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This study examines a team's chances of winning at the end of the game when an opponent is in potentially game-winning FG range. It's a dire situation for the team on defense because the offense could run enough time off the clock before its FG attempt so that there is no time to respond.

The Field Goal Choke Hold situation looks like this:  The defense has a lead of 1 or 2 points with less than 3 minutes to play. The opposing offense has just converted a first down inside FG (attempt) range. Through week 13 this season there have been 12 games that qualify, which makes this situation about as common as overtime. (There were 12 more games with similar circumstances except the game was tied, a situation nearly identical but that requires some slightly different math.)

Two strategies are compared. The first is playing for the stop and forcing the FG attempt. This may be dangerous due to the ability of the offense to burn clock. The second strategy is to allow an immediate TD. This strategy forfeits points to the opponent in exchange for enough time to respond with a game-winning TD drive.

There is no guarantee the offense will take the bait and score a TD. If the offense is cognizant of the strategy, they may take a knee close to the goal line. So strictly speaking, this analysis merely estimates which combinations of circumstances make an immediate TD preferable to forcing a FG attempt. Even if all offenses were prepared for this contingency and were inclined to take a knee, this analysis lays out when that would be the smart move.

Analysis Overview

There are many considerations in the analysis:

1. The current time on the clock
2. How many timeouts you have
3. When you would get the ball back given a series 'stop' and a FG
4. The current field position of your opponent
5. The accuracy of a FG attempt from the expected attempt distance
6. The chance of scoring either a FG or TD after the 'stop' and FG
7. The chance of scoring a TD from your own 20 given an intentionally allowed TD at the current clock time, minus the time for the TD play itself

The problem boils down to a single comparison: A defense would prefer to intentionally allow a TD whenever the probability of scoring a TD in response exceeds the total probability of the offense missing the FG attempt plus the possibility that either a FG to TD can be scored in response to a successful FG.

Assumptions

It's a very complex problem with many moving parts. To simplify the analysis, there are several assumptions needed. The intent here is to begin to get our arms around a seemingly impenetrable problem. First, for now, I'll only look at first downs as decision points. In other words, we'll decide whether to play for the stop or to allow a TD immediately following a series conversion by the offense.

Second, once within reasonable field goal range, the offense will only run the ball and will not make another conversion. Of course they could convert, but this would be the least preferable outcomes for the defense. The possibility of a series conversion would only add weight to the scale on the side of intentionally allowing a TD. As with other analyses questioning conventional wisdom, it's best to choose simplifying assumptions that count in favor of the conventional choice and against the unconventional choice. In other words, this analysis says, "Coach, even if you were to certain to get the stop, you would still want to allow the TD..."

Another assumption is that the team on offense will play smartly enough not to commit a significant penalty or turnover.

Also, the offense will gain a modest amount of field position on its three plays prior to a FG attempt. For the sake of simplicity, I'm going to say there will be 5 yards gained between 1st and 4th down prior to the FG attempt. Unless the offense is at a very long FG attempt range, a few yards in either direction will not make a large difference in the final analysis. Additionally, the offense will use 39 seconds between plays (whenever a timeout is not called or two-minute warning does not occur), and plays themselves will take 6 seconds.

Lastly, the analysis assumes that after a score, the subsequent drive will start very near a team's own 20-yard line. This is plausible because touchbacks are now so common that the average starting position following a kickoff is a team's own 22, and touchbacks would be preferred by the receiving team because no time elapses on the play.

Analytic Approach

If a team does allow a TD, it would need its own in response to win. The probability of scoring a TD is a function of only time remaining. The probability estimates for scoring are based on recent history where teams need a TD to tie or win on a final possession.

If the team on defense forces the FG attempt and it's successful, a TD or FG in response would be needed to win. The probability of scoring either a TD or FG is also a function of time and based on recent historical scoring rates for teams that need a score to tie or win in the endgame.

The main question boils down to three possibilities:

A. Forcing a stop and hoping for a FG miss
B. Given an opponent's made FG, getting your own FG or TD in response after the opponent has burned as much time as possible
C. Getting your own TD in response to an intentionally allowed TD at the current time remaining (minus the duration of the play).

