One thing that I think is important when tackling a complicated problem is to avoid rushing into it. Instead, I like to break it down and solve the simplest part first and then move on to more complicated parts.

The question today is, “What play should an offensive/defensive coordinator call?” Of course, NFL play books run 800+ plays. I’m not really prepared to consider the effects of 800 plays versus 800 (over a half a million combinations, each of which would have to be carefully considered). Instead, let’s suppose that each the offense and the defense have two plays. A rush style play and a pass style play. The defense wants them to be the same – that is, they do best when they select a rush style defense against an offensive rush play (and the same for pass plays). On the other hand, the offense wants them to be different – they want to catch the defense off guard.

Let’s select some reasonable values to start with. Let’s say that if they both select a rush play (R) then the offense gets 1 yard on average, and if they both select a pass play (P) the offense loses 1 yard on average (sometimes a sack, sometimes incomplete, and occasionally a short pass).

The top row is for “offense rushes” and the bottom row for “offense passes” while the left column is for “defense defends against the rush” and the right is for “defense defends against the pass”. Certainly the two diagonal results are what the defense was hoping for. Let’s fill out the table and suppose that the offense can rush for 5 yards on average when the defense is expecting a pass and can, on average, complete a pass for 12 yards (sometimes still incomplete – butter fingers) when the defense expects a rush.

When should each do which and what is the expected outcome? Well, plugging in the numbers and skipping over some yawny game theory, we get, for these numbers, that the offense should rush 76% of the time and the defense should defend against the rush only 35% of the time. On average, each play will yield about 3.6 yards.

I just made some numbers up here, but let’s take a look at the outcome anyways. 3.6 yards per play is just about enough to make a first down in three plays. But why the disparity in strategies? Well, the defense wants to defend strongly against the pass. That 12 yard option is a pretty big kill-joy, so they are going to target that 65% of the time. For the offense, on the other hand, this leaves their running game pretty wide open, so they decide to run a whopping 76% of the time. Now this is more than most offenses run,

so maybe I overestimated the sack attributed and we should shift that -1 value up a bit – but we can tweak numbers and formulas all day (which is what I did).

Let’s consider some more general cases. When will an offense rush more than pass? Well, your first instinct might be that it’s when the top row is larger (in some sense) than the bottom row, but actually it’s when the sum of the first column is greater than the sum of the second column (as is certainly the case for this example – hence the 76% selection of runs). Why does this make sense? In this example, when the defense defends against the rush they give up 1 or 12 yards, as opposed to defending against the pass where they give up 5 yards or a loss of 1 yard. So the defense is probably going to be more interested in defending the pass, which leaves the run wide open for the offense. Basically, the offense takes advantage of the weakest part of the defense.

Similarly, the defense will select to defend against the rush when the sum of the top *row* is greater than the sum of the bottom. That is, when the offense has a large rushing threat, then they are going to defend against it a lot. They don’t really in this example, so the defense defends against the pass more.

Ta-da! Now we’re all NFL playcallers! GMs: I’ll assume I just have a check in the mail.

Okay, this is all great, but these results probably aren’t that surprising. (Hopefully they illustrate a new way of looking at offenses versus defenses – and needless to say, this can be roughly extended to other sports as well, probably left and right handed pitchers and hitters for example.)

They do show one important thing though (again, that we all kind of know already) – and that is that randomness is important. For each the offense and the defense, it is important to select their R,P plays according to the given probabilities. If, say, the defense decided to blitz (a P defense strategy) every play, then the offense would simply run the ball every play and do quite well. Alternatively, if the defense decided to pack the box every play (guard against the run) the offense could just throw to poorly guarded receivers and fly down the field. The way to win is to mix it up *randomly*. Any predictable pattern by one team can be countered by the other. It makes me wonder (and hope more than a little) that playcallers are secretly rolling d20s to figure out what play to call.

Optional extra credit for the interested reader:

There are several details that I glossed over. One of which is that throughout all of this I assumed that the offense always gets more yards when they catch the defense unawares as opposed to the other way around. That is, that both B and C are greater than A and D below.

Last one of these buggers I promise.

The general results, given the figure above are that the offense will rush the ball (C-D)/(B-A+C-D) of the time and the defense will defend against the rush (B-D)/(C-A+B-D) of the time. The total expected outcome when both sides play optimally in this sense is (BC-AD)/(B-A+C-D) yards. Proving these formulas are left as an exercise to the reader.