## Your Volleyball Lineup – Optimizing The Order

A volleyball lineup is a weird thing; different from a “starting lineup” in any other sport. Every volleyball team is really made up of six unique units which often have very different strengths and weakness- even while the same players stay on the court. A team might go from juggernaut to pushover when their dominating outside rotates into the backcourt. Unlike any other sport (except perhaps baseball), these units (or rotations, as we tend to call them in volleyball) must all take their turns- rotating through in a predetermined order. Setting aside matchups, chemistry, and other intra-personal concerns, there is one thing that makes a volleyball starting lineup quite interesting: the order (or starting point) of the rotations can have a huge impact on a team’s success, particularly in a 15-point deciding game.

There are two reasons why the ordering of a lineup can have a huge impact on the game. First, different rotations often have significant differences in their ability to score points in a sideout or serving situation. Using data from my own team, I notice that (through 10 matches), there is a huge difference between our strongest sideout rotation (setter in 5, where we are siding out at 73%), and our worst (setter in 2, where that number is just 48%). The difference between our strongest serving rotations (setter in 1 or 2, where we score about 57% of the time in both), and our weakest (setter in 2 or 3, where we are score about 37% of the time) is large as well. This is not an uncommon, particularly in high school or juniors volleyball where there is often a much greater talent disparity between the strongest and weakest players on the court (and thus, the strongest and weakest rotations), as compared to collegiate or international volleyball.

The second characteristic that lends great importance to the ordering of the lineup is that, in volleyball, certain rotations may be “in play” for many more points than others. In baseball, it has been shown through simulation, that teams should score more points by batting their best hitters first in the lineup. However, since the difference between the batter who gets the most at bats and the batter (or position in the lineup, if pinch-hitters are used) who gets the least at bats can be no more than one, the difference (while statistically significant) is not all that great. However, imagine if Albert Pujols or Josh Hamilton were allowed to go up to bat again if they got a hit, and keep doing so every time they got on base. Well, this is exactly what happens in volleyball! Your strongest serving rotations get to keep playing every time they get a point, and your weakest sideout rotations have to stay until they get a point!

A good high school volleyball team will usually sideout somewhere between 50 and 60%. At this rate of sideout, the average number of rotations per game will be somewhere around 26 or 27, or 13 for each team. This means that, on average, each rotation will have about 2 serving chances (ie, 2 chances to score as many points with your serve before the opponent sides out), and 2 sideout situations each game. What makes this so important though, is that, while you have about a little better than 50% chance of your first server coming up to serve for a 3rd time, that chance dwindles below 5% for the odds of your last serving coming up a 3rd time. For any coaches reading this, check your stats: if you use the same lineup every time, it’s likely that your first server has served between 15 and 25% more balls than your sixth server. I sure hope they are a better server!

This effect becomes even more dramatic in a 15-point game. Again, using a rate of sideout somewhere between 50 and 60% (as typical for high-level high school women’s), the average number of rotations per game is going to be somewhere around 15 or 16. This means that most of the rotations will only get to go once, with (on average) 1 or 2 of them getting to go twice. This means that the lineups become even more important in a 15-point deciding game. The great thing about this is that the prepared coach now has a decisive advantage over the unprepared coach!

I will go into some of the mathematics and experimentation I used to arrive at the conclusions, and show you the methods I used, both to open them up to critique and to allow others to explore the topic in new (and hopefully better) ways. However, because I am sure that many of you don’t find the same sick joy in poring over Excel spreadsheets as I do, I will place the conclusions first, and then follow up with the data afterwards:

Conclusion: Look at your stats, find your highest-percentage rotation. Start there.

The majority of the time, this will be the correct lineup. For me, this means that, if I have my choice, I want to start by receiving serve with the setter in 5. In simulated games (more on that in a bit), this rotation won 73% of the time. If I started in the worst possible scenario (serving with the setter in 3), I only win 59% of the time. This means that, for my team, simply changing the lineup can result in me being about 25% more likely to win a 15-point deciding game! Even the difference between the “standard” choice for most high school teams (pick the serve, and start with the setter in 1), which is still one of my better rotations, is almost 5% weaker than my “optimal” lineup. How many practices, serve receive drills, or wash games, would your team need to do to increase their chances of winning by 5%?

