A model-based approach to football strategy.
|April 1, 2004|
In Week 16 of the 2003 NFL season, the game between the 49ers and Eagles in Philadelphia was tied at the end of regulation. It was cold and windy, and television commentator Cris Collinsworth asked an pertinent question: How strong would the wind have to be before it would be correct for the coin-toss winner to choose the wind rather than choose to receive?
Collinsworth and his colleague Troy Aikman recognized that an answer to that question would have no practical implications. Former Detroit coach Marty Mornhinweg was so thoroughly ridiculed for his decision to take the wind against Chicago in 2002 that it's doubtful a coach will ever again pass up an opportunity for the first overtime possession, even in a hurricane.
To be fair, it's not just an aversion to being second-guessed that motivates coaches. Undoubtedly they believe (correctly) that receiving the kickoff is better under ordinary conditions, and they may believe it's better under all conditions, even though we have seen no analysis to suggest that. In addition, having the first possession gives a feeling of control. As quarterback Matt Hasselbeck said when his Seahawks won the overtime coin toss versus the Packers in the 2003 playoffs: "We want the ball, and we're going to score."
Of course, as Hasselbeck discovered, scoring is just one of the two possibilities. You can't control the outcome; all you can do is try to make the probability of winning as high as possible.
So, the question posed by Cris Collinsworth is worth considering. In this article we will present a model with which we can analyze the effects of wind on a game. Using the model, we'll try to determine how strong the wind has to be in overtime to make choosing the favored end of the field preferable to electing to receive.
To analyze the probability that the receiving team wins the overtime, we built a simulation model. The main features of the model are as follows: Given a team's starting field position on a possession, the yardage gained on the drive (prior to a FG attempt or punt) has a probability distribution. If the drive ends in field goal range, there is a FG attempt, whose probability of success depends on its length. If the attempt is good, the game ends. Otherwise, the other team gets possession at the spot of the kick (or the 20-yard line).
If the drive ends short of FG range, the team punts. The punter either tries to kick long, or, when it's beneficial, attempts a "pooch" kick to pin the opponents deep in their own end. In either case, the exact length of the punt is a random variable. If the punt goes into the opponent's end zone, it comes out to the 20 yard line. Otherwise, there is a probability distribution for the yards gained on the punt return. The simulation continues until a team scores.
For those who are interested in the details, the MATLAB® source code for the simulation is otsim.m. The programs driveend.m, fg.m, and punt.m are also used. We have adopted a fairly simple structure to generate the conditional probability distribution for the yardage gained on a drive (conditional on the starting point), and have informally calibrated it to data from the 2003 NFL season. Punts are assumed normally distributed. For cases in which the punter is trying to kick long, we estimated the mean (43 yards) and standard deviation (8 yards) of the punt from 2003 data (excluding observations in which the line of scrimmage was beyond the punting team's 40-yard line). We assume the punter can select a smaller mean for a pooch kick, and that the standard deviation scales down proportionately.
From data on all FG attempts during the 2003 regular season and playoffs, we have estimated the probability that a FG attempt will be successful, as a function of the length of the attempt. From 53 yards, about 50% of kicks are good. As the distance decreases, the probability of success rises roughly linearly until we get to 42 yards, at which point about 75% of attempts are good. From there, the probability again rises roughly linearly (but at a different rate), reaching nearly 100% at 20 yards. In the model we assume that teams attempt field goals of no more than 53 yards, which we refer to as "field goal range."
We have modeled punt returns as having an exponential distribution with scale proportional to the length of the punt. For a 43-yard punt, the mean punt return is assumed to be 7 yards.
Using these distributions for the random variables in the simulation, we find that in the absence of any asymmetries caused by wind, the team that receives the overtime kickoff wins the game with probability 0.64. This is somewhat larger than the observed fraction: Since 1994, when kickoffs were moved to the 30-yard line, the team receiving the kickoff has won 59% of the games that were decided in overtime. However, there have been only 165 such games. Therefore, even if we assume that the true probability of winning has been constant over the last decade, the estimate of 0.59 is subject to sampling error, with a standard deviation of about 0.04.
In addition, the observed fraction of overtime games won by the receiving team is a downward biased estimate of their true probability of winning in the absence of wind, simply because many overtime games have wind. The team that loses the coin toss chooses the favored end of the field, which at least partly offsets the receiving team's advantage.
To quantify the bias, suppose that the average NFL game (including indoor games) is played in wind with the following observable effects: First, the starting field position following a with-the-wind kickoff is one yard worse, on average, than it would be in the absence of wind. Second, punts against the wind average two yards shorter than normal, and punts with the wind average two yards longer than normal. Finally, field goal range is shortened two yards (relative to normal) when kicking into the wind, and extended one yard when kicking with the wind. With these parameter changes, our simulation model says that the team that receives the overtime kickoff wins the game with probability 0.61. This still exceeds the observed 0.59, but is well within sampling error. (Of course, it's not a shock that the prediction of the model is consistent with observation, since the model's inputs were estimated from data.)
Two aspects of this calculation merit elaboration. First, when we speak of field goal range being shortened or extended, we don't mean that the only effect is a change in the maximum length of an attempt. Rather, in the model there is a rescaling, so that a particular probability of success is associated with a different distance than would be the case in the absense of wind.
In addition, the effects of the wind on field goals, punts, and kickoffs are not independent inputs. A wind that shortens FG range by two yards will have some definite impact on punts and kickoffs. The numbers we used represent our estimate of what that impact would be.
Consider a headwind strong enough to shorten field goal range by 10 yards. We will call this a "ten-yard wind." A wind of this strength will in addition have specific effects on field goals kicked with the wind, punts, and kickoffs. Assume that, for a ten-yard wind, field goal range is extended by 2 yards for kicks with the wind. Further assume that through a combination of touchbacks and other longer-than-average kickoffs, the starting field position following the kickoff averages the 20-yard line. Finally, assume that punts into the wind average ten yards shorter than normal, and punts with the wind average 10 yards longer than normal.
With these assumptions, it turns out that a ten-yard wind is enough to justify choosing the favored end of the field: According to the simulation model, if the coin-toss winner elects to receive, their probability of winning the game is only 0.48. More than a third of the reduction from 0.64 to 0.48 is due to the effect on field goals kicked into the wind.
Notice that we haven't even considered the effect of wind on the passing game. Though this effect is probably smaller than the others, its direction is unambiguous. Indeed, we can't think of a single aspect of the game that benefits from a headwind.
So, subject to the approximations inherent in the model, we have a tentative answer to the question posed by Cris Collinsworth: If the wind is strong enough to reduce your field goal range by more than ten yards, you should take the wind.
On November 24, 2002, in a game between the Lions and Bears at Champaign Illinois, the score was tied at the end of regulation. Detroit won the coin toss, and coach Marty Mornhinweg elected to take the wind. After receiving the kickoff, Chicago marched down the field and kicked the winning field goal. Mornhinweg was ridiculed for his decision, but was it really a historic blunder?
Certainly, the events that took place in OT show that the wind was a factor. Chicago's Ahmad Merritt fielded the kickoff two yards deep in his end zone. The Bears went for it on 4th and 3 rather than attempt a 48-yard FG. And Paul Edinger's 40-yard game winner cleared the crossbar by only three yards. So, the wind may indeed have been at the "ten yard" level that, our model suggests, justifies taking the wind.
Copyright © 2004 by William S. Krasker