Possibilities A and B comprise the win probability (wp) for forcing the stop and the FG attempt. Possibility C directly relates to the chances of winning after allowing an intentional TD. [I left wp in lowercase to distinguish it from the global Win Probability model I often use. This analysis estimates the chances of winning indirectly using scoring probabilities in the endgame. This is necessary because the number of timeouts is so critical.]

Or alternatively:

wp[force FG] = p(FG fail) + p(scoring | made FG)

wp[allow TD] = p(scoring own TD in response)

The decision should be whichever wp is higher:

Decision = max{wp[force FG], wp[allow TD]}

The next four parts of this series will estimate the time that the team on defense will regain possession (part 2), estimate the probability of a failed FG attempt (part 3), estimate the probabilities of the team on defense responding with its own score (part 4), and present the final results (part 5).
This is the second part of a five-part series on when a defense would prefer an intentionally allowed TD to forcing a FG. The first part laid out the analysis and assumptions. This part explains how to estimate the time remaining when the defense would regain possession following a forced FG attempt by the offense.

Time Remaining Following a Forced FG

The first task of the analysis was to create an algorithm to compute the time on the clock when the team on defense would get the ball back following a forced FG. This is a function of current time and time outs remaining for the defense. For example, suppose the offense has just converted a series so that the 1st down snap will happen at 1:20, and the defense has two timeouts. The offense will run three times, you'll call both timeouts, and following a FG, you'll probably get the ball back with 17 seconds remaining. The two-minute warning is factored in, which is more challenging than it might seem.

The time-you-get-the-ball-back algorithm assumes that the defense will use its timeouts at every immediate opportunity. The only exception will be when the play itself spans the 2-minute warning. For example, if there is 2:10 on the clock at the snap and the play duration is 6 seconds, the defense will call a timeout at 2:04 rather than allow the clock to wind down to the 2-minute warning.

However, there is a special case where the defense may want to allow the clock to run down to the two-minute warning rather than use all its timeouts. For example, if there is 2:10 remaining between 2nd and 3rd down and a team has 2 timeouts remaining, it may chose to allow the clock to wind down to 2:00. The third down snap would occur following the two minute warning, and the defense would call its 2nd timeout between 3rd and 4th down, at around 1:54. This would allow the defense to save one timeout for use on offense.

I did not account for this scenario in my analysis. The reason was that timeouts on offensive final drives did not have a discernible effect on scoring success. In fact, the number of timeouts available to the offense appeared to have a slightly negative effect. I believe this is an artifact of sample error, but the fact remains that I can't estimate the value of an offensive timeout with the current data. For that reason, I said teams would always use their timeouts while on defense, which definitely does show to have measurable value. (An alternative approach would be to notionally add time to the game clock for each timeout available to the offense.)

The graph below indicates how much time will be on the clock at kickoff (vertical axis) based on the time remaining at the 1st down snap (horizontal axis). This assumes the offense will attempt to keep the clock running on 1st, 2nd and 3rd downs, and the defense will use its timeouts whenever a play does not span the two-minute warning. It also assumes the offense will consume 39 seconds between plays whenever the clock is allowed to run and that the duration of each play is 6 seconds.

Each color represents a number of timeouts remaining to the defense. The sharp vertical segments indicate the effect of the two-minute warning.



Here are two examples to help understand this chart. Suppose the defense has two timeouts and the 1st down will occur with 1:40 (100 seconds) to play. On the horizontal axis find the 100 mark and move upward until you hit the green (2 timeouts) line. Look directly leftward from that point to the vertical axis to see that there will be 37 seconds remaining. Here's the breakdown from the algorithm for this example:

1st down snap: 100
Timeout
2nd down snap: 94
Timeout
3rd down snap: 88
4th down snap: 43
Get ball back at: 37
Timeouts left: 0

Now suppose the defense has one timeout and the 1st down will occur with 2:10 (130 seconds) to play. On the horizontal axis find the 130 mark and move upward until you hit the red (1 timeout) line. Look directly leftward from that point to the vertical axis to see that there will be 67 seconds remaining. Here is that breakdown:

1st down snap: 130
Timeout
2nd down snap: 124
Two minute warning
3rd down snap: 118
4th down snap: 73
Get ball back at: 67
Timeouts left: 0

These numbers are critical to the analysis because they will largely determine likelihood of scoring in response to either a forced FG or an allowed TD, which in turn will determine which is the better option.

