When You Realize You're Not Even Close
Volleyball in Light of Skinner and Goldman's Optimal Strategy in Basketball
Citation:
Skinner, B., & Goldman, M. (2015). Optimal Strategy in Basketball. ArXiv:1512.05652 [Physics]. http://arxiv.org/abs/1512.05652
You know the feeling when you taste a new dish that is so good that you keep repeating to your friends how good it is after each bite?
You know the feeling when someone shows you the difference between what you’ve been doing and the way it should be done?
Have you ever felt both of those at the same time?
That weird mish-mash of thoughts and feelings is the result of reading Skinner and Goldman’s “Optimal Strategy in Basketball”. Add to that the dismay of knowing that they wrote it in 2015 and I finally discovered it in 2020. The only thing that eases that dismay is the knowledge that I wasn’t anywhere near prepared enough then to do anything about what I read. While I still may not be prepared enough now, I can at least see how to get there from where I currently am.
So what was so good about the paper that I couldn’t stop telling my friends? The introduction was all it took. The authors clearly summarized what my friends and I have been fumbling around for months, if not years.
The score difference fluctuates somewhat randomly throughout play.
That net score difference is determined by the skill levels of the teams and the scoring strategies they use during play.
There are three main principles that go into determining what the optimal use of scoring strategies should be.
Those ideas may not seem so earth-shattering but that is the amazing part. If those ideas are so plain to see, why hadn’t I seen anyone else nail these down in some formal, empirical way until this paper? It’s not until you try to explain something complex to others that you realize just how little you truly understand it.
That’s where math starts to come in. The first of the three principles to consider is “allocative efficiency”, or determining the frequencies at which certain players/plays should be used in order to maximize scoring. The authors demonstrate that the answers are not as simple as you think they are. It isn’t as simple as using the average points scored by each player/play. You need to understand the marginal points scored. Marginal rates are derivatives. They explain how large the changes are between data points. The question shouldn’t be “how many points do you expect to score running this play?” The question should be “how many more points would you expect to score if you ran that play one more time?” There’s an element of diminishing returns because repeating the same play over and over will lead to fewer expected points over time for a variety of reasons.
The second principle is “dynamic efficiency”, which the authors explain as an “optimal stopping problem”. To maximize dynamic efficiency, teams must choose which shots to take and which ones to pass up within a given possession. Teams should shoot when they reach a point at which the expected value of the current shot is greater than the average expected value of continuing the possession. The expected value of a possession keeps going down as the shot clock keeps winding down so earlier shots are typically better than shots in identical circumstances later in the possession. This explains, at least to some extent, why NBA teams take a lot of 3-point shots early in the shot clock.
The third principle is “risk and reward”. You can easily understand there is a trade off between these two but the authors quantify this trade-off in an insightful way. Increasing “risk” means you decrease your chances of scoring your “average” number of points in order to increase the chances you score a number of points much higher than your average. The downside of increased risk is not only the decreased chances of scoring your average but also an increase in the chances of scoring far less than your average number of points. So why would you risk, given how it negatively impacts your scoring? If you remember back to the introduction of the paper, you should care about the difference in scores more than how many points you score. If your opponents are likely, based roughly on average scoring, to outscore you then you have to take some chances to possibly increase your scoring. In statistics-speak, you are trying to fatten the tails of your scoring distribution. In COVID-speak, you are trying to flatten the curve.
So how do I think these ideas can be applied to volleyball?
I’m very early on in developing my thoughts on this but I view this post as a chance to sort of think out loud, which I find helpful.
When it comes to “allocative efficiency”, I think it encompasses both offensive play selection as well as setter choice. While there may be some play selection in transition offense, I think that, just by sheer volume of opportunities, I’m mainly considering side out offense. Setter choice is always going to be constrained to some extent by reception/dig quality, which is only the beginning of the deep complexity involved in figuring this out. Start by considering a given play in a three-hitter rotation. You shouldn’t just consider the attack efficiencies of the three front row attackers when running that play, you should compare how efficient each attacker is as they are set more and more in that situation. You’re looking to determine how to get the most expected scoring by leveraging each attacker as much as possible before their skill curve deteriorates too much1. The better the individual attacker, the more the distribution will skew towards them. To some extent, you set different options to “keep your opponents honest” but mathematically, you are trying to maximize expected scoring. This is also an example of how the score fluctuations in a match are somewhat random. There are many factors that contribute to these situations and who the setter chooses to set can vary randomly within your maximal scoring scenario. You can achieve your allocative goals without strictly scripting what happens on every play. I think allocative efficiency can also influence setter choice on less-than-perfect passes. To further complicate your thinking, this would mean not just knowing how to maximize scoring on a certain play with certain attackers but also how that efficiency may change as reception quality deteriorates. The abilities of the attackers determine this to some extent but there is always an element of allocation to consider.
