Braess’s Paradox and “The Ewing Theory”

I spend a silly amount of time thinking about basketball. Usually this works directly to my detriment, since it means that I easily get distracted when I’m supposed to be working. And when you’re trying to make living “theorizing”, focus can be at a premium. But after Tuesday’s post on the price of anarchy, I can’t help thinking about what these ideas could do if they were applied to basketball. If you haven’t read the “price of anarchy” post yet, please do, or the things I say here might not make a lot of sense.

Like the study of traffic patterns, basketball is a network problem. Each possession can be thought of as a journey from point A to point B, where point A is the beginning of the possession and point B is a shot attempt. The team can go from A to B by any number of routes. The “path” they take is the sequence of plays and passes leading up to the shot. As a simple example, here’s what a pick and roll between the point guard and the power forward might look like: One possible outcome of the play is that the point guard keeps the ball the whole time and takes a shot: this would be the top routes. You could also have the point guard pass to the power forward after the screen and have the power forward take the shot: this would be the bottom routes. Or you could have the point guard pass to the power forward after the screen, who then hits the cutting small forward for the shot. Each of these plays (there are six possibilities) represents a different path, and each path will have a different efficiency.

If you’re a basketball coach or player, the question you want to know is: which plays should I run, and how often? In network language, the question would be: what is the optimal usage rate for each path? The question can be an extremely tricky one, as we saw with traffic patterns. It may be that shutting down a link or a node (elminating a possible pass or player from the play) improves the overall efficiency. In the study of traffic, this was Braess’s Paradox: closing a road can improve traffic. In basketball, it gets called “The Ewing Theory”, and its implication is this: eliminating a scoring option can improve the efficiency of your offense.

If you’ve never heard of The Ewing Theory, please check out this article by the always-hilarious Bill Simmons of ESPN.com. Simmons coined the term to describe the observation that the New York Knicks always seemed to play better when their best player, Patrick Ewing, was out. (Just to be pedantic, the “Ewing Theory” isn’t really a theory; it’s an observation. It should probably be called the “Ewing Paradox”).

My goal for this post is to give a plausibility argument for the Ewing Theory. I want to show that it is truly possible for a team to improve when their best player and primary offensive option is removed. What’s more, it can happen without any psychological effects (like the team being extra motivated to play hard without their star player). The answer has everything to do with the price of anarchy, and the fact that making the highest-percentage play every time down the court is not the same as playing your best possible game.

The team with a dominant big man

Basketball may be a network problem, but it is an almost hopelessly complicated one. There are so many possible variations of so many possible plays, that it would be silly to try and draw them all out in one diagram. So I’m going to simplify the network very drastically, and assume there are only two points: the beginning of the possession and the shot attempt. This is equivalent to assuming that the only thing that matters for efficiency of a play is who takes the shot. It’s a big assumption, but it allows us to make definite predictions. I’m going to assume that the team has one star player, whom I’ll call Patrick, and four “average” players. The diagram for their offense will look like this:

Patrick may be far-and-away the best offensive option on this team, but that doesn’t mean that he should shoot every time. That’s because, as a scorer, he will become less efficient the more he is used. The more shots Patrick takes, the more the defenses will focus on him and make it hard for him to score. As a result, Patrick’s teammates should take some significant fraction of the team’s shots, even though they are not as good as he is. We can make a guess as to how Patrick’s offensive efficiency might decline with the number of shots he takes:

If Patrick is used only very rarely, then the defense will not adjust to him and scoring will come easy. As a result, he’ll shoot nearly 80% from the field. In contrast, if Patrick takes every shot his shooting percentage will drop to nearly 20%.

It’s not terribly easy to find justification for this assumption: a good player never takes only 2% of his team’s shots, and no one in their right mind would ever try to take every single shot for his team (*insert Allen Iverson joke here*). But there is definitely evidence for a negative correlation between a player’s field goal percentage and the number of shots he takes. Consider this data for “dominant big men” Shaquille O’Neal and Kevin Garnett:

Each data point on this graph represents a different season in their respective careers (see Data Footnotes at the bottom). The message is fairly clear: when your big man is used as the primary offensive option, his efficiency suffers. When he is used more rarely, to clean up misses and take advantage of low-post mismatches, he is significantly more effective.

Just to show that this effect isn’t limited to low-post players, here is a similar data set for guards Ray Allen and Kobe Bryant:

Clearly, Ray Allen the occasional sharpshooter is more effective than Ray Allen the go-to offensive creator. And the story is similar for Kobe Bryant.

