Single-Point-Failure Model of the Passing Game

Baseball has long been considered the easiest of professional sports to model and analyze mathematically. It’s certainly far simpler than football. One reason baseball is easier to model is that the sport isn’t really a team sport, at least in the most mechanical sense. It’s an orderly series of one-on-one match-ups between pitchers and hitters. Fielding and base-running certainly matter at the margins, but it’s the pitcher-batter interaction that dominates most outcomes.

In contrast, every football play seems like a desperate, chaotic scramble of 22 players. Where baseball is a series system, football is more of a parallel one. In a very simplified way, much of a football play can be modeled as several simultaneous one-on-one match-ups. Take a simple pass play. Each pass blocker matches-up with a pass rusher, a back picks up a blitz or dog, and each receiver matches-up with a pass defender. (At least this would be the case with a man-on man pass defense. Zone defenses can be thought of in a very similar way as I’ll describe below.)

This kind of system is similar to a chain. If any one link fails, the entire system fails. No matter how well the other offensive lineman are blocking, if one lineman misses his block there’s probably going to be a sack. And if one pass defender blows his assignment, either by being beat in man-to-man or being in the wrong place in a zone, there’s a good chance for a big pass completion. This is why a football play can be thought of as a “point-failure” system.

Just like each player has a batting average, each offensive lineman could have a core probability of allowing a pass rusher to beat him and either pressure or sack the quarterback. Likewise, each pass rusher has a core probability of beating a blocker and getting to the QB. These baseline probabilities could be very low, but because it only takes 1 of the 5 pass rushers to be successful on any given play, the resulting chance of a hurry or sack grows considerably.

This is why having a world-class, Hall-of-Fame worthy tackle might not mean that much for a team’s overall pass protection, especially if there are weak blockers elsewhere on the same line. The math works out so that it’s better to have a line full of average blockers rather than a line of one all-pro and four slightly below-average colleagues.

For simplicity’s sake, say each pass rusher has a 5% chance of beating his blocker (within the likely time period before the throw). With 5 pass rushers on a pass play, the chance of any one of them getting to the QB would be 1 – (1-0.05)5), which is 0.23. So in this very simple model, the chance of any 1 of the 5 pass rushers hurrying, hitting, or sacking the QB would be 23%.

The receiver-defender match-ups would work similarly. Say there is a 5% chance a pass defender will either be beaten man-on-man or blow his zone assignment. It only takes one blown assignment for a failure to occur. No matter how well the other members of the secondary are doing, a single failure can lead to a big pass. With four defensive backs in coverage, this would put the overall chance of a wide open receiver at 1-(1-0.05)4) = 0.19.

So, in a very simple way, a passing play is like two chains under strain. One chain is the pass protection, and the other is the pass defense. Each link is a player vs. player match-up, and it has its own probability of breaking based on the abilities of the respective players. The first chain to break loses.

Can you imagine a football team with a starting player who is a point-failure in nearly every play? He'd be a lineman who always gets beat by a pass-rusher or a defensive back who always gets beat by a receiver. It would be ugly. Can you imagine any sport where this could be the case every game? Consider the National League, where pitchers are nearly always an easy out. In baseball, the failure of a pitcher at the plate is confined to his at bat.

So far, I’ve left out the most important player. The quarterback has to see open receivers and throw accurately to make big plays. He has maneuver in the pocket, and scramble from pass rushers. The QB is a big wildcard in my chain analogy.

I imagine this is how football video games like Madden are modeled, at least at the core. The game designers need to know what probabilities of allowing a pass rusher to beat a blocker should be to yield a realistic sack rate. Just looking at sacks alone, we can estimate a ballpark individual “sack allowed” rate is for individual linemen. Overall, the NFL sack rate is about 6.5%, so to solve for the baseline individual rate we can say:

6.5% = 1-(1-x)5
-whole bunch of algebra-
x= 1.1%

Remember that’s an extremely rough figure because there are lots of other factors to consider, such as overload blitzes that linemen can’t handle or don’t control, or quick out passes that allow almost no chance of a sack. Plus we’re only counting sacks, not hits or hurries. So I’m only demonstrating a process, not declaring an answer, or even claiming there is a worthwhile answer. With such a low baseline rate and the NFL’s small sample sizes, it would be difficult in the extreme to grade a lineman purely statistically.

I'm only offering this analysis as a way of thinking about the sport. The only conclusion I’ll draw is a simple one. Ask yourself which is stronger, a chain with 10 links, or a chain of 20 links? It’s the shorter chain. If each link has a certain chance of breaking, you’d want the one with the fewest links.

Offensive passing systems that are heavy on multiple-receiver sets have a mathematical advantage. The more pass defense match-ups and the fewer the pass-rush match-ups an offense can create, the better. An offense would generally want the pass-rush match-ups to be the like the chain with fewer links, and the pass-defense match-ups to be the chain with more links. This way, there is a greater chance of a single point failure in the secondary and a lesser chance of one in the offensive line.

Again, I'm not proposing any sort of statistic to grade individual players. I'm just stepping back and examining why football is sometimes called the ultimate team sport.

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11 Responses to “Single-Point-Failure Model of the Passing Game”

  1. Anonymous says:

    This is a good article, and it's funny you posted this since just today I posted an article about the difficulties with modeling football (and the resulting easy-ways-out certain statisticians seem to take).

