Run/pass, continued

(note: axes labels in second plot are screwed up - it's actually "run fraction" and "pass D fraction")

The previous post gave a simple model for play sequencing that results in unequal run/pass payoffs, but that's not the only way for things to be complicated in play-calling. In the previous example, you've still got a stable Nash equilibrium, located at the minimax point of the two-play sequence. Given the variation we see in NFL playcalling, it seems unlikely that a stable Nash equilibrium exists at all - coaches almost seem to choose the fraction of times that they run based on personal preference, rather than some optimal mix.

Competition dynamics can provide more complicated behavior than a simple stable Nash equilibrium, however, but essentially require nonlinear behavior in the "utility function" - that is, the function that gives the payoff for a choice as a function of the opposing player's choice. One possibility is a fixed point which is unstable - that is, one of the players has incentive to move away - but which admits oscillations about that point. You see behavior like this with predator-prey dynamics.

A simple, non-linear model

Consider the following game:

1. Each team chooses a number of plays, either run or pass.


2. The plays are then presented, one against each other, randomly.


3. The payoff for a run vs. run is -1-(number of previous run vs. run plays)/(number of previous plays)


4. The payoff for a run vs. pass is 1-(number of previous run vs. pass plays)/(number of previous plays)


5. The payoff for a pass vs. pass is -0.5-(number of previous pass vs. pass plays)/(number of previous plays)


6. The payoff for a pass vs. run is 2.5-2.5*(number of previous pass vs. run plays)/(number of previous plays

This may seem very bizarre, but basically the idea is that as the various personnel see a certain type of play more, they react quicker to it rather than the other option. One note about this model is that things jitter around a lot early on (as the fractions jump around a lot) but then settle as the game goes on and approach an equilibrium behavior. I'll only talk about the equilibrium behavior.

Assume that the fraction of runs the offense attempts is x, and the fraction of pass Ds the defense attempts is y (yes, continuing my ridiculously silly column order mixup). As the number of plays gets large, the utility function for run and pass approach:

p(Run) = (-2x)y^2 + (2x+2)y - (1+x)
p(Pass) = (3.5x-3.5)y^2 + (2-5x)y + 2.5x

As you can see, the utility function is clearly non-linear, and moreover, depends on the offense's choice. Passing or running 100% of the time is a poor choice regardless of what the defense does : for a pass, the payoff is zero if the opponent plays run all the time, and -1.5 if the opponent plays pass all the time.

This model isn't quite as silly as the previous one - in my mind, it's just as viable a model for football as the simplistic zero-sum, constant payoff game. Even if the defense is in a pass formation or in pass personnel, if an opponent constantly runs similar plays at them, they'll get better at recognizing it and will perform better against it. Not as good as the proper formation would, but better.

However, the "optimal strategies" for this kind of a game are extremely different. One way to see this is to look at the average payoff per play, as a function of pass fraction and run D fraction. Here's the plot for the simple zero-sum game, with coefficients ((0.5,-1.5),(-0.5,1.5)).



I've compressed the scale and shifted it a bit to make it easier to see the main structure, which is that cross-shaped structure around (0.5,0.75). That's the minimax point, and the Nash equilibrium - note that at a run D fraction of 0.75, the defense has no incentive to change its playcalling, since the result is always the same. Similar for the offense at 0.5. And, as we expect, at these points, the payoffs for the two options are the same. Thus, the equilibrium is stable - if either player changes his strategy at this point, that player's payoff will decline or stay the same, and the other player will have a better strategy available as well.

But what about for our new "learning defense" game?



Wow, that looks completely different. There is something "kinda like" the equilibrium structure, at about (0.75, 0.75) here, but it's tilted almost 45 degrees. That is, past this point, "more passing/more run D" is strictly better for the offense, "more passing/less run D" is strictly better for the defense, and so forth. Note that this point is already weird - it's saying "run 75% of the time, but play pass defense 75% of the time."

This is not located at the minimax point for both players - the offense minimizes the defense's maximum payout at about ~60% running, whereas the defense minimizes the offense's maximum payout at about ~70% pass D.

That point is also not stable! At (0.75,0.75) the defense thinks it can do better by playing either run D more or pass D more. The offense thinks it's perfect. But when the defense plays, say, more run D, the offense can then improve by playing more pass. What you end up with are orbits around that point - in fact, if you model each team's behavior as "if you can do better, you try it" you get what's called a limit cycle, where defenses and offense continually chase each other's tail.

And that's the key to having unequal payoffs between runs/passes here: the "learning D" game model results in a situation where defenses always can do better, but offenses can always counter, and the method by which the two do better results in them oscillating over time between "run heavy/pass heavy." However, in this situation, over time, sometimes runs would be better, sometimes passes would be better - in a long term average, they would be somewhat close to equal. So the problem here may be that coaches aren't stupid, and over any small timeframe, passes and runs wouldn't be equal, but long term, things would somewhat balance out.