The next three installments will estimate the probability of a failed FG attempt (part 3), estimate the probabilities of the team on defense responding with its own score (part 4), and present the results (part 5).
This is the third part of a five-part post on when an intentionally allowed TD is preferable to forcing a FG in the endgame. The first two parts of the post laid out the analysis and assumptions, and estimated the time remaining when the team on defense would get back possession. This installment looks at the probability of an unsuccessful FG attempt to take the lead.

Field Goal Success Probability

The simplest way to win for a defense caught in the field goal choke hold is to pray for an unsuccessful FG attempt. This is a mostly straightforward calculation but has a couple wrinkles. The success rates here include all possible causes of a FG ultimately being unsuccessful. That includes blocks, botched snaps, and even penalties that force longer attempts which turn out unsuccessful.

Also, as mentioned previously, an estimate of the expected field position on 4th down is required. For now, I will say that the offense will gain 5 yards during its 1st, 2nd and 3rd down plays. This is essentially a plausible placeholder for now, and a more detailed analysis can be done to confirm or adjust this.

The graph below shows three things. The jagged blue line is the actual raw FG success rate by line of scrimmage. (Add 17 or 18 yards for the commonly used 'kick distance'.) The red line is the estimated true probability of success. This estimate was computed non-parametrically, using locally weighted regression. The green line is the same curve as the red line, but offset 5 yards closer to the uprights. (Inside the 10, the 5 yards are progressively curtailed.)




But what really matters is the inverse of the chart above. What we care about is the expected probability of failure, that is, one minus the probability of success. The chart below provides a different perspective, emphasizing how the FG failure rates dramatically increase with range.



The final two parts of this post will look at the probabilities of a responding scoring drive (part 4) and will present the final results of when a defense would prefer to allow a TD to forcing a FG (part 5).
 
This is the fourth part of a five part series on when a defense would prefer an intentionally allowed TD to forcing a FG. The previous three installments laid out the analysis and assumptions, computed the time remaining when the defense would regain possession, and estimated the probability of a failed FG attempt. This part will estimate the probabilities of the team on defense responding with its own score.

The Probability of Responding to a Successful FG

If the defense plays conventionally, and the opponent's FG is successful, a score will be needed to win. Either a FG or TD will do. We can assume that the drive will began at or very near the offense's own 20-yard line for a couple reasons. First, the average starting field position for all drives is the 22. And second, it's very likely that, with time at a premium, the offense would prefer a touchback so that no time expires on the kickoff.

For this estimate, I looked at all game situations in which an offense needed a score to survive and had a 1st down at or very near its own 20. Success is defined as any drive that results in a TD or FG.

The blue line is the raw success rate as a function of seconds remaining at the 1st down snap. The red line is the smoothed estimate of the probability of scoring based on a local regression. Because there were several bins of data with very few cases (which caused the large noisy swings in the raw averages), I used a regression method that weighted each case by how often it appeared. In other words, if there were 10 cases where a team gained possession with around 50 seconds to play and 20 with around 60 seconds to play, the regression weighted each bin of cases proportionately.


The probability of success drops precipitously under 30 seconds to play. That appears to be the very least amount of time for a team to get into FG range from a team's own 20. But time in excess of 30 seconds is only marginally more valuable as time remaining increases.

The Probability of Responding to a TD

In this case, the offense now needs a TD of its own. Like the situation that requires a response to a FG, the team now on offense is assumed to gain possession at or very near its own 20-yard line. The preceding TD play was estimated to consume six seconds.

