Thursday, January 7, 2010

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 DRun D

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 DRun D

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.


  1. "Offenses have a strong advantage over defenses: they choose the plays."

    So true and not thought about enough. Even "attacking" defenses like Jim Johnson's or Rex Ryan's are still, at some level, reactive.

    I hope you keep it up over here. Very interesting stuff.

  2. This'll probably be a permanent home for me - sparsely updated, but it will be updated occasionally. I just don't have enough time in my life now to do everything I want to do.