And now for the nerdy stuff:

I basically used a 3-part method to do this research:

Part One: Determine the “points per rotation” or “points per service opportunity” based on the sideout percentages. I’m sure there is a way to calculate an exact value, but I must have slept through that lecture in grad school and even the Wikipedia page for “Bernoulli Process” is scary, so I decided to just get an experimental estimate. I just used Excel to generate random numbers and did 500 “service runs” for each value to get the following chart:

Sideout Percentage ; Average Points per Service Opportunity

75 .32

70 .38

65 .52

60 .59

55 .78

50 1

45 1.13

40 1.69

35 1.68

30 2.49

25 2.83

As you might be able to tell, even with 500 trials each, the results aren’t perfect (what’s up with 35% eh?), but I believe they give you a pretty decent estimate. To find the in-between values, the fitted line:

P=8.7849*e^(-4.42*x)

Gives you an r-squared value over .9 to the experimental data, so it’s a pretty good fit. “P” would be “points per service opportunity’ and “x” would be your sideout percentage.

Part Two: Create an “average” or “expected” game, starting with each of the 12 possible starting lineups (serving or receiving with setter in 1-6).

Basically I did this by using the chart above to go rotation by rotation. Remember that the team receiving serve will always (except when I note below) be expected to get exactly 1 point, while the serving team will be expected to get either a bit less (if the receiving team’s sideout % is above 50) or a bit more (if the receiving team’s sideout % is below 50) than 1. Since it’s generally impossible to know (an impractical, for this exercise) your opponent’s sideout percentage, you just set it to be (1-X), where X is your point-scoring percentage when serving. Then, you simply go rotation by rotation until one team gets to 25 “expected points.” Below I show you an example with my team starting by receiving serve with the setter in 5:

Us Them

1 .361 (since we sideout at about 73% in this row)

1.212 (since we score at about 55% here) 1

1 .694 (since we sideout at about 57%)

.574 (since we score about 38% here) 1

1 .623 (since we sideout about about 60%)

.539 (since we score about 36% here) 1

1 1.04 (since we sideout about 48% here)

1.35 (scoring at about 58%) 1

1 .803 (siding out at about 55%

1.33 (scoring at about 57%) 1

1 .816 (about 54% sideout)

.852 (scoring at about 47%) 1

1 .361 (back to the start)

1.212 (since we score at about 55% here) 1

1 .694 (since we sideout at about 57%)

.574 (since we score about 38% here) 1

1 .623 (since we sideout about about 60%)

.539 (since we score about 36% here) 1

1 1.04 (since we sideout about 48% here)

1.35 (scoring at about 58%) 1

1 .803 (siding out at about 55%

1.33 (scoring at about 57%) 1

1 .816 (about 54% sideout)

.852 (scoring at about 47%) 1

1 .361 (so this rotation comes up a 3rd time)

.279 .23

If you notice, the numbers for the final row are a little weird. On the left is (1.212 * 0.23), which is the expected points for next rotation (serving with the setter in 4), but modified so that the “expected score” ends at about 25 points, rather than something like 25.8. In turn, the other number is simply (1 * 0.23) or just 0.23, to keep the same ratio for that last rotation. I don’t think this is totally necessary, but it makes the next part a bit more accurate. This was for a 25-point game. If you just want to do it for a 15-point game, you would obviously stop a bit earlier.

At this point, you have everything you need to determine your starting rotations, by simply doing these “games” for each rotation and each starting point. If you sideout better than 50%, you will most likely be best starting in serve receive, unless you have one particularly dominant point-scoring rotation. You can then come up with your “expected point differentials” for when you start in each rotation. This is simply done by taking your expected points (which will mostly be above 25 or 15 if you have been winning most of your matches, or below, if you have been losing most of them) and subtracting the other team’s points.