For this estimate I looked at all endgame situations in which the offense needed a TD to survive and had a 1st down very near its own 20. The graph below plots the success rate by time remaining. As mentioned previously, timeouts remaining for the offense do not have a measurable effect. The jagged blue line is the raw data, and the red line is the smoothed estimate.



Without the possibility of a FG, the curve appears smoother than the probability of either a TD or FG, at least according to the regression. I intuitively suspect that the true probability is closer to initial steepness that the raw numbers indicate, but for now I'm leaving the regression as is. It's another factor I'd be willing to revisit.

Comparing the two charts (FG or TD needed vs. TD needed) suggests that teams play differently, and perhaps irrationally, based on the score situation. When time is not too pressing (with about 2 minutes or slightly more to play), offenses that need a TD appear slightly more likely to be successful than those that only need a FG. I suspect this is due to the illusion of FG "range" as well as risk-reward balance for both the offense and defense.

Next, the fifth and final part of this series will put everything together and present the results.
This is the fifth and final part of the series on when a defense would prefer an intentionally allowed TD to forcing a FG. The previous four parts laid out the analysis and assumptions, computed the time remaining when the defense would regain possession, estimated the probability of a failed FG attempt, and estimated the probabilities of the team on defense responding with its own score. Now it's time to put all the parts together and present the results.

Putting It All Together

Taking a step back, the goal is to compare two strategies for the defense. The first is to play conventionally and force a stop and a FG attempt, hoping it will either fail or that there is enough time to match it with a counter-score. The second is to intentionally allow a TD immediately and use the time remaining to respond with a counter-TD.

So far, we have estimates for the key inputs:

-When the team on defense would get the ball back
-The probability of failure on the FG attempt
-The probability of responding to a made FG with a score
-The probability of responding to an allowed TD with a TD

The allow-the-TD strategy is the simpler one to value. We can account for the time of the intentional TD play and plot the probability of responding with another TD as a function of time. However, there is one wrinkle. If the team on defense is only ahead by one point, the offense would be smart to go for the two point conversion following a TD, allowed or not. If the offense converts the two point conversion, a response TD only ties. If the two point conversion fails, it's no different than kicking the extra point. A response TD wins either way. The offense therefore has nothing to lose by going for the two point conversion.

For when the team on defense is ahead by two points:

wp[allow TD] = p(scoring own TD in response)

But for when the team on defense is ahead by one point, and the offense would go for the two point conversion to take a 7-point lead:

wp[allow TD] = p(scoring own TD in response) * p(2-pt conv fails) + 0.5 * p(2-pt conv succeeds)

The value of forcing the FG is slightly more complicated because it combines the possibility of a failed FG with the possibility of responding with another score.
 
wp[force FG] = p(FG fail) + p(scoring | made FG)

This is a total probability computation, much like we do for typical 4th down decisions. The probability of scoring is a function of time remaining, and the probability of a failed FG is a function of field position. The calculation becomes:

wp [force FG] = p(FG fail) *1 + p(scoring) * (1 - p(FG fail))

Final Result with a Prominent Example

There are too many variables to show in a single illustration, so presenting the results required some creativity. There are timeouts, field position, time remaining, plus the result variable, win probability. To simplify things, the results are broken out into separate graphs for each possible number of timeouts remaining. Also, field position is represented by multiple lines on each graph, with each color denoting a 5-yard increment.

I'm going to go out of order so I can illustrate the results with a prominent example from Super Bowl 46 between the Giants and Patriots. Up by 2 points, the Patriots defense took the field to stop a final Giants drive that started on the New York 12 with 3:46 to play. It took only three plays for the Giants to make it inside New England's 35.

The graph below shows the win probability for defenses with two timeouts remaining. The horizontal axis is time remaining at the 1st down snap. The vertical axis represents the wp for the various situations described in the curves. The black line is the wp for allowing an intentional TD. The colored lines are the wp for forcing the FG attempt and each one represents the field position at the 1st down snap. Wherever the black "allow TD" line is higher, an immediate offensive TD would be preferable to forcing a FG.

