A model-based approach to football strategy.
|May 29, 2004|
Several times in a typical game, a coach must decide whether to punt or go for it on fourth and short, or whether to attempt a two-point conversion after a touchdown. In this article we will present Tables, computed from the footballcommentary.com Dynamic Programming Model, that provide guidance for making those decisions.
On their opening drive in the 2003 AFC Conference Championship game, the New England Patriots went for it on 4th and 1 at their own 44 yard line. What makes this decision noteworthy isn't that it was wrong—it wasn't—but that coaches in that situation almost always punt.
To defend their unwillingness to go for it on fourth down when every thoughtful analysis says they should, coaches often cite the loss of momentum, and the harm to the players' morale, that purportedly results from a failed fourth-down attempt. However, David Romer, who has urged coaches to go for it more often, has reportedly actually examined data on momentum swings, and found no evidence that momentum is a factor. If anything, Romer says, the data suggest that teams play harder after giving up the ball following a failed fourth-down attempt or a turnover. Our own opinion is that few things could be more demoralizing to players than a coach who doesn't give them the best possible chance to win the game.
Elsewhere we have discussed the limitations of the model David Romer uses to analyze fourth-down situations, namely that as a steady-state model rather than a backward-induction model, it doesn't take into account the score or the time remaining in the game. And indeed, the results of the footballcommentary.com Dynamic Programming Model don't support Romer's more sweeping conclusions. However, we agree that coaches often punt when they should go for it. In particular, according to our Model, teams should always go for it on 4th and 1 if they trail, are not in field goal range, and are not too deep in their own end. Closer to either goal line, or when leading, or with more than a yard to go for a first down, it is still often correct to go for it, but these cases must be considered on an individual basis.
We worked through the arithmetic for a couple of 4th down situations in our analysis of Super Bowl XXXVIII. One of those situations arose with 9:04 remaining in a scoreless first half, when New England faced 4th and 1 at the Carolina 38 yard line. We found that New England's decision to go for it raised their probability of winning the game by about three percentage points.
To provide some guidance, we have prepared a series of Tables that show what the probability of picking up the first down has to be to justify going for it on fourth down. As an example of how to use the Tables, suppose we lead by 3 points with 21:00 remaining in the second half (i.e. 6:00 left in the 3rd), and face 4th down at the opponent's 40 yard line. We go to the Table labeled "Opponent's 40 yard line, second half," and find that the entry corresponding to a 3-point lead, and 21:00 remaining in the second half, is 0.46. The interpretation of this number is that if our probability of making the first down exceeds 0.46, we should go for it, and otherwise we should punt. So we should certainly go for it if it's 4th and 1 or 2 yards to go. With 4 or more yards to go for a first down, we should punt.
Certain patterns emerge from the Tables. For example, to justify going for it, we need a higher probability of success if we're ahead than if we're behind. Moreover, this disparity is more pronounced late in the game. Specifically, if we are ahead, it is more likely to be correct to go for it early in the game than late. Conversely, if we trail, it is more likely to be correct to go for it late in the game than early. So, the rule of thumb "Kick Early, Go For It Late" captures some of the flavor of the Model when we're behind, but not when we're ahead.
Although a decade has passed since it was brought into the NFL, the two-point conversion attempt remains an unusual event. Many coaches consider going for two only near the end of the game when the situation is very clear. Kicking is the "default." This preference is largely arbitrary; we suspect that if the NFL had originally allowed only two-point conversions, and introduced kicked extra points in 1994, then current teams would go for two far more often.
Only 68 two-point conversions were attempted during the 2003 regular season and playoffs (of which 31 were successful), although there were many more situations in which teams should have gone for two. According to the footballcommentary.com Dynamic Programming Model, during Week 1 alone there were 13 cases in which a team should have gone for two, although the team did so on only six of those occasions. (And in the proverbial exception that proves the rule, Baltimore went for two when the Model says they should have kicked.)
Using the Model, we have produced a Chart that can be used to help decide whether to go for two. Rather than simply stating whether to kick or go for two, the chart shows what the probability of a successful two-point conversion has to be to justify going for it. For example, suppose that with 18:00 left in the game (i.e. 3:00 left in the 3rd) we have just scored a TD to go ahead by 5 points. In the column corresponding to 18:00, and the row corresponding to a lead of 5, the Chart entry is 0.27. This says that it is correct to go for two if the probability of making it exceeds 0.27. Since NFL teams are successful on about 40% of their two-point conversion attempts, or perhaps a bit more, the correct decision in this case is to go for two.
Those who don't like to think about probabilities can replace each Chart entry that is less than or equal to 0.4 by "2," meaning go for two, and replace each Chart entry that exceeds 0.4 by "1," meaning kick. However, presenting the results as we did might have an interesting consequence. Just as in Garrison Keillor's mythical Lake Wobegon, where all the children are above average, it's possible that every NFL coach thinks his offense has a higher-than-average probability of success on a two-point conversion. So, in the unlikely event that coaches start to pay attention to the Model, they may go for two quite often.
Certainly, on occasion a coach makes a decision that could only be correct if he believes his team has a high probability of success on two-point conversions. For example, in Week 15 of the 2002 season, the Vikings
went for two
against the Saints when trailing by 1 point with 0:05 left to play. Since kicked extra points succeed with probability around 0.985, and the probability of winning in OT is about 0.5, going for two makes sense only if Minnesota's probability of success exceeds
We have not included Chart entries for the first half of the game. To be sure, even in a close game, there are certain score differentials—down by 2, or leading or tailing by 5—for which it is correct to go for two with most of the second quarter left to play. However, the more time that remains in the game, the more likely it is that kicking an extra point is the right thing to do. Indeed, in the limiting case of an infinite game, only expected points matter. Therefore, one would go for two only if the probability of success exceeds half the probability of a kicked extra point—about 0.4925. But to paraphrase Woody Allen, an infinite game is very long, especially near the end.
One question that arises is whether the observed percentage of successful two-point attempts is subject to a selection bias. Specifically, it's possible that coaches only try for two when they think their chances of success are high. We don't believe this is a problem, essentially because coaches go for two so seldom. With few exceptions, coaches try for two only when the situation is so compelling—for example leading by 1 late in the game—that every team would make the same decision. Nevertheless, selection bias will become an issue if teams begin paying attention to models, and attempt two-point conversions in many more situations.
Copyright © 2004 by William S. Krasker