For me, it looks like this:

Rotation 25pt +/- 15pt +/- Rotation 25pt +/- 15pt +/-

Rec, Set in1 3.29 2.19 Serve, Set in 1 3.36 2.23

Rec, Set in 6 3.19 2.24 Serve, Set in 6 3.35 1.89

Rec, Set in 5 3.71 2.65 Serve, Set in 5 3.15 2.24

Rec, Set in 4 3.12 1.71 Serve, Set in 4 3.26 1.74

Rec, Set in 3 3.16 1.54 Serve, Set in 3 2.87 0.96

Rec, Set in 2 3.05 1.97 Serve, Set in 2 2.53 1.41

One interesting thing to notice is that the optimal lineups for 25-point games and 15-point games can be different. There’s two explanations for this: one, I screwed up my math somewhere. Entirely possible. But I believe the likely reason is that, in 15-point games, the effect of one dominant rotation is magnified. Our sideout offense is so good in rotation 5 that it is worth starting with our 4th-best server (if we have to serve in a 15-point game), just to get to that rotation sooner- and thus be more likely to have it come up at the end of the game. I was also surprised in that it seems like the effect is driven more by the sideout rotations than in the serving rotations. I suspect this is because the spread between my best and worst sideout rotation (about 35 percentage points) is greater than the spread between my best and worst serving rotations (about 22 percentage points).

Part Three: For curiosity’s sake, I used these point differentials to calculate win expectancies. For example, being +2.65 points in a 15-point game means you can be expected to score about 15/(15+12.35) or .54.8% of the points in a 15-point game. This corresponds, experimentally, to winning about 73% of 15-point games. By experiment, I simply generated 28 random numbers (between 0 and 1) in Excel. If 15 or more of them were less than 0.548, I counted that as a “win” and if 13 or less of them were less than 0.548, I counted that as a “loss.” I ran 500 trials for being +2.65, (my optimal rotation, receiving with the setter in 5), +2.23 (the “usual” rotation of starting with serve with setter in 1), and +0.96 (my worst rotation, serving with the setter in 3). Since at some point I needed shovel off the foot of snow piling on my truck, I have not yet done this for all the point differentials (or calculated a best-fit equation, as I did with the service runs), but I may at some point. It is really not very important though, as it is more of a curiosity-satisfier (ie, “just how important is it?”) as opposed to something that has practical information for coaches to use.

Final Questions:

If you’ve made it this far, I both commend you, and feel a bit sorry for you, as you must also live on the East Coast and must be trapped inside by the snow with nothing to do other than read a 2,500 word examination of optimal lineup construction. I think there are several other avenues of exploration within this topic:

It would be nice for a mathematician well-versed in Stochastic Processes to give an exact calculation of the points per service opportunity. I think the experimental numbers are good, but they obviously are not perfect.

The point-differential or “+/-“ numbers I came up with are all based off averages. Obviously, volleyball games are often made up of runs, so the concept of standard deviations comes greatly into play. Since the “games” I played were not made up of actual randomized “service opportunities”, but instead based on mean values, it’s not quite as accurate. A piece of software that could accept the 12 inputs (sideout and point-scoring %’s for all 6 rotations) and simulate 1 million games of each starting position would be awesome.

I haven’t discussed matchups or anything yet. I do this for two reasons: One, I haven’t really seen any compelling evidence that suggests the power of matchups. Ie, is it better to play your best-blocking opposite against their best leftside (because it will slow them down), or is the effect of that blocker pretty much equal (because if they can slow down the top leftside, they could probably shut down the other one) across the board? Two, the risk for adjusting your volleyball lineup for a matchup is that the other team will adjust and thus leave you with a sub-optimal starting lineup AND the wrong matchups. “A bird in the hand…” they say.

And finally, you may have noticed there are some instances where the math doesn’t totally match up. Particularly with regards to the best-fit line not matching exactly to the numbers I show. This is because I did the calculations for my team, then thought it would be a pretty cool thing to share, but wanted to do some more rigorous experimentation to give more accurate numbers when I share. I don’t think it’s a huge deal, but the number for the best-fit line is the most accurate, while the examples I share from my team (and their derivations) are a bit less accurate.

I hope this helps when it comes to optimizing your lineup with your team!

Joe Trinsey

Gold Medal Squared