You'll notice two abrupt vertical inclines in the colored curves for the force-FG option. The leftward one is due to the rapidly increasing probability of responding to a made FG with a score with respect to time. The second is due to the two minute warning. The force-FG option curves are so irregular because the time the defense would get the ball back is so irregular. The allow-TD curve is smooth because the time the defense would get the ball back is nearly immediate.


The Giants had three first downs inside FG range. The first (1) was at 2:52 at the NE 34. The second (2) was at the NE 18 immediately following the two minute warning. The third (3) was at 1:09 at the NE 7.

As the chart shows, 1st down #1 was well above the choke-hold zone. The probability of winning by forcing a stop and a FG attempt was greater than for trying to match an intentionally allowed TD in that situation. However, 1st down #2 was barely outside the choke-hold zone. My original analysis had suggested the TD be allowed at this point, but that's partly because retaining two timeouts on defense in that situation is so uncommon that the general Win Probability model discounted it. If NE had only one timeout left, it would have been a no-brainer to allow the TD on 1st down #2 (see below). The other reason is that the Giants played unconventionally and with abandon, passing the ball aggressively even inside FG range.

Curiously, Patriots coach Bill Belichick did not call a timeout between play #2 and play #3. Not until following a 1-yard gain on 1st and goal from the 7 did Belichick call his second timeout. On the very next play, Ahmad Bradshaw was (by most accounts) allowed to score the TD. Ultimately, the Patriots got the ball with 57 sec to play and one timeout remaining.

Had NE called a timeout prior to 1st down #3 and the identical events unfolded--NYG scoring a TD on their subsequent 2nd down--NE would have had an estimated 20% chance of winning instead of the 6% or so they had when they actually took possession. It's possible Belichick was hoping that somehow time would run out on the Giants. But it's more likely that, with two timeouts in his pocket, Belichick chose to wait to see how the first down play turned out before deciding to use them. If his defense held, he would use one, but if his defense allowed a conversion, he would wait to see how the subsequent first down play went. I think this was a mistake because at any time the very next play could be a touchdown, and he'd rather have the extra 39 seconds than an extra timeout on offense.

General Results

Here are the resulting charts for when an immediate TD is preferable to forcing a FG. (Suitable for lamination, coaches!) With no timeouts remaining, the situation is very dire, and there is a relatively large window for preferring to allow a TD (or for taking a knee). The solid black line is the wp for when the team on defense leads by two points. The dashed black line is the wp for when the team on defense leads by one point. Note: I chose a value of 47% for the chance of the offense converting a two point conversion in the case of the 1-point lead for the defense.

As a reminder, wherever the first down situation is above the appropriate black line, the preferred option is to force the stop and FG attempt. Wherever the situation is below the appropriate black line, the preferred option is to allow the TD.

 

With a single timeout, the window gets smaller as the team on defense's ability to respond to a made FG comes into play.

 


Here is the chart for two timeouts remaining, which we saw earlier in the example from Super Bowl 46.

 


With all three timeouts available to the defense, the immediate TD is almost never preferable to forcing the FG. There's just a tiny window with about a minute left and the ball inside the 15.

 

Conclusion

There's more work to be done. As pointed out by a commenter, if the offense misses its FG attempt but still has timeouts and time on the clock, the probability of winning by making a stop would be lower than I've estimated here. We also want to know the numbers for when the game is tied, or when the defense is up by three.

This is why football is uniquely compelling. In what other sport would it be better to allow your opponent to achieve a major score? When would you prefer that your opponent score a goal in hockey or soccer or lacrosse? When would you want your opponent to ever hit a three-pointer? What about baseball or cricket? Sure, you'd prefer to walk in one run to save four runs, but that's instinctively intuitive, the same way a football defense would normally prefer to give up 3 points instead of 7.

This may be the most complex, most challenging, and most counter-intuitive analysis I've done. There were some assumptions made in this analysis that could use some refinement, but I think we've got our arms around the problem, and we have a framework for further research. We also have a clear way of presenting the results in a way a coach can look up quickly in the heat of battle.

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