 0.689 |
| First to 8 (with 3 point margin) | 231 | 0.874 | 0.825–0.914 | 0.784 |
| First to 16 (sets 1-4) | 508 | 0.843 | 0.808–0.873 | 0.790 |
| First to 16 (with 3 point margin, sets 1-4) | 321 | 0.938 | 0.905–0.962 | 0.885 |
| First to 20 (sets 1-4) | 508 | 0.890 | 0.859–0.916 | 0.850 |
| First to 20 (with 3 point margin, sets 1-4) | 349 | 0.971 | 0.948–0.986 | 0.937 |
And the Italian Superlega:
| Indicator | Number of sets | Set win rate | Confidence interval | Set win rate from simulations |
|---|---|---|---|---|
| Served first | 506 | 0.532 | 0.487–0.576 | 0.478 |
| First to 1 point | 506 | 0.583 | 0.539–0.626 | 0.551 |
| First to 8 points | 506 | 0.692 | 0.649–0.732 | 0.689 |
| First to 8 (with 3 point margin) | 206 | 0.835 | 0.777–0.883 | 0.784 |
| First to 16 (sets 1-4) | 471 | 0.826 | 0.789–0.859 | 0.790 |
| First to 16 (with 3 point margin, sets 1-4) | 276 | 0.917 | 0.878–0.946 | 0.885 |
| First to 20 (sets 1-4) | 471 | 0.864 | 0.830–0.894 | 0.850 |
| First to 20 (with 3 point margin, sets 1-4) | 322 | 0.957 | 0.928–0.976 | 0.937 |
In the PlusLiga, the team serving first won almost exactly half the time. The team first to 8 points won around 57% of sets, but if that point was reached with a 3 point margin or better, the win rate rose to 70%. In the Superlega, serving first appeared to be a slight advantage, corresponding to wins in around 53% of sets. Furthermore, teams that served first and who won the first point went on to win the set 66% of the time in the Superlega, and 60% of the time in the PlusLiga.
The team first to 8 points won the set (in simulations) at about the same rate as was observed in the real data. However, the other indicators were stronger in the real data than in the simulation. This reflects — at least in part — the fact that sideout rates won’t be equal between teams in general, and that many other factors determine the outcome of a set.
2.1 Closely-matched teams
From the data, being the first team to serve does not appear to be as much of a disadvantage as the simulations would suggest. Why is this? Perhaps the picture is clouded by the fact that many of these sets will be between two mismatched teams (i.e. one team won rather easily). In these sets, serving first will have little role in determining which team will win. If we restrict our attention to sets in which the winning margin was three points or less, what do we see?
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 209 | 0.459 | 0.390–0.529 |
| First to 1 point | 209 | 0.541 | 0.471–0.610 |
| First to 8 points | 209 | 0.545 | 0.475–0.614 |
| First to 8 (with 3 point margin) | 60 | 0.700 | 0.568–0.812 |
| First to 16 (sets 1-4) | 191 | 0.696 | 0.626–0.761 |
| First to 16 (with 3 point margin, sets 1-4) | 72 | 0.764 | 0.649–0.856 |
| First to 20 (sets 1-4) | 191 | 0.723 | 0.653–0.785 |
| First to 20 (with 3 point margin, sets 1-4) | 70 | 0.857 | 0.753–0.929 |
The team serving first in close sets in the PlusLiga won around 46% of sets. The remaining indicators (first to 8 points, etc) were also weaker predictors of set win, as one would expect given that these sets were all won by only a slim margin.