    In any event, though I agree that a great deal of the game could be modeled in this way, as a series of single-point models, some of that gets more complicated for plays like options/reads and certain pass against zones. Good offenses look for 2 on 1s and 3 on 2s. It's not always a blown assignment that leads to a completion: the correct pass combination (or receiver read to get into it) should put a defender in an untenable bind -- do I cover the guy in the flat or the curl?

    And option-run game football is designed to avoid single-point matchups, since you look to get (a) double team blocks at the point of attack, which then frees up your runner and QB to (b) get a two on one read to option off a defender.

    Of course, your point is still valid and interesting, and in many ways it just reinforces what a team sport football is. Keep up the great work.

  2. Brian Burke says:

    Chris-Thanks. I saw your post earlier and it motivated me to dust this off. I wrote this a few weeks ago, and thought it wasn't terribly interesting once I got it down on paper.

    But you're right. I imagine 90% of football tactics is breaking free of the 1-on-1 match-up somehow and getting an advantage--saturating a zone, an overloaded pass rush, or a double-team on a run block.

    Incidentally, here is a case in point from a Fangraphs post about Barry Bonds. His '01/'02 seasons were so good, you could literally put a papaya in the lineup to hit behind him, and you'd come out with a replacement-level duo. In other words, you could bat me behind Bonds, and the Giants would have won 70+ games each year.

    But if you put me (or a papaya) in at right guard for New England last year, they'd go 0-16 instead of 16-0.

  3. Anonymous says:

    you guys are way smarter than me but I enjoy the work that is done on this site for more than just the unbelievable edge it provides me in my office pool :) Keep up the good work!

  4. Anonymous says:

    that post sat in the drawer?

    I thought that was one of the best post of the year

    I think if you knew what plays were being called you could come up with very accurate effectiveness rating for players, obviously we don't have that info though

  5. Anonymous says:

    Can you imagine a football team with a starting player who is a point-failure in nearly every play? He'd be a lineman who always gets beat by a pass-rusher or a defensive back who always gets beat by a receiver. It would be ugly...

    It sure as heck was ugly -- last year at LG for the Jets as they plugged various nobodies into the void after they got rid of Kendall for having a bad attitude, without bothering to first get a replacement for him.

    This is not a thought experiment any more.

    Look at Thomas Jones's stats last year and this.

  6. Anonymous says:

    I would also think that the large number of players involved in determining the success/failure of each play would be a major contributor to the perception of "parity" in the NFL.

  7. Anonymous says:

    It gets even more complicated regarding max protection schemes. Remember that when an eligible receiver blocks in pass pro, he remains a threat to catch a pass because he could block and release. Defenses still have to account for him. [In some ways, this is a tactical counter to the zone blitz by the defense.]

    The chip and the delayed release provide blocking help while still requiring defensive coverage.


  8. Anonymous says:

    you've just opened up a serious rabbit hole. God bless anyone who tries to model this properly (including all the unknowns others have mentioned). I'd speculate that although increasing the number of WRs adds additional failure opportunities for the defensive secondary it may also have a negative impact on the success rate for the WRs (the field is more cluttered, WRs have less room to maneuver / QB has more information to process before making a decision). It is highly unlikely that these effects would cancel out, and I have no idea which would be weighted more heavily.
    Play calling is also huge here, a play with a go-to target decreases the pr a QB will recognize a 2nd, 3rd,...,nth receiver even when they've won their X% of point matchups.

    Totally meaningless aside: in madden i rarely run more than 2 WR sets because I tend to throw more INTs when there are more DBs on the field...

  9. Anonymous says:

    Your premise would lead coaches to think that a strong running game should lead to more wins than a strong passing game and yet in the NFL it is the opposite. The reason is that while having many more links in the chain leads to a higher rate of failure, it is also harder to defend. If you know who has the ball and there can only be one, for the most part the defense can focus all of its attention on that one player. The passing game mitigates this advantage by shrouding who will get the ball and when at the cost of a higher risk of failure. Now the question becomes whether the risk of failure outweighs the chance of success. The answer is of course, "It depends".

    Further, unless you are playing single wing, a significant portion of your players are not involved in a running play. For example, the receivers on the opposite side of the ball, the QB after they hand off (if they hand-off), the opposite side tackle and guard (unless they pull) etc. In a pass play, far more players are involved in the play.

  10. Anonymous says:

    The tail end of your article reminds me of usually successful running offenses failing more than one would expect when they go to jumbo packages. In essence, the offense is creating a larger chain that needs to hold and prevent defenders from getting to the ball carrier before the vital one or two yards are obtained. Recently, we've seen offenses adjust by putting stronger links in the backfield as lead blockers, such as Dockett and Branch for Arizona against Green Bay and Cody for Alabama.

  11. Sean says:

    This corroborates Parcells' team building approach. Whenever Parcells inherited another terrible team, he would immediately sign a bunch of veteran players who were known as "hold the fort" players. These signings weren't popular with fans, as it often seemed like Parcells was overpaying mediocre players just because he was comfortable with them, but the idea was to remove matchup weaknesses as quickly as possible. It allowed each of his teams to be competitive almost immediately.

    It should be noted, however, that while the basic theory here is perfectly sound, it doesn't follow that you are always better off with average players across the board, as you need to match up on a position-by-position basis. A defense is most likely to put its biggest matchup problem at the blind side pass rushing position, which means that you are more likely to need an above-average player there to cancel out the advantage.

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