# What we know about the theory of optimal strategy in basketball

A few months ago I was asked by Mark Glickman to write a book chapter about optimal strategy in basketball. (Mark is a senior lecturer in the Department of Statistics at Harvard, and is currently in the process of setting up a sports analytics laboratory.)

That was a sort of daunting task, so I recruited Matt Goldman as a co-author (Matt is currently working in the chief economist’s office at Microsoft, and has done some great work on optimal behavior in basketball), and together we put together a review that you can read here. The Chapter will appear in the upcoming *Handbook of Statistical Methods for Design and Analysis in Sports*, one of the Chapman & Hall/CRC Handbooks of Modern Statistical Methods.

Of course, it’s likely that you don’t want to read some academic-minded book chapter. Luckily, however, I was given the opportunity to write a short summary of the chapter for the blog Nylon Calculus (probably the greatest of the ultra-nerdy basketball blogs right now).

You can read it here.

A few things you might learn:

- An optimal team is one where everyone’s
*worst*shots have the same quality. - An optimal strategy does not generally lead to the largest expected margin of victory
- NBA players are shockingly good at their version of the Secretary Problem

(Long-time readers of this blog might see some familiar themes from these blog posts.)

* An optimal team is one where everyone’s worst shots have the same quality.

Isn’t this a bit like saying an optimal poker strategy is one where every raise has the same quality?

Let’s suppose that a basketball possession is a 2-agent zero sum game with limited information. So the offence uses some scheme to pick the player that shoots, and the defence chooses some counter-scheme. Then, if *the defence* is playing optimally, it might be indifferent to shot choices. However the defence could easily be indifferent but not optimal. For example, this is true if the defence makes such stupid choices so it’s totally ineffectual.

In addition, you’re assuming that the ‘usage curve’ is continuous. So a player could have some small rate of 80% shooting, and 20% shooting the rest of the time. Then the team might be maximally exploiting if it got all of the players 80% shots, and none of the 20% shots.

The most technically rigorous (and general) way to express the allocative efficiency criterion is that the total scoring should be maximized with respect to everyone’s shot choices (under the constraint of a finite number of shots available). This condition ends up implying that all teammates should agree on how good a shot should be before you take it.

You’re right that the line about “everyone’s worst shots have the same quality” is assuming a continuous usage curve. Optimization with piecewise functions is messier to write down, but the principle is the same.

Being easily lost on many higher-level analytical tools, I chose to follow the ‘secretary problem link… which consumed several hours of reading and pondering. Impressed by how applicable the concept is to, well, most of life’s executive choices. ‘As Good as it gets’ comes to mind; never saw th film but the title has always resonated. Add: your posts are consistently captivating and clearly written; I always read them first among the Reader options. Bravo

Thank you!