Hit me with your best shot

Choosing and designating some models as Great has been causing me anxiety and so here’s a model that I’m going to write about just for fun. Thanks to Titia Praamsma for sending me this mathematical model of shot selection in basketball by Dr. Brian Skinner of the University of Minnesota. Skinner even acknowledges the parsimony issue:

While the complex nature of decision-making in basketball makes such a description seem prohibitively difficult, it is nonetheless natural as belonging to the class of “optimal stopping problems”…

Fig. 1. Occassionally, I have wanted to derive a mathematical model for basketball, however, the complex nature of the decision-making process in basketball makes it prohibitively difficult for me.

Now, I follow basketball fairly closely, yet I’ve never been able to come up with any good ideas for applicable mathematical models (see Fig. 1). After reading Skinner’s work I realize that this is because I was framing the question so that I would have to tackle it using complicated and possibly uninformative methodology.*

Let me explain. When I think about basketball, I think that as a coach your job is (partly) to invent set plays that will increase your team’s chance of getting a high quality shot. This problem is spatial, game theoretical (i.e., it depends on whether the other team is playing a zone or a man-to-man defense) and it’s probably necessary to coordinate the movement of all five players (i.e., n=1 or 2 is fundamentally a different problem and so studying these simple cases is likely uninstructive). I know these types of problems are hard and so it seemed like too much work for me to derive a mathematical model for basketball.

The above discussion is meant to illustrate that anything can seem complicated if you choose to look at it that way.**  After acknowledging the complexity of basketball, Skinner goes on to come up with a simple model of shot selection. I think that his conceptual approach is quite clever and I hope this is underscored by my admission that I had (naively) written basketball off as not something that was amenable to simple models.

Fig. 2. Tiffany Hayes (with the ball) of the University of Connecticut: dropping 35 points, shooting 11-for-15, and lending support to the controversial "hot hand" phenomenon (see Skinner (2012)).

Certainly, part of what Skinner does that’s clever is come up with a good question. The question is this:

Given a shot opportunity of quality, q, should a player take the shot?

The answer is that it’s a good idea to shoot when q > f where f is a threshold quality value that depends on n, the number of shot opportunities remaining in a possession and the article explains how n might depend on the team’s turnover rate, the existence of a shot clock and how fast the team moves the ball.

Interestingly, the article reports that a team that moves the ball well has a higher n, and therefore a higher f, and would want to execute more passes (until the shot opportunity q > f arises) than a team with poor ball movement (i.e., lower n and lower f). This is counter-intuitive because you’d think that if a team passes the ball quickly, then they can shoot sooner. This is true, but the result isn’t about whether a team with good ball movement can shoot earlier in the possession and win, it’s about what’s optimal: it’s optimal to expect a better shot to arise during a possession if shots are created at a faster rate – and so the team with good ball movement makes more passes and they might even wait longer to shoot.

Let’s revisit the model construction. You might say that if Brittany Griner gets the ball in the post then she should dunk it. That’s a good decision because this would be a high quality shot where q > f, but what I like about the model is that it abstracts away the name of the player, where the player is on the court, and what type of offense was run to generate the shot opportunity, and simply summarizes all this complexity into the one variable, q, shot quality.

Another aspect of this paper that I like is the comparison with data from the NBA. In fact, section 4. of Skinner’s paper is solely dedicated to recasting  f in terms of a shooting rate under the assumption that shot opportunities arise at a constant rate τ, so that the number of remaining shot opportunities n at time t is Poisson distributed. This is a nice final step because how would one ever know if q > f without these addition assumptions? My point is that q (shot quality) is not an especially useful quantity because how would one measure that? On the other hand, the shot rate can be estimated from a play-by-play box score which reports when shots were taken during the course of a game.

So that’s basketball. Some time in the future we might talk about soccer or coffee, but I have some other posts to get to before that. There’s also a homework problem if you continue to scroll down.

* I think I’m suggesting if the only ideas that you have involve deriving a complicated model, the solution might be to refine the question.

** I stated that there’s always a complicated way of looking at a problem. What I mean is that you can take a complex phenomenon at face value and then obviously it will appear complex. Today’s philosophical question is:

Given a complex process does there necessarily exist a simple way of looking at it that will yield productive insight?

This entry was posted in Fun, Great models, Just simple enough by Amy Hurford. Bookmark the permalink.

I am a theoretical biologist. I became aware of mathematical biology as an undergraduate when I conducted an internet search to learn about the topic. Now, twelve years later, I want to know, what is it that makes great models great? This blog is the chronology of my thoughts as I explore this topic.

8 thoughts on “Hit me with your best shot”

1. Brilliant! Now if only I can “translate” this for the 12-year olds that I coach …

2. Love this post! You’ve inspired me to work on a machine that will monitor the movements of players and the ball on the court and emit a tone when q>f! We can use it at the next 3-on-3 tournament!

• Wow, those are GREAT posts! That’s a really neat result concerning the 250+ moves to checkmate. In units of moves, it sounds likes there’s no simple way of expressing the solution to that problem. Taking note of your other post on synthesizing, do you think the conclusion that there’s no simple solution to that chess problem could be a matter of how we’re looking at it (regarding your comments on contingency in community ecology)? Perhaps, there could be general patterns among the 250 moves if we could figure out the right way of looking at them. (Maybe?)

• No. I don’t think there’s any ‘angle’ from which humans can look at lengthy chess endgames in order to come to some kind of general or synthetic understanding of the winning plan. If there was any pattern or “strategy” to be seen in the moves that lead to a win in, say, a rook+bishop vs. rook endgame, I think grandmasters would’ve seen it by now.

The only thing I think we can hope for (and that we mostly have) is a general understanding of the features that make chess endgames more or less understandable. For instance, lengthy, impossible-to-understand endgames only occur when one side is slightly stronger than other, and when there are pieces besides just kings and pawns on the board. If one side is much stronger than the other, or if there are just pawns on the board, the best moves are simple enough that humans can discover and understand them. Analogously, I think we can hope to explain why the assembly of some ecological communities is ‘assembly rule driven’, and why the assembly of other communities seems to be intractably complex and ‘rule-less’.

3. I love this post Amy. I love that it’s mathematically proven that it’s more important to make good passes than it is to shoot often. I can see this with my own team – we definitely need to get this drilled into our heads.

• I love that the little kids at half time of the Raptors games are being optimal. High turnover rate? Just take the first shot you see – who can blame you, it’s the smart thing to do!