How to best use your superstar

Back to our hypothetical Patrick. Suppose that Patrick’s “average” teammates all shoot 45%. (Since none of them will ever be called upon for more than a fifth of the team’s offense, we don’t need to worry about how their efficiency declines with usage.) Patrick is clearly a better shooter than his teammates, so how many of his team’s shots should he take?

The answer depends on your level of strategy. You might think that the best option is to feed the ball to Patrick until his field goal percentage drops to the level of his teammates. Only then should his teammates shoot the ball. And indeed, it’s hard to argue with this logic: if Patrick is shooting 60% and his teammates are only shooting 45%, then clearly Patrick should be taking more shots, right? In game theory, this strategy is called the “Nash Equilibrium”; it is the result of considering each play individually, and looking for the best possible outcome. It is also completely equivalent to the “selfish” optimum in the traffic problem, where each driver takes the path that is in their own best interest. As a result of the strategy, Patrick takes about 44% of the team’s shots (more than three times more than any of his teammates) and the team shoots 45% from the field.

But something seems wrong, doesn’t it? You have a superstar player who is capable of shooting as well as 80% from the field, but still the team is only making 45% of its attempts? There must be a better strategy. And, of course, there is. Here is a plot of the team’s overall shooting percentage as a function of the number of shots that Patrick takes:

As you can see, the team is most efficient when Patrick takes only about 21% of the team’s shots, just slightly more than everyone else. It seems ridiculous at first: in such a game Patrick would be shooting 60% while his teammates shot only 45%; surely he should be getting more shots. But the added benefit of keeping Patrick more poorly-defended pays off, and his team’s shooting percentage improves to about 48.5%.

This is the price of anarchy in basketball. A team that looks for the best play each time down the floor will shoot only 45%, whereas a team aware of its “global optimum” can do as well as 48.5%. They just have to purposefully refrain from going to their superstar, even when he is the best option.

I think about this sometimes when I watch the Magic use Dwight Howard sparingly, or when the Lakers use Pau Gasol only as a third option, even though he’s clearly their most efficient scorer. Maybe that’s not bad teamwork; maybe it’s good strategy.

What happens when Patrick goes out?

So what happens when Patrick injures his Achilles tendon and must be replaced by an “average” teammate? Certainly, the global optimum decreases. The team no longer has Patrick’s 60% shooting to bolster them, and must rely on a team of five 45% shooters. But if the team was going by the “one play at a time” strategy (the Nash Equilibrium) and not the best possible strategy (the global optimum), then losing Patrick doesn’t hurt them at all. Having five equally “average” players forces them to share the ball and prevents the defense from focusing in on any on player. As a result everyone shoots 45%. They shot 45% when Patrick was on the court anyway, so the team loses nothing by having him sit out.

The arguments of this post have been a little tricky, but the final conclusion is fairly straightforward. When the team’s best player is not on the court, the team is forced to make harder, lower-percentage plays and to share the ball. And when a team shares better, it plays better: the defense is kept off balance and has no room to focus in on any one guy. These sorts of improvements can make up for the loss of direct offensive production.

The point of this post was not to make a quantitative prediction. There were a lot of unjustified assumptions here, and you shouldn’t take any of them too literally. The point was to show that the “network” of an offense can have surprising and unpredictable properties. For a team to play its best possible game, it might need to intentionally take lower-percentage shots. When a team loses a high-percentage option, its overall offense might adjust in surprising ways that can make the team better. Flow becomes re-routed in the offensive network, and the result might be to push the team closer to its true global optimum. The Ewing Theory, in my mind, is completely plausible.

Data Footnotes

The data above is taken for full seasons from the careers of the respective NBA players. I omitted the first two seasons of each player’s career (while they were still improving significantly), and any season where the player missed more than a third of the games due to injury.

Kobe Bryant had two seasons of anomalously low field goal percentage during the rocky seasons of 03-04 and 04-05 (where he missed 33 games because of the Colorado rape trial and a severe ankle sprain). These are shown on the graph but are not included in the fit line.

UPDATE: To those readers who objected to my use of FG% as a measure of “scoring effectiveness” rather than something like TS%, maybe this will be slightly more convincing:

http://i1012.photobucket.com/albums/af249/gravityandlevity/skill_curves-1.png

The open circles/squares represent the first three years of the players’ careers.

UPDATE #2: A more detailed example of Braess’s Paradox in basketball.