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Understanding balanced, effective play-calling with Game Theory and Nash's Equilibrium

A theory on why a good running game does and does not "open up" the passing game, and why balanced play-calling might be best.

Kyle Terada-USA TODAY Sports

In the past five seasons, Pete Carroll has stressed utilizing a balanced approach with regards to their offensive input, stressing that, ideally, Russell Wilson would've passed as many times as Marshawn Lynch carried the ball at the end of the day. We sort of have a general understanding as to why this is logical, but most of the time everything ends up speaking in circles - especially when you consider that the defense knows exactly what you know too.

During a game, for example, If a team finds out that running the ball is working better than passing, the offense will run the ball more. The defense knows this, so they will focus on defending the run more. Now the offense sees that the defense is focused on run support, so they decide to switch to the passing game. But now the defense reacts to this too, so they start defending the pass, and the offense goes back to the run...and the cycle repeats again until we realize we sound like John Madden.

In reality, the numbers do not follow with Carroll's approach, because selecting passing and running plays entirely depends on the particular scenario your offense is facing. Your strategy is dictated by what the defense gives you, and you adjust.

This adjustment where game theory comes in. Semantically, it's impossible to identify one specific strategy in football that works due to a multitude of factors that you have to consider. But as Brian Burke states, we can utilize mathematical strategies to quantify what is the best outcome for either teams under these conditions.

Let's look what this means through a payoff matrix. Note that, for simplicity, I have only quantified two possible results for the teams - running the ball (and defending it), and passing the ball (and defending it):


As such, if an offense calls a run play and their opponent defends it, then it's a win for the defense because it's most likely stopped for little to negative yardage. On the other hand, if the offense passes while the defense focuses on the run, then only the offense benefits. There is no play in football where both the defense and offense benefit from one particular scenario - if one team wins, then the other does not. It's a zero-sum game.

At the same time, the scenario above poses no clear-cut strategy, or pure strategy, for the offense to succeed. Both running and passing the ball only yields a 50% success rate. Therefore, the best possible outcome for the offense is to utilize a mixed strategy and choose the outcomes at random, or in this case, deciding to pass or run with a flip of the coin. This is why a balanced approach for the offense is best, as the defense only has a 50% chance of guessing and defending the right play.

Now this would all be well and good, but we all know that the game isn't exactly like this. As Danny Kelly points out, depending on philosophy, the weight of a run play can differ from a pass play, so these teams would want to run more than others, even potentially running more than they pass. Similarly, an offense will more than likely have a better running back than a quarterback and favor one aspect over the other. The average expected yards of a passing play is also much larger than a running one.

So how should the payoff matrix change to reflect these realities? Well, we can add numerical values in place of (+) and (-) signs. For now, I will update the chart to reflect a more realistic scheme, like the Seahawks offense:


As indicated above, most of the payoff matrix remain the same, with the exception of offense running vs. defense passing value. Lynch, as a better running back, creates better positive outcomes than someone who is only league average. So whereas the latter will only gain 5-10 yards, Beast Mode will rumble his way for 15-20 instead. We indicate this increased benefit with two (+) instead of one.

In this scenario, the best way to combat the defense is to actually pass more - even if your greatest asset is your running back. It seems counter-intuitive at first. You might argue that because there are more beneficial payoffs to running the ball instead of passing, the Seahawks will more likely use Marshawn Lynch in their gameplan, shifting the proportion of play-calling from a 50% split.

Yet the defense too knows that Lynch is dangerous too, and as such will more likely focus on him before the game even starts. So in reality, while the payoff matrix shows that you should run the ball more with a better running back, it fails to take into account the interaction of your intuitive opponent. Lynch is only more effective than the average running back if the defense treats him as an average running back.

This is what the Nash Equilibrium is; that just knowing what strategy your opponent is using should not affect your own strategies to win. In other words, if the defense does know that Lynch is a dangerous weapon, they should not solely focus on run defense.

Another example would be offenses passing against the Legion of Boom. Many offenses that have played the Seahawks and Richard Sherman have avoided throwing to his side - and still ended up with poor performances. Nash's equilibrium shows that opponents should not play into the Seahawks' intuition that they will avoid throwing to Sherman's side because that's exactly what Carroll and co. wants you to do.

As you can probably guess, we can extend this logic over and over again as we did earlier. In the Lynch example, the offense might know that the defense is not focusing on Lynch and may elect to continue to run with him, falling into a double blind and exactly what the defense wants the offense to think. Game theory, therefore, suggest that we should not fall into leaning towards one side of the ball and instead remain unpredictable with balanced play-calling.

To sum, I think Chris B. Brown of SmartFootball wraps this up very well:

I remember someone asking Hal Mumme when he was at Kentucky about how his teams' yards per carry had dropped around a yard or so from the season before. The reporter was incredulous and turned red faced at Mumme's response: Mumme told him that he saw the same thing, and that to fix it he would throw the ball more.

The reporter cut him off and essentially called him an idiot, mentioning that everyone knows you run better by simply running more (wear them down!). I'm pretty sure Mumme's point was that he coached a passing team, and if his yards per carry was going down, at least one reason was that the defense was spending too much time on the run and that he, as playcaller, was not taking advantage of passing game weaknesses defenses were leaving open.

As theoretical as game theory is, it plays an important role in understanding play selection when it is not situationally-based (even if most times they are).

And if we are to believe that football can be sequenced in certain payoffs, then even with particular advantages in players, coaching and scheming, we can assume that offenses should remained balanced if that are to be at their best.