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Thoughts on Thresholds Models of Collective Behavior

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:

thresholds 1

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.

  • Josh Sanford

    Are any of you familiar with the story of Pulaski Academy (HS), here in Little Rock, Arkansas? They never punt and they almost always kick on-side. Fascinating. Not a lot of copycats yet…. https://www.washingtonpost.com/news/sports/wp/2015/08/13/the-highly-successful-high-school-coach-who-never-punts-has-another-radical-idea/
    Or here: http://fivethirtyeight.com/features/the-high-school-football-coach-who-never-punts/

    • Tom

      I like how at the end of the WP article Kelley says “I don’t want anybody doing this”; he’s that confident it gives him an advantage.

      • gary bajillion

        I think Brandon Marshall read this article last summer.

  • Does anybody have any good examples in the NFL of teams being pushed over their respective thresholds and adopting something that previously just wasn’t done? (In baseball, I think we are seeing this right now with defensive shifts.)

    Perhaps if the Steelers start going for two all the time, as they’re hinting about, this will have a domino effect on the rest of the league.

    • My initial thought was the leagues attitudes towards race. You can see the thresholds effect take place with players, then with QBs, then coaches.

      • Yes, African-American quarterbacks came to mind for me also. It’s crazy to think that Warren Moon — a QB most football fans today remember well — couldn’t even get a gig as an undrafted free agent coming out of college. Perhaps we saw this with short quarterbacks more recently.

        The victory formation might be another example. I’m sure it crossed early football coaches minds that downing the ball immediately was safer than running a play. But to my knowledge, the victory formation didn’t became popular until the late ’70s (after the Miracle at the Meadowlands), because it was considered cowardly or something like that.

        • Kneels also weren’t protected plays until 1985. Any QB who kneeled still ran the risk of getting blown up by a defender before the play was called dead. That eliminated the only remaining disincentive to running a victory formation.

          • Richie

            I just read that arena football requires a play to go for positive yardage for the clock to keep ticking with under a minute left in a half. I think I would favor that rule in the NFL. The kneeldown play is no fun.

            • I am on the opposite side. If you’ve built a lead, I think you deserve to nurse that lead in whatever manner you choose. If you don’t want an opponent taking a knee to run out the clock, score more points or stop them from scoring as many. I do, however, prefer the CFL’s method of documenting kneels as team yards lost rather than negative QB rushing yards. That’s super dumb, in my ultimately unimportant opinion.

              • Adam

                The NFL’s record keeping is still in the stone age compared to other sports, and the kneel = rush attempt nonsense is one of the more egregious examples.

            • Adam

              While I agree that kneeldowns are boring, I am fundamentally against any rules that artificially boost the chance of a comeback. If you can’t attain the lead by the end of the 4th quarter, you don’t deserve to win. The college rule of stopping the clock after every first down makes me cringe, and IMO it cheapens the accomplishment of a game winning drive.

          • Interesting. I didn’t know that about kneels. This explains some old footage I once saw where the QB (Terry Bradshaw in a Super Bowl?) takes the snap and curls up on the ground like a pill bug — he couldn’t just take a knee.

        • Adam

          The league has made progress with black QB’s even over the last five years. Not too long ago, black quarterbacks were automatically considered “running QB’s” and were valued for their athleticism more than their intelligence or work ethic. The Jameis Winston #1 selection is evidence of evolution, as he fits the mold of the prototypical pocket passer, once the exclusive domain of white quarterbacks.

  • Tom

    This is a fascinating take on the whole “coaches are too conservative” cry we’ve been hearing for the past number of years, and I’m inclined to think that this explanation plays a part in what we’re seeing. Although I’ve always agreed with the numbers and analysis (regarding 4th down decisions), like you, I’ve never been totally comfortable with the notion that coaches just don’t know what they’re doing or are afraid for their jobs. There’s definitely good coaches and not-so-good coaches (I don’t think I’d ever call one of these guys “dumb”), and aggressive and not-so-aggressive coaches, but there’s something else at work here and this really sheds some light on it. Pretty cool, I’ll check out Granovetter’s paper.

  • Josh Sanford

    This is only tangentially related, but after this discussion, I can’t help thinking about the Broncos’ play-calling the last SB. It seems like they looked at the odds and thought could win if only they “didn’t try to win” on offense. They called incredibly non-winning plays on offense in order to try to win on defense. That’s a “risky” (or non-traditional) approach that paid off. They were aggressive and non-conforming by being super non-aggressive? Can I say that?

    • Adam

      My impression is that Kubiak wasn’t even trying to convert on third down, much less score touchdowns. I agree that this wasn’t your run-of-the-mill conservative game plan; the Broncos took the very unorthodox approach of letting the defense do the scoring (directly or indirectly), and it’s almost as if they wanted the offense to come off the field as fast as possible. In a convoluted way this was a risky strategy, especially to attempt on the sport’s biggest stage. I wonder if any teams will try this in 2016…

      • Tom

        Yep. 3rd-and-9 on your own 26, 5:42 left in the game and you’re only up by 6? Give it to C.J. Anderson. You could say that by that point the Broncos’ D had demonstrated their mastery over the Panthers’ O, even so, that’s a pretty (calculated) risky move.

    • Tom

      Agreed, I think you can say that. It seems that we were basically relying on their defense, and considering the “offensive nature” of the game today (meaning, more points are scored per game than 30 years ago), it was pretty risky move. It’s easy to say after the fact, “Oh yeah, they had VON MILLER! Of course you just punt the ball away!”

      • Josh Sanford

        Of course, it really, really helped that the Panthers simply refused to alter their game plan: so that Miller, Ware and the others new exactly where to go if they wanted to hit Newton in the face and/or knock the ball out of his hand.

  • John McSweeney

    I wrote a paper using a similar model to explain how much of a crossword puzzle you can solve – we can think of each answer as having a “threshold” consisting of the number of letters (i.e. crossing words) needed to find it. One day you might solve a whole puzzle, and the next day, for a different puzzle of ostensibly equal difficulty, you may get next to nowhere. It doesn’t mean that the puzzle was appreciably harder or that you got dumber, it may just be that the second puzzle didn’t quite have the critical number of low-threshold (or “instigator”) words, even though the general distribution of thresholds was similar from one day to the next.

  • Adam

    This paper is quite fascinating, and I definitely believe there is validity in threshold theory. I’ve had fun pondering what my threshold is for various activities, and speculating as to what other’s thresholds might be. The comment section of FP is an example of thresholds at work – a handful of us have a comment threshold of 0, which leads us to making comments whether or not anyone else has. There are another subset who only comment in reply to others, but will not start the conversation. And still others who only comment when a large number of readers have already participated.

  • Ken Adams

    I don’t think this theory provides an alternative to the “coaches want to keep their jobs”, I think it reinforces it. If coaches think being aggressive is better than conservative, yet don’t do it because others don’t, what other mechanisms could be causing that behavior?