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.

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.

Table 1. Simulated set win rates according to various indicators
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:

Table 2. Polish PlusLiga: set win rates according to various indicators. The rates observed in the simulations are included for comparison
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:

Table 3. Italian Superlega: set win rates according to various indicators. The rates observed in the simulations are included for comparison
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?

Table 4. Polish PlusLiga set win rates according to various indicators, only considering sets that were decided by three points or less
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:

Table 5. Polish PlusLiga set win rates according to various indicators. 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:

Table 6. Polish PlusLiga set win rates according to various indicators. 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:

Table 7. Polish PlusLiga set win rates according to various indicators. 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?

Table 8. Italian Superlega set win rates according to various indicators, only considering sets that were decided by three points or less
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:

Table 9. Italian Superlega set win rates according to various indicators. 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:

Table 10. Italian Superlega set win rates according to various indicators. 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:

Table 11. Italian Superlega set win rates according to various indicators. 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):

Table 12. Polish PlusLiga set win rates according to various indicators, only considering sets that were decided by more than five points
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
Table 13. Italian Superlega set win rates according to various indicators, only considering sets that were decided by more than five points
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?

Table 14. PlusLiga: set win rate for teams serving first, by set number. “Close sets” were those won by 3 points or less; “one-sided” sets were those won by more than 3 points
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
Table 15. Italian Superlega: set win rate for teams serving first, by set number. “Close sets” were those won by 3 points or less; “one-sided” sets were those won by more than 3 points
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.

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