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June 12, 2004

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An Analysis Of Tempo

In a game between evenly matched teams, the team that leads should slow down the pace of play on offense, and the trailing team should employ a hurry-up offense. Exceptions to this rule can arise toward the end of the game, but always in situations that are easy to recognize.

The more interesting case is that in which one of the teams is a clear favorite. In his comprehensive book Football Clock Management (reviewed at Football Outsiders ), John T. Reed tries to cover this case by proposing a general principle that refers not to the score, but to the probability of winning the game. Clock Management Rule 1.10 states that you should use a hurry-up offense whenever your probability of winning the game is below 0.5, and use a slowdown offense when your win probability exceeds 0.5. In particular, according to Reed, the underdog should begin the game in a hurry-up offense, and the favorite should begin in a slowdown.

However, this prescription is incorrect. Although trailing and being the inferior team both lower your probability of winning the game, they actually have opposite implications for the proper tempo. Intuitively, since teams alternate possessions, a hurry-up offense by either team creates more possessions for both teams. Essentially, it "lengthens" the game. However, if you are the inferior team, lengthening the game is undesirable; indeed, in an infinite game you would lose for certain. Instead, you want to "shorten" the game, to give random variation the best chance of nullifying your opponent's superiority. Therefore, the underdog should begin the game in slowdown tempo, while the favorite employs a hurry-up offense. (John Sterner points this out in his book The Football Coaches' Guide to Clock Management.)

Moreover, since the underdogs have a clear preference for a slowdown offense to start the game, a continuity argument suggests that they should stick with the slowdown even if they fall behind early by a small amount. The analysis described below confirms that this is the case. Of course, if the underdogs fall behind by a sufficient amount, they should switch to a hurry-up. The model described in the next section allows us to quantify the qualitative features we have just described. Specifically, the model tells when you should be in a slowdown, and when you should be in a hurry-up, as a function of the point differential, the time remaining in the game, and the relative abilities of the teams.

Description Of The Model

To analyze tempo, we built a model that is in most ways a simplification of the footballcommentary.com Dynamic Programming Model, but can also be regarded as an enhancement of the model of Harold Sackrowitz. Teams alternate possessions. The probabilities of scoring a touchdown or a field goal on a possession can be different for the two teams. On each possession, the team with the ball decides whether to use a hurry-up offense or a slowdown offense. For each of these tempos, there are two possible outcomes for the amount of time consumed during the possession. Of course, the expected time consumed is smaller for the hurry-up offense. The objective is to maximize the probability of winning the game.

In the model, the probabilities of scoring a touchdown or a field goal on a possession are the same regardless of whether the offense is using a slowdown or a hurry-up. (The only exception is the trivial one, that at the end of the game, the slowdown offense is more likely to run out of time.) In addition, since the slowdown and the hurry-up are the only tempos permitted, we are implicitly assuming that neither of these offenses incorporates so large a change in the usual mix of pass and running plays as to materially reduce the probability of scoring below what could be achieved with an in-between tempo. So, when we say "slowdown," we really mean being careful to let the play clock run down, without necessarily using significantly more running plays. Similarly, "hurry-up" means being careful not to use too much of the play clock, but not necessarily using more pass plays. As Sackrowitz and Sackrowitz have shown, it will rarely make sense to employ a clock-management strategy that significantly lowers the probability of scoring on a possession.

The model solves for the choice of tempo (hurry-up or slowdown) optimally by backward induction. Similarly, decisions regarding two-point conversions following a touchdown are made optimally, by backward induction. Those who want a more precise description of the model can examine the source code, which is written in MATLAB®.

One of the inputs to the model is the expected number of points by which the favorite will win the game. We suggest using the point spread applicable for placing bets. As the market-clearing price, this represents a sort of consensus, at least among people who bet on the game. (In principle, the point spread represents the median of the distribution for the point differential, rather than the expected value, but the difference won't be important.) At any rate, regardless of how one comes up with an estimate, we will refer to the expected winning margin as the "spread." To avoid confusion, we will never use the word spread to refer to the actual point differential at some time during the game.