I think it’s necessary to understand how marginal scoring rates change in all of these situations before efficient allocation can be properly understood. In my experience, coaches only understand average scoring so when they consider complicated situations, their thinking reverts to “set the attacker who is most efficient on average” which will not result in the best allocative strategy.
I think making direct “dynamic efficiency” comparisons between basketball and volleyball is a bit tougher, mainly because time constraints in volleyball are much different than in basketball. The rules of volleyball prevent lengthening possessions. With that said, I think there are some interesting comparisons to be made. First, there is a constraint in terms of how close teams are to the end of a set. I have seen this referred to as the “red zone”, or after 20 points. I have also seen research suggesting that 18-15 is a kind of point of no return, in that the trailing team is very unlikely to win the set under normal circumstances. It is possible to study dynamic efficiency as a principle that governs decision making as teams approach the end of sets. How should decision making change as teams near 25 points? Does attack efficiency change in some important way as teams near 25 the way expected points change as the shot clock or game clock nears zero? What impact is there on dynamic efficiency when a team is leading versus trailing? (Is that question really more about risk and reward?)
What about considering rotations as part of dynamic efficiency? Should teams weigh how to shorten time spent in weaker rotations and lengthen time in stronger ones? Is this a different question from allocative efficiency? Is attack choice (line/angle, hit/tip, etc.) a dynamic choice, allocative choice, or both?
“Risk and reward” is probably the most straightforward because it is a meta-strategy, so it isn’t tied to the rules that structure how the game is played. “Risk and reward” is about choosing strategies and about risk profiles. There are plenty of stories in the sports canon about underdogs adopting radical strategies to overcome heavy odds. I don’t think women’s college volleyball coaches consider themselves separate from such ideas. How does a college coach think about risk and reward? How willing is any given coach to gamble on less-certain strategies? It’s not that the strategies don’t work (you need to be wary of dualistic thinking and resulting), it’s that the strategies are much more variable. Do coaches who are willing to risk have any data to support how much variance accompanies a given strategy? How frequently can a coach utilize such strategies and keep their job? Variance means getting a much wider variety of results (in terms of scoring, not necessarily outcomes) and that means living with much less certainty, which can be exhausting.
But what if you coach a college team that can usually make it to a postseason tournament but has very little chance of advancing far? Would you consider risking more, especially given there may be little chance of overcoming the gap between you and your opponent? What if you coach a team that makes the tournament every year, advances a round or two and then needs some luck? How many years would you be conservative and hope that the stars align to make it to the finals?
And then there’s the question of how do you risk? Which strategies are available to your team that you aren’t currently employing to score? Aren’t you trying to practice the things you can’t control well enough yet so you can control them better? Is risking all about tailoring strategies to fit certain opponents beyond the small tweaks that you usually make? Is it about putting more eggs into a single basket than you feel comfortable doing? Do you have to risk in all facets of the game (serving, attacking, blocking, etc.) or is deploying a large change in one area enough? Can you risk with certain personnel decisions rather than with team strategies?
And how would you measure the risks we would take? You need to employ strategies that you trust will either fatten tails or shift curves in a favorable direction. Blindly risking is still risking but you can do better than that. You should gather some kind of data on the potential strategies, whether those come from other teams or your own team in situations in which you are favored enough to experiment. It is a lack of rigor that can give risk a bad name.
There’s a lot of research to be done, data to be gathered, and numbers to be crunched to get a handle on these ideas. But before you do the math, you still need to decide how to translate between what you know about volleyball and what Skinner and Goldman have taught you about basketball. Are there areas unique to volleyball that the authors didn’t have to account for? Are there arguments they made that won’t hold up in volleyball? There is some substantial (but manageable) theoretical work to do before the applied work. Skinner and Goldman did the theoretical work and began the applied work but then ended their paper with this massive caveat:
At a practical level, the biggest hindrance to quantitative basketball strategy is usually the difficulty of accurately estimating the efficiency of different offensive options. The usage curves f(p) are particularly difficult to estimate from easily-measurable statistics, and are necessary for a quantitative determination of optimal allocation. What’s more, the usage curves are really only robustly defined relative to a particular defense, and can vary strongly depending on the quality of the team’s opponent. A major advance in their determination may therefore provide the most important step toward enabling quantitative optimization of basketball strategy. (p. 14)
So it’s going to be a long haul. We’re not even close. But we can get there from here.
There’s an element of game theory to consider here as well, but the authors demur on this point: “Some previous studies have examined the question of optimal play selection in soccer, football, and baseball through the paradigm of the ‘minimax’ rule in game theory (Palacios-Huerta, 2003; Kovash and Levitt, 2009). The minimax rule generally dictates that an optimal game-theoretical strategy (say, for the defense) is the one that minimizes the worst case scenario (the number of points given up). These studies, however, evaluated adherence to the minimax strategy only by examining the average return from different offensive options, rather than the marginal return. Such studies are therefore unlikely to be able to gauge whether basketball teams are exhibiting optimal allocation between plays.” (p. 6)