We should probably get similar results if we consider teams that ended up in similar positions on the league ladder (i.e. were of similar strengths):
Matches between top 4 teams only:
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 45 | 0.556 | 0.400–0.704 |
| First to 1 point | 45 | 0.511 | 0.358–0.663 |
| First to 8 points | 45 | 0.689 | 0.534–0.818 |
| First to 8 (with 3 point margin) | 16 | 0.812 | 0.544–0.960 |
| First to 16 (sets 1-4) | 41 | 0.805 | 0.651–0.912 |
| First to 16 (with 3 point margin, sets 1-4) | 24 | 0.917 | 0.730–0.990 |
| First to 20 (sets 1-4) | 41 | 0.902 | 0.769–0.973 |
| First to 20 (with 3 point margin, sets 1-4) | 25 | 0.960 | 0.796–0.999 |
Matches between teams 5–8 only:
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 22 | 0.409 | 0.207–0.636 |
| First to 1 point | 22 | 0.636 | 0.407–0.828 |
| First to 8 points | 22 | 0.818 | 0.597–0.948 |
| First to 8 (with 3 point margin) | 13 | 0.846 | 0.546–0.981 |
| First to 16 (sets 1-4) | 21 | 0.905 | 0.696–0.988 |
| First to 16 (with 3 point margin, sets 1-4) | 15 | 1.000 | 0.782–1.000 |
| First to 20 (sets 1-4) | 21 | 0.857 | 0.637–0.970 |
| First to 20 (with 3 point margin, sets 1-4) | 15 | 1.000 | 0.782–1.000 |
Matches between teams 9–12 only:
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 33 | 0.515 | 0.335–0.692 |
| First to 1 point | 33 | 0.485 | 0.308–0.665 |
| First to 8 points | 33 | 0.697 | 0.513–0.844 |
| First to 8 (with 3 point margin) | 13 | 0.923 | 0.640–0.998 |
| First to 16 (sets 1-4) | 29 | 0.793 | 0.603–0.920 |
| First to 16 (with 3 point margin, sets 1-4) | 12 | 1.000 | 0.735–1.000 |
| First to 20 (sets 1-4) | 29 | 0.828 | 0.642–0.942 |
| First to 20 (with 3 point margin, sets 1-4) | 15 | 1.000 | 0.782–1.000 |
What about close matches in the Superlega?
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 221 | 0.570 | 0.502–0.636 |
| First to 1 point | 221 | 0.566 | 0.497–0.632 |
| First to 8 points | 221 | 0.534 | 0.466–0.601 |
| First to 8 (with 3 point margin) | 69 | 0.623 | 0.498–0.737 |
| First to 16 (sets 1-4) | 203 | 0.665 | 0.596–0.730 |
| First to 16 (with 3 point margin, sets 1-4) | 77 | 0.727 | 0.614–0.823 |
| First to 20 (sets 1-4) | 203 | 0.685 | 0.616–0.748 |
| First to 20 (with 3 point margin, sets 1-4) | 80 | 0.825 | 0.724–0.901 |
And matches between closely-ranked teams?
Matches between top 4 teams only:
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 51 | 0.431 | 0.293–0.578 |
| First to 1 point | 51 | 0.510 | 0.366–0.652 |
| First to 8 points | 51 | 0.608 | 0.461–0.742 |
| First to 8 (with 3 point margin) | 21 | 0.905 | 0.696–0.988 |
| First to 16 (sets 1-4) | 45 | 0.911 | 0.788–0.975 |
| First to 16 (with 3 point margin, sets 1-4) | 24 | 0.958 | 0.789–0.999 |
| First to 20 (sets 1-4) | 45 | 0.867 | 0.732–0.949 |
| First to 20 (with 3 point margin, sets 1-4) | 34 | 1.000 | 0.897–1.000 |
Matches between teams 5–8 only:
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 46 | 0.457 | 0.309–0.610 |
| First to 1 point | 46 | 0.630 | 0.475–0.768 |
| First to 8 points | 46 | 0.696 | 0.542–0.823 |
| First to 8 (with 3 point margin) | 21 | 0.762 | 0.528–0.918 |
| First to 16 (sets 1-4) | 42 | 0.810 | 0.659–0.914 |
