Aspects Of Endgame Strategy

Taking a knee late in the game is standard practice for the team that's ahead. However, a team that's behind by 1 or 2 points will sometimes also find it correct to "waste" one or more downs, deliberately not scoring, in order to take time off the clock in preparation for a game-winning field goal. This is covered in great detail in John T. Reed's book Football Clock Management (reviewed at Football Outsiders ). Reed recommends taking the clock down to 0:03 or less before calling timeout or spiking the ball. In the first part of this article we will examine whether, with excess downs and timeouts, it's better to leave a little more time on the clock, to protect against the possibility of a bad snap on the field goal attempt. As part of this analysis, we will estimate the probability distribution for the clock usage on a successful field goal.

In the remainder of this article we will investigate whether it can ever make sense for a team that trails by more than a field goal to waste a down in order to run down the clock. We will see that such a situation is possible, but extremely rare. Much more common are cases in which a team trailing by more than a FG has a preference for taking time off the clock (although not a strong enough preference to justify deliberately not scoring). For these cases we will describe the equilibrium strategies for the offense and defense, and compare them to the equilibrium strategies we derived in an earlier article for the case in which the clock is not a factor.

Clock Usage On Successful Field Goal Attempts

It's impossible to determine how much time to leave on the clock in preparation for a game-winning field goal without first analyzing how much time a field goal consumes. We will describe the results of such an analysis in this section.

Table 1
Seconds	Freq
< 3	0.03
3	0.18
4	0.41
5	0.31
6	0.05
> 6	0.02

We examined a sample of over 400 successful field goal attempts from the 2003 season to get information about the probability distribution for the amount of time that a FG runs off the clock. Table 1, at left, shows the frequency distribution for the elapsed time. For example, 18% of successful FG attempts used 3 seconds. The most common usage, 4 seconds, occurred 41% of the time. Note that 3% of successful field goals took fewer than 3 seconds off the clock, presumably because the clock operator was slow starting the clock after the ball was snapped. At the other extreme, some successful field goals consume a substantial amount of time. According to Note (3) to Rule 4-3-1-k, the clock operator is supposed to stop the clock once the kick has clearly traveled out of bounds, even without a signal from an official. Nevertheless, in our sample, 2% of successful field goals used up more than 6 seconds. One notable example came at the end of Super Bowl XXXVI, when Adam Vinatieri's walk-off field goal used 0:07.

Strictly speaking, the expected clock usage on a successful FG should be a linearly increasing function of the length of the attempt. It might also depend systematically on whether the clock is already running when the ball is snapped. To investigate this we regressed the clock usage C on the length L of the attempt, and on a dummy variable D that equals 0 or 1 according to whether the clock was stopped or running prior to the attempt. The estimated regression is

E(C) = 2.34 + 0.053 L + 0.241 D

where E(C) is expected clock usage. (The standard errors of the coefficients are 0.18, 0.005, and 0.1 respectively.) This suggests that an extra yard on the length of the field goal increases the expected clock usage by 0.053 seconds, and can be regarded as a crude estimate of the inverse of the speed at which the kick travels. In addition, the estimated regression suggests that on average, a FG consumes nearly an extra quarter second when the clock is already running prior to the snap, compared to when the clock is stopped. This might reflect a lag between the snap and the restarting of the clock.

Table 2
Seconds	Freq
< 3	0.04
3	0.32
4	0.53
5	0.10
> 5	0.01

Because of the apparent dependence of the clock usage on the length of the attempt and on whether the clock was stopped, we show in Table 2 at left the observed frequencies of clock usage if we restrict the sample to field goals under 30 yards, with the clock stopped prior to the attempt.

Setting Up A Game-Winning Field Goal

John T. Reed recommends stopping the clock at 0:03 in preparation for a game-winning field goal. However, if the FG attempt will be on 4th down, or if the offense has no timeouts (other than the one they will use to stop the clock prior to the field goal), there is no reason to leave even three seconds. Doing so introduces a small chance, roughly 4% according to Table 2, that the offense will be forced to kick off to the opponents, and provides no compensating benefit.

On the other hand, if the offense has a spare timeout and a spare down, then there is at least in principle a reason to leave a few extra seconds on the clock: If there is a bad snap, the offense might have time to call timeout and retry the kick. This was in fact the strategy Miami used in their week 17 game against the Jets. Trailing 21-20, with 3rd and goal at the Jets 6 yard line, the Dolphins let the play clock run all the way down before using their second timeout with 0:07 left in the game. Miami's go-ahead field goal left 0:03 on the clock, and the Dolphins were forced to kick off and give the Jets a final opportunity. Since a FG almost never consumes 7 seconds, it's actually easy to show that Miami erred; they should have taken a knee on 3rd down and used their last timeout to stop the clock at 0:01. Roughly speaking, Miami's strategy replaced the risk of a bad snap with the risk of a kickoff return. Over the past two seasons, about 0.7% of kickoffs were returned for touchdowns, which is not very different from the probability of a bad snap. However, that percentage is relevant only for normal kickoffs. The probability is surely larger when the receiving team, knowing that they either score or lose, laterals as often as necessary to keep from being tackled, as New Orleans did in its game versus Jacksonville. (As it happened, though, the Jets went quietly; Kevin Swayne received the kickoff and was quickly tackled.) Moreover, even with the clock stopped at 0:01, a bad snap isn't always fatal: There is a chance of scrambling for a touchdown.

