Freelance Friday: Optimal Basketball Theory

Freelance Friday is a regular series at Nylon Calculus where we solicit contributions from the community at large to broaden the reach and scope of basketball analytics writing. Today’s piece comes from Brian Skinner concerning optimal basketball theory. Brian is a postdoctoral researcher in physics at MIT.  He blogs at Gravity & Levity, including this excellent post concerning the value of improved offensive rebounding. You can follow him on twitter @gravity_levity. 

Pitches, suggestions, questions of comments about Freelance Friday should be directed to thenyloncalculus at gmail dot com. 

Is there a theory of optimal basketball strategy?

Here I mean to use the word “theory” in the scientific sense – a set of quantitative principles that describes reality in a self-consistent way – rather than in the popular sense – a conjecture or unproven assertion. So my question is really this: do we have a quantitative basis for understanding what optimal behavior is in basketball? If so, what can we say about how close actual NBA teams are to it?

These questions are the basis of a recent review written by myself and Matt Goldman (who is now working in the chief economist’s office at Microsoft). Our review tries to give an overview of some of the major things that we know about what optimal strategy is and is not.

You can read a preliminary draft of the chapter here, but let me try to give you the executive summary, with most of the math and the technical details cut out. I’ll go through the three major principles associated with “optimal strategy”, and for each principle I’ll try to give you (1) a brief explanation of what it means, (2) an example of the principle in action, and (3) a summary of what we know about how well NBA players adhere to it.

Principle #1: Allocative efficiency

• What it is:  
  The principle of allocative efficiency says that in an optimal offense, all offensive options should have the same marginal efficiency.  This means, for example, that the worst shots taken by any given player should have the same quality as the worst shots taken by any of his teammates.
• An example:  
  At present, Stephen Curry is a significantly more efficient scorer than any of his teammates.  But that doesn’t necessarily mean that he should shoot more frequently than he does.  That’s because the average effectiveness of a Steph Curry shot declines the more often he shoots, since achieving a higher shooting rate would require Curry to start taking lower quality shots (the ones he would normally pass up).   
  
  The actual optimum shooting rate for Curry is the one where his mental cutoff for when a shot is good enough to take (the quality of his marginal shots) is equal to that of his teammates.
  
  In general, the optimal strategy will involve the most skilled players having a better average scoring efficiency than their teammates.

• How good are NBA players at it?  
  This turns out to be a difficult question to answer statistically.  What we record from data is only the average quality of all shots that are taken.  We don’t directly observe the quality of the shots that are passed up (that is, how many passed-up shots would have been made, had they been taken).  This makes it hard to know whether a given player should have taken more.
  
  The key missing ingredient to gauging the optimality of players’ shot selection is called the “usage curve” (or, in the language of Dean Oliver from Basketball on Paper, the “skill curve”), which tells you the relation between scoring efficiency and shooting rate.  Some clever early work was done on the problem of measuring usage curves by Matt and his collaborator Justin Rao.  But finding a robust way to measure it for individual players (or individual plays) remains a big challenge.
  
  I personally suspect that NBA teams have a propensity to over-use their best offensive options.  But this remains to be proven in a rigorous way.

Principle #2: Dynamic Efficiency

• What it is:  
  The principle of dynamic efficiency concerns the optimal time to shoot, given a dwindling shot clock and uncertainty about future opportunities.  It says that a shot should be taken only when its expected number of points scored exceeds the expected return of the remainder of the possession.
• An example:  
  With 16 seconds left on the shot clock, a player should probably pass up the opportunity to take a weakly-contested long 2.  Such a shot scores less than one point on average (depending on who’s shooting it), while a live possession with 16 seconds left is worth more than one point on average.
  
  But if the possession continues to the point where there are only 3 seconds left, and if the chance for a weakly-contested long 2 reappears, the player should probably take the shot.

Preliminary evidence suggests that NBA players are shockingly good at satisfying dynamic efficiency.  At least, they are in aggregate.

Take a look, for example, at the plot below (with data taken from

this paper

).  The blue circles show the point value of the lowest quality shot that NBA players tend to take when there are

seconds remaining on the shot clock.  The open red circles show the average point value of continuing the possession.  For an optimal basketball team, the two curves should coincide with each other.  And, for data averaged over the whole league, they virtually do.

Principle #3: Risk/Reward Tradeoff

• What it is:

When there is a lot of time left in the game, the best offensive strategy is the one that maximizes the expected number of points scored.  But with relatively little time left, it is often better to adopt a strategy that reduces your expected score but which either reduces the probability of an unlikely upset (if you are favored to win) or increases the probability of an unlikely comeback (if you are the underdog).

This idea can be called the “risk/reward tradeoff”.  Graphically, you can illustrate it something like this:

The blue lines in this picture represent the distribution of possible final scores for a strategy by team A (which is the underdog), and the red lines are the same for team B (which is favored to win).  

In the top picture, the dashed line shows a strategy that gives team A a better chance of winning.  Even though the dashed line corresponds to a smaller final score on average, it has more overlap with the red line, and therefore has a higher chance of leading to an unlikely upset.  (i.e., to an unlikely event where team A happens to score on the high side of its range of possibilities and team B happens to score on the low side of its range of possibilities)

In the bottom picture, the dashed curve represents a better strategy for team B.  Even though it has a smaller mean score than the solid red line, it has less overlap with the blue line, and therefore corresponds to a smaller chance of an unlikely loss.

The general formulation of this principle (in simplified form) is that the optimal strategy maximizes the quantity

• An example:  
  Imagine a team that is not particularly good at shooting 3-pointers.  It might choose to take them sparingly early in the game, understanding that its expected score will suffer if it shoots too many.  However, if the team finds itself with a 10 point deficit and 2 minutes left in the game,  it probably makes sense to increase the number of 3-pointers taken.  Victory is unlikely in this scenario, and a larger proportion of 3’s  increases the variance in the outcome.  

• How good are NBA players at it?  
  NBA players’ intuitive feel for the risk/reward tradeoff seems to have a number of deficiencies.  Generally, NBA teams do respond correctly to “underdog” situations: they increase the proportion of 3’s taken when they have a big deficit and little time remaining.  But they respond incorrectly to situations where they are favored.  Where they should play conservatively (making safe, low-variance plays), they actually play less conservatively: taking more 3’s, for example.
  
  In situations where the game is very close, NBA teams seem to become highly risk averse.  For example, in such situations teams tend to take a suboptimally small number of 3-pointers.

Of course, there is virtually no end to strategic considerations in the game of basketball.  I’m sure that future research will add significantly to this list of principles, and give them much more predictive power.  But for now there are some surprisingly general, and surprisingly useful, things that we do know.  

And, judging by the behavior of current NBA players, they are perhaps not as widely known as they should be.