But! There's another equilibrium here, and it's stable: at roughly 90% passing, 10% running, with defense playing 100% run D (at 0.1, 0). The offense thinks it's doing the best it can, as does the defense for small changes - it can't see that if it played 40% run D (a huge change) it'd get the same results, and even less run D would give it even better results (pushing back to the limit cycle).

At this point, the run/pass payoffs aren't equal at all! Runs produce -1.1, and passes produce 0.25. Think about how insane this seems: the offense is passing 90% of the time, but you're playing 100% run defense - because it's 'good enough,' and playing the pass 'a little bit' makes things worse. From the offense's point of view, runs are god-awful, but mixing them in even a little boosts the output of your passing game a lot (from zero to 0.25 in this case).

Sounds like a plausible description of the current situation.

So now we have two ways that we can have unequal run/pass payoffs without stupid coaches:

1. Offenses may be optimizing multi-play sequences, rather than one play at a time.


2. Defenses may play better when exposed to the same situations.

In the second situation, there may be a better option available (the league could be 'stuck') but the coaches aren't being stupid, because the 'better option' doesn't really appear 'better' unless you drastically change; small changes just make things worse.

In this case, the game theorists could be both right and wrong; it may be that the current situation isn't ideal, but the league is trapped in a local optimum away from a global optimum.

Game theory and run/pass imbalance

There've been a few papers (here) and blogs (here) that have been investigating team playcalling - namely, the "run/pass" decision. The articles usually revolve around the fact that the run/pass payoffs are not equal. At face value, it seems like each play in football should be a zero-sum game - whatever the offense gains, the defense loses. And if each play is a zero-sum game, then, if the play calling is ideal, any grouping of plays you come up with should, on average, produce the same amount (assuming they were against an equal mix of defenses).

We observe that this isn't true: namely, passes produce significantly more than runs. Therefore, the conclusion they reach is that play calling is not ideal. This conclusion, however, isn't the only thing that could explain the observation.

Zero-sum games in general

The idea of a "zero-sum game" is that you've got two players, each of whom can make a choice - say, an offense choosing "run" or "pass," and a defense choosing "run D" or "pass D." The payoff for those choices is then summed up in a matrix:

	Pass D	Run D
Run	0.5	-1.5
Pass	-0.5	1.5

These numbers are just arbitrary, of course. Von Neumann proved a theorem that says that the optimum strategy is for each player to randomly choose one of the choices, with a probability such that the combination minimizes the maximum payoff that your opponent can get (and thus maximizes your minimum payoff). At this optimum strategy, the payoffs of the two options are equal.

It's very hard to get around this final conclusion, as it's intuitively very simple: if runs produce so much less than passes, why not pass more? Then, the defense will play run D less, runs will produce more, and passes will produce less, and we'll all hit a nice equilibrium.

The thing is, we don't actually know what the payoff matrix is for football, of course. We just know what we see, and when we group passes together, we see one average payoff. When we group runs together, we see another average payoff.

Now consider this option: an offense chooses two choices, and presents them in whatever order they wish. The defense chooses two choices, but must randomly present them. If the offense chooses "run, run" or "pass, pass" the payoffs are the same as before. If they choose "run, pass" the payoffs for the second pass are drastically different (1, -1) - yes, good against a pass defense, and bad against a run defense (bear with me if this makes no sense). If they choose "pass, run" the payoffs for the second run are (0, -2).

The math for solving this is actually exactly the same as before, because both the choices on defense are independent of each other. The payoffs for each choice now depends on how often the defense plays 'run' or 'pass' so we can't write it in matrix form easily. We'll call the fraction of time that a defense plays 'pass' "pD%"

1. Run, Run: 2*(0.5*pD% - 1.5*(1-pD%))
2. Run,Pass: (0.5*pD%-1.5*(1-pD%))+(1*pD%-(1-pD%))
3. Pass,Pass: 2*(-0.5*pD% - 1.5*(1-pD%))
4. Pass,Run: (-0.5*pD%+1.5*(1-pD%))+(0*pD%-2*(1-pD%))

which simplifies to

1. Run,Run: 4*pD% - 3
2. Run,Pass: 4*pD% - 2.5
3. Pass,Pass: -4*pD%+3
4. Pass,Run: -pD%-0.5

This actually looks exactly the same as a new zero-sum game. Two of the choices are terrible - "run, run" is clearly worse than "run,pass" and "pass,run" is clearly worse than "pass,pass". We can reframe the values into a payoff matrix again (that's just a bit of rewriting) and get:

	Pass D	Run D
RP	1.5	-2.5
PP	-1	3

The solution, in this case, is for the defense to play pass D 69% of the time, and run D 31% of the time, and for the offense to choose "pass, pass" 50% of the time and "run, pass" 50% of the time.

