Previous research on the balance between running and passing in the NFL has largely missed the mark because it has focused on only half of the equation. Opposing defenses have been ignored. In this article, I'll illustrate how game theory can be applied to football play calling, and how it explains the balance between various strategies.
The research to date, notably the Alamar and Rockerbie papers along with my own articles, typically follow the same approach. They calculate the average expected yardage gain by running and by passing, accounting for various risks such as incompletions, turnovers, or sacks. The difference between the average expected gains for running and passing is then presented as a puzzle. If for example, passing yields a higher average gain than running, why don’t teams pass more often?
Further, if passing is far preferable to running, why should a team run at all? Hopefully that's obvious. An opposing defense could exclusively focus on defending the pass, dramatically limiting any expected gain.
Yet the better play choice remains the pass. So I should pass more often than run, right? But the defense knows this too, so I know he'll certainly play a pass defense. Therefore, I should run. But he knows I know that he knows that...and around we go. This is where game theory comes in. Game theory can actually quantify the optimum mix of strategies mathematically.
Offenses in the NFL do not consist of simple run or pass decisions. In reality there are a myriad of formations, plays, and variations available as options. Likewise, defenses have a similar number of alternatives. But for now, just to illustrate the potential for game theory to help understand play calling, let's limit offenses to two strategies: the run and the pass. Defenses are limited to three: run defense, pass defense, and the blitz.
The example below represents a single NFL play as a "game." The number in each cell is the offense's "utility" given the combination of offensive and defensive strategy employed. The utility is not necessarily the yardage gained or expected points, but simply a proportionate scale of how preferred an expected outcome is to the others. And since football is a zero-sum game, the defense's utility is always the negative of the offense's.
|Run D||Pass D||Blitz|
For example, if the offense chooses a run play and the defense chooses a run defense, then the outcome would be "-3" to the offense. This doesn't mean the offense would always lose 3 yards on such a play, but that the outcome is typically a setback, say a gain of only 2 yards or less. The utility to the defense would be +3 because what's bad for the offense is equally good for the defense.
In this example a run play against a pass defense yields a positive result of 4 units. A pass against a run defense yields an even larger payoff of 9. A blitz against a pass play could set the offense back by -5 units. But the blitz is risky because if the offense had called for a run, it would likely be fairly successful. Each strategy option for both the offense and defense has its advantages, disadvantages, and risks based on which strategy the opponent has chosen.
From the offense's point of view, it appears that running is the safer bet. Two of the three possible outcomes are positive, and the worst possible outcome is no worse than -3. Passing has the highest possible payoff, but two of the three outcomes are negative. There is no clear-cut "dominant" strategy choice for the offense.
The best approach is to unpredictably sometimes run and sometimes pass, which game theory calls a mixed strategy. This is also what we see in reality. In fact, game theory methods can actually "solve" the game, providing the optimum proportion of running and passing. The solution is what's known as a Nash equilibrium, named after the mathematician made famous by the movie A Beautiful Mind. The Nash equilibrium is the mix of strategies where each opponent, knowing the strategies of the other, has nothing to gain by changing his own strategy. In other words, an offense is choosing the best proportion of play calls taking into account the defense's strategy mix, and the defense is choosing its best proportion of strategies knowing the offense's strategy mix.
On the football field, the equilibrium point is found by trial and error. Countless plays are recorded and remembered along with corresponding opponent tendencies. Coaches tend to find the optimum mix of plays subjectively based on a combination of experience, intuition, and tradition. Game theory, however, can provide the true optimum mix of strategies, assuming the strategy choices are clear and their outcome distributions are known.
Recalling the example strategy matrix above, let's look at how a mix of running and passing would fare against a single defensive strategy. According to our example, if a defense always blitzed and the offense ran 100% of the time its expected utility would be +6 for every play. And if the offense passed 100% of the time its expected utility would be -5. We can plot those two points on a graph, and because we've defined utility as a linear function of preference, we can also draw a straight line between them (below). This line represents the expected utility of a given offensive strategy mix, from 100% run on the left to 100% pass on the right.
