I would like to recommend Revisionist History, a new podcast from Malcolm Gladwell. His third podcast, The Big Man Can’t Shoot, nominally covered Wilt Chamberlain and his struggles at the free throw line. But, as is often the case with Gladwell’s work, it’s about so much more than that.

Of particular interest to me was the academic paper Gladwell cited, which formed the meat of the podcast. It was written by Mark Granovetter, back in 1978, and is titled Threshold Models of Collective Behavior. Here’s a link to the paper, which I recommend reading if you have the time.   But a couple of Granovetter’s examples resonated with me as being particularly relevant to us, and I would like to reproduce them here using a football analogy.

[Note: You may wonder why am I copying his work here? I find the application of this idea of threshold models of collective behavior to be worthwhile for our broader discussion, and I think the best way to encourage discussion of it here is to reproduce it in our world, rather than just telling you to go read a link.  Full credit, of course, belongs to the author.]

Many analysts, myself included, think that NFL teams are way too conservative when it comes to going for it on 4th down. In general, coaches do not call plays in an optimal way, and we have long understood that part of the problem is no coach wants to take the heat for failing unconventionally. So we just assume that “the NFL” is overly conservative on this point.

Now, let’s make some assumptions. “Being aggressive” is not a binary thing — there are hundreds of aggressive/conservative options/decisions that come up in a season — but to simplify things, let’s assume that coaches can either be aggressive or conservative. And, let’s assume that right now, all 32 head coaches are conservative.

However, let’s assume that all 32 coaches think being aggressive is better than being conservative, but they also have resistance to switching from being conservative to being aggressive. And these resistances are not uniformly held: these 32 coaches each have different thresholds on when to make that switch. Let’s say the Patriots would be willing to be aggressive as long as just one other team was aggressive first. This would mean New England is very eager to be aggressive, but just doesn’t want the spotlight solely on them. And let’s say the 49ers would be aggressive as long as two other teams became aggressive first. And the Ravens would be aggressive as long as three other teams were aggressive first.

In model terms, we would say that the Patriots have a threshold of 1: they need to see just one other coach/team be aggressive. The 49ers have a threshold of 2, the Ravens have a threshold of 3, … and, finally, Tennessee has a threshold of 31 and Washington has a threshold of 32. This gives us a uniform distribution of thresholds.

If we were to chart each team’s threshold on the choice of switching from being conservative to being aggressive — i.e., how many other teams would need to be aggressive before they chose to be aggressive – it would look like this:

Now, how aggressive would we say this version of the NFL is? That depends on your perspective. In some ways, it’s pretty average: there is one team at each threshold, so it’s not as though the majority of teams need to see 75% of the NFL switch before they do. On the other hand, what would we observe in practice? Absolutely nothing. No team would appear aggressive at all, because no team will switch from conservative to aggressive under this model. In fact, if all 32 teams had a threshold of 31 (in what, I would argue, would be a much more conservative NFL), we would observe the exact same thing.

Using this example, you can see that it’s tricky to draw inferences about each team’s behavior based on group action: here, these 32 teams have distinctly different thresholds towards aggression, but in practice, all 32 look the same.

Now, let’s assume that Washington hires a new head coach who happens to be very aggressive. In fact, he’s so aggressive that he has a threshold of 0: he doesn’t need to see any other team become aggressive, he’s ready to make the switch. We can call him the ‘instigator.’ Let’s further assume that nothing else changes; how does each NFL team’s threshold look now?

That looks pretty similar to the last graph, of course, but think about what will happen. The instigator, with threshold 0, becomes aggressive. This activates New England and their threshold 1; the activity of these teams activates San Francisco and their threshold 2, and they collectively activate Baltimore and threshold 3, and so on, until all 32 teams have switched and become aggressive. This is what Granovetter refers to as the ‘bandwagon’ or ‘domino’ effect, or in NFL circles, the “copycat” effect.

Now, all the sudden, the NFL looks very aggressive. All 32 teams are aggressive, and the narrative would be that the NFL is now a very aggressive league. Even though 31 of 32 teams are just as sensitive to being aggressive as they were before, when the narrative was that the NFL was a very conservative league!

But wait, let’s tweak the above example just slightly. Let’s say that when Washington hires its instigator with threshold 0, New England’s head coach retires. They replace him with someone almost exactly as aggressive, a coach who has a threshold of 2. He is just slightly more cautious than his predecessor, so he needs to see one other team go out on a limb before he will.

By all of our usual ways of describing teams, these two hypothetical leagues are essential identical. One wouldn’t think this should change the league as a whole, but take a look at the new model:

The instigator in D.C. makes his switch, but there is no team with a threshold 1. That means the domino is stopped, with just 1 switching team. This is a simple example, of course, but it illustrates Granovetter’s point: “it is hazardous to infer individual dispositions from aggregate outcomes.”

Now, the NFL is back to being a conservative league with one loony coach. Unretire New England’s head coach, and the entire NFL is aggressive. We know, since we made this hypothetical universe, that the two leagues are almost identical in composition; the difference here results from the “process of aggregation.”

Granovetter notes that thresholds models may be of “particular value in understanding situations where the average level of preferences clearly runs strongly in favor of some action, but the action is not taken.” In other words, we know that coaches should be aggressive, but they’re not, and it confuses us. But the threshold model at least offers an alternative explanation other than “coaches are dumb” or “coaches are just concerned about keeping their jobs”, two answers that have never been wholly satisfactory to me.

Ironically, if there were 5 very aggressive coaches in the NFL, we might be able to more clearly determine that the NFL is full of conservative coaches. After all, then we would know that at least 27 coaches need to see at least 6 coaches switch before they would. But, perhaps, the fact that there aren’t really any aggressive coaches could be a sign that the situation isn’t as bleak as you think. Perhaps there are 32 NFL coaches with a threshold of say, 3. In practice, we would see what we see today, but that doesn’t mean the pendulum couldn’t swing very quickly.

Anyway, that’s it for now. I encourage you to read Granovetter’s paper, and to leave your thoughts in the comments.