| First to 16 (with 3 point margin, sets 1-4) | 20 | 0.900 | 0.683–0.988 |
| First to 20 (sets 1-4) | 42 | 0.929 | 0.805–0.985 |
| First to 20 (with 3 point margin, sets 1-4) | 24 | 1.000 | 0.858–1.000 |
Matches between teams 9–12 only:
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 52 | 0.654 | 0.509–0.780 |
| First to 1 point | 52 | 0.538 | 0.395–0.678 |
| First to 8 points | 52 | 0.615 | 0.470–0.747 |
| First to 8 (with 3 point margin) | 16 | 0.625 | 0.354–0.848 |
| First to 16 (sets 1-4) | 47 | 0.851 | 0.717–0.938 |
| First to 16 (with 3 point margin, sets 1-4) | 22 | 0.955 | 0.772–0.999 |
| First to 20 (sets 1-4) | 47 | 0.787 | 0.643–0.893 |
| First to 20 (with 3 point margin, sets 1-4) | 31 | 0.968 | 0.833–0.999 |
2.2 Poorly-matched teams
What if we look in the other direction — at one-sided matches that one team won easily? We would expect serving first to be of little importance (because the stronger team will win regardless of whether they serve first or not):
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 197 | 0.503 | 0.431–0.574 |
| First to 1 point | 197 | 0.589 | 0.517–0.658 |
| First to 8 points | 197 | 0.858 | 0.801–0.903 |
| First to 8 (with 3 point margin) | 113 | 0.973 | 0.924–0.994 |
| First to 16 (sets 1-4) | 191 | 0.990 | 0.963–0.999 |
| First to 16 (with 3 point margin, sets 1-4) | 172 | 1.000 | 0.979–1.000 |
| First to 20 (sets 1-4) | 191 | 1.000 | 0.981–1.000 |
| First to 20 (with 3 point margin, sets 1-4) | 190 | 1.000 | 0.981–1.000 |
| Indicator | Number of sets | Set win rate | Confidence interval |
|---|---|---|---|
| Served first | 166 | 0.494 | 0.416–0.573 |
| First to 1 point | 166 | 0.639 | 0.560–0.712 |
| First to 8 points | 166 | 0.910 | 0.855–0.949 |
| First to 8 (with 3 point margin) | 102 | 0.971 | 0.916–0.994 |
| First to 16 (sets 1-4) | 158 | 0.981 | 0.946–0.996 |
| First to 16 (with 3 point margin, sets 1-4) | 140 | 1.000 | 0.974–1.000 |
| First to 20 (sets 1-4) | 158 | 1.000 | 0.977–1.000 |
| First to 20 (with 3 point margin, sets 1-4) | 157 | 1.000 | 0.977–1.000 |
Yep.
3 Patterns with set number
Do these patterns vary by set number?
| Set number | Win rate (all sets) | Win rate (close sets) | Win rate (one-sided sets) |
|---|---|---|---|
| 1 | 0.469 | 0.439 | 0.488 |
| 2 | 0.531 | 0.440 | 0.581 |
| 3 | 0.455 | 0.439 | 0.465 |
| 4 | 0.544 | 0.444 | 0.596 |
| 5 | 0.571 | 0.667 | 0.471 |
| Set number | Win rate (all sets) | Win rate (close sets) | Win rate (one-sided sets) |
|---|---|---|---|
| 1 | 0.466 | 0.484 | 0.449 |
| 2 | 0.618 | 0.625 | 0.613 |
| 3 | 0.450 | 0.537 | 0.390 |
| 4 | 0.641 | 0.677 | 0.617 |
| 5 | 0.514 | 0.611 | 0.412 |
In both the PlusLiga and Superlega, using all matches (see the “Win rate (all sets)” columns), serving first appeared to be a disadvantage in sets 1 and 3, but an advantage in sets 2 and 4. In set 5 in the PlusLiga serving first appears to be an advantage, and perhaps a very slight advantage in set 5 in the Superlega.
Looking at close sets only, the set 1-4 patterns disappear in the PlusLiga: serving first is consistently a disadvantage. Serving first in the fifth set is an advantage in both leagues for close sets.