Equilibrium Strategies on Fourth and Short

An interesting article at Football Outsiders presented evidence that teams pass too often in short-yardage situations. Using data from the 2003 season they find that on third or fourth down, with one or two yards to go for a first down or a TD, running plays pick up the required yardage more often than pass plays. Specifically, 69% of running plays were successful, compared to 47.3% of pass plays.

Unless the offense is very close to the opponent's goal line, these percentages are potentially misleading, because pass plays gain more yards on average than running plays. Teams should therefore be willing to pass even if the probability of picking up the first down is somewhat smaller. But how important is this caveat? We can get some insight using the footballcommentary.com Dynamic Programming Model. Strictly speaking, the importance of the extra yards for pass plays will depend on the score, the field position, and time remaining in the game; but as an example we will suppose it's early in the fourth quarter, the score is tied, we're at the opponent's 40 yard line, it's 4th and short, and we've decided to go for it. Assume also that on average, a completed pass in this situation picks up 2 more yards than a successful running play. If a running play succeeds in picking up the first down 69% of the time, what does the probability of a successful pass play have to be to make us indifferent between running and passing?

According to the version of the Model in effect as this is being written, if we fail to make the first down, our probability of winning the game becomes 0.4361. If we run and pick up the first down, our probability of winning the game becomes 0.6083. Finally, because of the 2 extra yards on pass plays, if we complete a pass for the first down, our probability of winning the game becomes 0.6139.

It follows that if we choose to run, our probability of winning the game is 0.69*0.6083+(1−0.69)*0.4361=0.5549. A pass play needs only a 66.8% chance of success to give the same probability of winning the game, since 0.668*0.6139+(1−0.668)*0.4361 also equals 0.5549. Although the difference between 69% and 66.8% is not enormous, it's not negligible either. Therefore, if we want to look only at the probability of picking up the first down or TD, and not consider average yards gained, we should restrict attention to cases in which we are near the opponent's goal line. For the rest of this article, we will assume that it is 3rd or 4th and goal from the 1 or 2 yard line. For these cases, running plays were successful 57.5% of the time in 2003, compared to 36.5% for pass plays. (Football Outsiders did not examine two-point conversion attempts, which for the purpose of this analysis are equivalent to 4th and goal from the 2.)

If it's true that, near the opponent's goal, running plays have a higher likelihood of success, then offenses should run more often in this situation. In fact, they should run all the time. Of course, if offenses begin running all the time, it won't be long before defenses will crowd all their players near the line of scrimmage, at which point passing will be the more attractive option. But then defenses will start to focus on the pass, and so on. Maybe things keep going in circles forever.

On the other hand, perhaps there is an equilibrium in this situation. By "equilibrium" we mean Nash equilibrium, a concept developed by John Nash (the subject of the book and movie A Beautiful Mind). A Nash Equilibrium is a pair of strategies, one for the offense and one for the defense, with the following properties: The offense's strategy is its best possible strategy, relative to the strategy chosen by the defense; and similarly, the defense's strategy is its best possible strategy, relative to the strategy chosen by the offense. So, in an equilibrium, each side knows the other's strategy, but neither side has any incentive to change its own strategy.

It's clear that, in equilibrium, the offense's strategy can't be to run the ball 100% of the time. Faced with that strategy, the optimal strategy for the defense is to focus exclusively on the run; in which case running all the time isn't optimal for the offense. The same remark applies to a strategy of passing the ball 100% of the time.

So the equilibrium strategy for the offense has to be the kind of strategy we actually observe: a randomized strategy, in which the offense runs the ball with some probability p and passes with probability 1−p. Now, it can't possibly make sense for the offense to choose randomly whether to run or pass, unless the offense is indifferent between those two actions; in other words, unless running and passing are equally likely to succeed. Therefore, in an equilibrium, the strategy for the defense must be the one against which running and passing have equal probabilities of scoring.

Keep in mind, this doesn't mean that the scoring probabilities for running and passing are 50%, or any other particular value. They just have to be equal to each other. Against a good defense, both probabilities might be 40%. Against a poor defense, perhaps 60%.

Now let's turn again to the equilibrium strategy for the offense. As we have seen, this will be a randomized strategy, running with probability p and passing with probability 1−p. Since running and passing are equally likely to work, the offense doesn't care what p is. It turns out though, that there is only one value of p for which it is optimal for the defense to use the designated strategy, against which running and passing have equal likelihood of scoring. So, in equilibrium, this will be the randomization probability used by the offense. Although we will leave the derivation of the equilibrium p to the mathematical appendix, we can give a rough description of what determines it. Suppose the defense were to alter its strategy, tilting it slightly more toward the run. How much effectiveness would this cost against the pass? If the answer is "very little," then the equilibrium p is near zero, meaning that the offense passes the ball most of the time. If the answer is "a lot," then the equilibrium p is near one, meaning that the offense generally runs.

To see why this makes sense, remember that in equilibrium, the defense's designated strategy of being equally effective against the run and the pass must actually be optimal for the defense. If (for example) the defense could improve its effectiveness against the run at very little cost against the pass, but chooses not to do so, it can only be because most plays are pass plays.

If the data paint an accurate picture, then it's clear that NFL teams in 2003 were not using equilibrium strategies, because running plays were more likely to score than pass plays. In the short term, offenses might indeed benefit by running more often. However, if Nash Equilibrium has any predictive power in this context, we should expect defenses to focus more on the run. It's an open question whether the equilibrium strategy for the offense entails more or fewer running plays than we currently observe.

A weakness of Nash Equilibrium in general is that a team whose equilibrium strategy is randomized is indifferent over randomization probabilities, and so has no obvious incentive to choose the one that equilibrium requires. Unfortunately, in most cases (including this one), no stronger kind of equilibrium exists.