But if 0:07 is clearly too much, what about stopping the clock with a bit less time left, say 0:04? If the FG is good, there is about a 64% chance that time will expire, according to Table 2. But in case of a bad snap, 4 seconds will usually be enough for the kicking team to down the ball and call timeout. In a mathematical appendix we work through the formulas, and find that this strategy gives about the same probability of winning the game as stopping the clock at 0:01. Specifically, if you have excess downs and timeouts, then in preparation for a game-winning FG it makes little difference whether you stop the clock at 0:01, or stop it at 0:04 with the intention of calling time in case of a bad snap. Different coaches can come to different decisions depending on their opinions about the probabilities of certain rare events, such as a bad snap, or a kickoff return for a touchdown.

Setting Up A Game-Winning Touchdown

Can it ever make sense for a team that needs a touchdown to win to deliberately not score in order to run time off the clock? To try to answer this question, we will consider the conditions that are most favorable for "wasting" a down.

First, we assume we trail by 6 points. A touchdown and an extra point will put us in the lead, but any subsequent score by the opponents, even a field goal, costs us the game. This creates the maximum possible incentive for running time off the clock.

Second, we want to assume that it's first down, and we are very close to the opponent's goal line, so that we have a high probability of scoring even if we choose to waste a down. Third, we assume that we have at least two timeouts remaining, so that we can stop the clock if necessary after our second- and third-down plays. (With fewer timeouts, we might run into the problem the Colts faced at the end of their regular-season game against the Patriots, in which the Colts were essentially forced to pass on third down.) We also assume that the opponent's have no timeouts, so that they can't frustrate our efforts to run time off the clock.

Finally, there should be an appropriate amount of time left: With too little time remaining, there is no need to waste a down, and with too much time, it would be pointless.

So, suppose we trail by 6 points, and we have first and goal at the opponent's one-half yard line. The clock is stopped with 0:55 remaining in the game; we have two timeouts left, and the opponents have none.

If we score on first down, we will then kick off with about 0:50 remaining. We estimate that the opponents will have about a 10% chance of scoring (although you are invited to plug in your own estimate). So, if we score on first down, our probability of winning the game is about 0.90.

On the other hand, if we run on first down but deliberately don't score, and let the play clock run down, we still have three downs. On a play from the half-yard line we have about a 65% chance of scoring, so our probability of scoring on one of the three remaining downs is

1 - (1 - 0.65)³ = 0.957.

If we do score we will then kick off, but less than 0:10 will remain in the game, so the opponents will have only a very small probability of scoring, say 0.02. Our probability of winning the game if we deliberately don't score on first down is therefore

0.957 (1 - 0.02) = 0.938,

which exceeds the probability that we win the game if we score on first down. So, in this specific situation, it really is correct to waste a down. Of course, this situation will come up very rarely. Indeed, if the probability of scoring on a single play is 0.57 (corresponding to roughly the 1 yard line) rather than 0.65, there is no advantage to wasting a down. But even if it is highly unusual for a team that trails by more than a FG late in the game to want to waste a down, there will be many other cases in which such a team will have a preference for running time off the clock. Therefore, there are implications for play selection. In the next section we will study the equilibrium strategies for the offense and the defense in this situation, and show how those strategies compare to the equilibrium strategies for the case in which the clock is not a factor.

Equilibrium Strategies

In a previous article, we examined the situation in which we are very close to the opponent's goal line and have decided to go for it on fourth and goal. When time is not a factor, we showed that in a Nash equilibrium, the opponents will choose a defense against which a run play and a pass play are equally likely to score. Our equilibrium strategy is a randomized strategy in which we run with probability p and pass with probability 1-p.

The existence of an equilibrium actually requires that defense against the run be subject to "diminishing returns." To understand what we mean by this, suppose that for a particular defensive formation, the probability of stopping the run is 0.40, and the probability of stopping the pass is 0.38. Further suppose that by focusing more on the run, the defense can increase the probability of stopping the run to 0.43, but at the cost of reducing the probability of stopping a pass play to 0.34. So, an improvement of 0.03 against the run requires a 0.04 loss against the pass. Finally, suppose that by focusing even more on the run, the defense can increase the probability of stopping the run to 0.46; but in this case the probability of stopping the pass falls the 0.28. So, the increase in effectiveness against the run is again 0.03, but now the loss of effectiveness against the pass is 0.06 rather than 0.04. This is an example of diminishing returns. Formally, diminishing returns means that a series of constant improvements against the run must be accompanied by ever increasing losses of effectiveness against the pass. This seems like a very reasonable assumption.

The strategies just described cannot constitute an equilibrium if the offense has a clear preference for running time off the clock. A run play that scores is equivalent to a pass play that scores, but a run play that fails to score is better than an incomplete pass, because the clock continues to run. So if the opponents use a defense against which running and passing are equally likely to score, the offense will run 100% of the time, rather than randomize.

To get an equilibrium in this case, the opponents have to use a defense against which run plays and pass plays give equal probabilities of winning the game. In a mathematical appendix we show that, not surprisingly, this defense is more focused on stopping the run than would be the case if the clock were not a factor.

In equilibrium, against this defense, the offense will use a randomized strategy, running with probability p' and passing with probability 1-p'. Will p' be larger than p? In other words, in equilibrium, will the offense be more likely to choose a running play than it would if running time off the clock were not beneficial? In the mathematical appendix we prove that if run defense is subject to diminishing returns, then p' will be larger than p. In other words, in equilibrium, the offense will indeed be more likely to select a run play than it would if taking time off the clock were not a consideration. This is true even though the defense is focusing heavily on the run. One could check to see if Nash equilibrium has predictive power in this situation by looking at the choices made by coaches in actual games.