But what's the payoff for runs alone? They're actually completely negative: -0.125, whereas passes are significantly more: 0.292. Note that both two play sequences do have equal payoff. The problem comes in not knowing that the "run" was required to set the defense up so that the second "pass" would work.

Edit: It's important to note that the entire reason this works is because the offense can sequence its playcalling, whereas the defense cannot. This may not make much sense, since as shown in the above table, a run is always followed by a pass - so it'd be pretty stupid to play the worse defense against a play that you know is coming.

So is this just an example of a stupid model? No, not really - because the two play sequence here does not have to be in order. The first play could be the first in the game, and the second could be 35 plays in. In that case, the defensive playcalling must be random because it has no idea when that (1,-1) pass will show up. For simplicity, we treat it here as a two-play sequence by the offense and random behavior by the defense.

A purist will note that I've essentially introduced two new "classes" of plays: the (1,-1) pass and the (0,-2) run. Why can't those be run on their own? That's because we're introducing the concept that the offense can manipulate the defense. (As an aside, it's easy to show that a R,P combination which is 'classical' run and 'classical' pass here is less preferable - it produces no net benefit versus any mixed strategy). The (1,-1) pass simply doesn't exist on its own.

Yeah, but does that game make any sense to model football?


That specific game? No, almost certainly not. But the question is "does it make sense to believe that plays can change the way a defense reacts to future plays?" and the answer to that is unquestionably yes. From here:

In fact, Schlereth said many were added to see how a defense would react to something the Broncos did, for example, putting four wide receivers on the field. Once the defense showed how it was going to handle such a situation, Shanahan would add plays accordingly.

Offenses have a strong advantage over defenses: they choose the plays. A defense simply cannot get an offense to attempt a deep pass to a WR to find out if they need to slant a safety to his side. An offense, however, can run a play to see how a defense reacts - and the defense can't choose not to react. A corner can't simply ignore the WR running a fake go route that the offense inserted specifically to see how the WR gets off the line.

Moreover, the actions of a defensive player aren't entirely planned - they react to the offensive player, who already knows what he's going to do, for the most part. That reaction time is human, of course, and manipulating that reaction by play sequencing is completely believable. After all, I've watched Bugs Bunny.

But the basic point here is that the offense has an advantage in that it can choose what information a play will reveal. If a wide receiver is tired, they simply call a play to someone else. A defense has no fool-proof way to avoid a player being exposed without pulling him off the field.

Could this explain the excess runs, given their poor success? Probably. Runs are low-risk, and if you're running a play primarily to gain information or manipulate a defense, risk-minimization is probably what you want to do.

Same idea, different reasons

There's another possibility, too: what if running a play on offense changes the way the will perform? I don't need to cook up another game for this, as it's almost exactly the same. What if running a passing play reduces the effectiveness of the next passing play?

This possibility is more than plausible: a passing play, specifically, likely resulted in the wide receivers running 20-30 yards, and then quickly getting back to the line. Running another passing play would likely be much less effective - possibly completely ineffective - as the receivers could easily be exhausted. The offensive coordinator, knowing this, wouldn't be so silly as to run an additional pass.

Final thoughts

It's funny to read this conclusion from the Kovash & Levitt paper.

Kovash and Levitt then look at the order of plays, and again find patterns that minimax theory would not predict. Conditional on other factors, a team that has passed is 10 percentage points less likely to pass on the next play.

The fact that patterned play calling exists supports the idea that play sequence is important, and that plays are not sufficiently independent on their own.

One final thought: I'm not saying that it's not true that coaches should pass more. What I am saying is that I think the information imbalance between offense and defense makes any idea of a minimax solution to playcalling pointless. The comment in the Shanahan article stresses that.

Welcome, blah, blah

I needed somewhere to write up random observations and opinions about football and the growing field of football statistics, so here we are. I'm an Eagles fan, hence the "Eagles By The Numbers," but most of the stuff I post will be non-Eagles specific. It'll probably be also dry and boring. Whee.

I should also note that I don't have a ton of time to explore in detail certain problems, but part of the reason I'm creating this blog is because there are a lot of people whose opinions on football are built on seriously flawed notions of the game. The idea that running keeps your defense off the field. The idea that a team that doesn't score a lot of points must have a bad offense.

A lot of those are cliches, but even some of the nouveau footballthink ideas are built on flawed assumptions. The idea that coaches should be passing a ton more because the average payoff for a run is so much lower. The idea that coaches should go for it a lot more on 4th down.

It doesn't take a lot to discredit those ideas, since most of them are built on theoretical assumptions and you can quickly show that they're flawed. So that's what we're here for. Which means that this site will contain a lot of me criticizing other people. Which I'm sure will make me look whiny, but please keep in mind the point of the blog.

Other sites you should read:

Football Outsiders
Advanced NFL Stats
Pro-Football-Reference (specifically the blog)