If the blitz were the only strategy choice for the defense, then the offense's solution would be obvious. It should run 100% of the time, and it would expect a +6 payoff each play. But the defense has other options. The chart below adds the run defense and the pass defense as potential strategies.
This chart of all possible strategies reveals the optimum strategy mix for both the offense and defense. Notice how the resulting plot forms a 'tee pee' shape between the y=0 line and the intersections of the defense strategy lines. This is where the offense's optimum strategy mix lies.
Think of the chart from the defense's perspective. The defense can hold the offense down to at most y by choosing a certain strategy or strategy mix. From the offense's perspective, it can maximize its payoff by selecting a mix of strategies at x. The version of the chart below identifies the solution.
To recap, the x-axis is a proportion of running and passing by the offense, from 100% running on the left to 100% passing on the right. The y-axis is the payoff to the offense based on the strategy mix of the defense.
To have any positive outcome, the offense needs to select a pass/run proportion somewhere between point a and point d. To maximize its utility, it should select the proportion at point b. (Keep in mind that the left side of the chart represents 100% run, so the fact that point b is closer to the left than the right indicates the offense should run more often than pass.)
Similarly, the defense should choose a mix of run defense (blue) and pass defense (red). The blitz strategy does not intersect at the offense's optimum (point b), so it plays no part in the defense's optimum solution assuming the offense optimizes. If the defense were to mix in some blitzes, the offense would be able to increase its expected utility to somewhere above point b. Only when the defense believes that a pass is more likely than at point c should it consider blitzing.
Further, we can use algebra to compute the equilibrium point. We know that the equilibrium lies at the intersection of the run defense and pass defense lines. They can be described as follows:
pass defense: yPASS D = 4 - 7x
At the equilibrium (point b), yRUN D= yPASS D. Therefore:
19x = 7
x = 7/19 = 0.37
Given the example, the offense should pass 37% of the time and run 63% of the time. Knowing this, we can also find the expected utility at the equilibrium point using either of the relevant defensive strategy lines. I'll use the run defense line:
y = -3 + 12(0.37)
y = 1.4
The point at which the defense should begin to blitz (point c) is where the pass defense line and the blitz defense line intersect.
blitz: yBLITZ = 6 -11x
yPASS = yBLITZ
4 - 7x = 6 -11x
4x = 2
x = 2/4 = 0.5
Only when the offense's probability of passing is >= 50% should the defense blitz. We can find points a and b similarly by setting y = 0 for the run defense and blitz equations.
Even though the example here is artificial, the principles of game theory apply throughout every football game. Of course, offenses aren't limited to two play choices. Their playbooks are famously complex. And defenses are not limited to three options. In fact, they're not limited to any number. In reality a defense has an infinite continuum of biases between run and pass, from goal line defense to prevent defense and everything in between.
These problems are really only a matter of scale. The algorithms of game theory can solve games with large numbers of strategy options. It can even handle strategy sets like a defense's, without discrete "pure" strategies. Plus, the problem lends itself to great simplification because much of every offense's playbook is composed of subtle variations of a few virtually universal NFL plays.
Application of game theory hinges on true measures of utility. The real difficulty in applying game theory to football on a practical level would be to develop a valid measure of the preferences of the various payoffs in the game matrix. Yards or points would seem like suitable measures, but a 3-yard gain is far more useful on 3rd and 2 than on 3rd and 9. And a 3-point field goal is fairly useless down by 7 late in the 4th quarter. In a forthcoming article I'll attempt to solve this problem with a proposed utility function for football.
But next, I'll look at what effect a star running back might have on offensive strategy. Should it pass more often or less often? How will the passing game's effectiveness change. How should a defense react? We have intuitive answers, but we'll see what John Nash says.