Another model input is the expected total points scored in the game. We suggest using the "over-under" that is familiar to bettors. However, the results of the model are not very sensitive to this input, for reasons that the mathematical appendix helps explain. To generate the results shown in the Tables in the next section, we used 42 for the expected total points, which is the average from the 2003 NFL regular season.

Results Of The Model

14-Point Spread
Time Remaining
60:00 45:00 30:00 15:00
Underdog Use hurry-up if
trail by at least
15 12 9 7
Favorite Use slowdown if
lead by at least
14 10 7 1

The Table at left summarizes some of the results of the model for the case in which the spread is 14 points. The entries in the row labeled "Underdog" show how many points the underdog must be trailing by in order for it to be correct for them to employ a hurry-up offense. The row labeled "Favorite" shows how many points the favorite must be leading by in order for it to be correct for them to go to a slowdown.

For example, with 45:00 remaining in the game (i.e. three quarters), a 14-point underdog should use a hurry-up offense only if they trail by 12 or more points. Trailing by fewer, or leading, they should use a slowdown offense. For another example, with 30:00 remaining, a 14-point favorite should use a slowdown offense only if they lead by 7 or more points. Leading by fewer, or trailing, they should use a hurry-up offense.

7-Point Spread
Time Remaining
60:00 45:00 30:00 15:00
Underdog Use hurry-up if
trail by at least
8 7 7 3
Favorite Use slowdown if
lead by at least
7 4 3 1

The Table at right contains the same information, but for the case in which the spread is 7 points. For example, with 15:00 remaining in the game, a 7-point underdog should use a hurry-up offense if they trail by 3 or more points. Trailing by fewer, or leading, they should use a slowdown offense. With 45:00 remaining, a 7-point favorite should use a slowdown offense only if they lead by 4 or more points. Leading by fewer, or trailing, they should use a hurry-up offense.

Since it's rare for the spread for an NFL game to exceed 14 points, interpolation within and between these Tables should be adequate for most situations. The exception is when the game is well into the fourth quarter. At that point, though, the numerous approximations inherent in the model make it a less useful tool -- and a coach's intuition becomes more reliable.

The Tables show no situations in which a hurry-up offense will be used regardless of which team has the ball. However, there are cases in which a slowdown offense will be used regardless of which team has the ball. For example, suppose that the spread is 14 points, and that with 15:00 to play, the favorite leads by 4 points. According to the Table at left, if the underdogs have possession, they will use a slowdown; but if the favorites have the ball, they will use a slowdown also. Although this is counterintuitive, it's actually quite familiar, even when the teams are evenly matched. Indeed, suppose 4:00 is left in a game and the teams are separated by less than one score. If the team that leads has the ball, they will use a slowdown to try to run out the clock. But if the team that trails has possession, they will also use a slowdown, to try to score near enough to the end of the game to deny their opponents another opportunity.

Notice that with almost the entire game left to play (the Table columns corresponding to 60:00), both the deficit that induces the underdog to use a hurry-up offense, and the lead that induces the favorite to go to a slowdown, are equal or nearly equal to the spread. This is not a coincidence. In a mathematical appendix we give an explanation.

Conclusions

In a game between teams of unequal ability, the underdogs should use a slowdown tempo unless they trail by a sufficient amount, and the favorites should use a hurry-up tempo unless they lead by a sufficient amount. The amount that is "sufficient," in each case, depends on the time remaining in the game and on the extent of the mismatch, and can be approximated by the model described in this article.

Coaches are often reluctant to use strategies that conflict with conventional wisdom, due to an aversion to being second guessed. For the strategies described here, there is an additional impediment: One of the coaches would have to admit, in effect, that he's coaching the inferior team. So we shouldn't expect to see teams following this prescription any time soon.


Copyright © 2004 by William S. Krasker