## When to Intentionally Allow a Touchdown, Part 5

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.

### 27 Responses to “When to Intentionally Allow a Touchdown, Part 5”

1. Nate says:

Are you assuming that the defense can force a FG at will? If the offense has a significant chance to convert to a new set of downs, that will impact the defense's win chance outside of 1st and goal situations. (Though, for some unfathomable reason, conventional thinking is apparently loathe to advance the ball once 'field goal range' is achieved.)

Does anything interesting happen further to the right on the zero timeouts left chart before you run out of model?

>We also want to know the numbers for when the game is tied, >or when the defense is up by three.

I'm not sure it's going to be all that complex unless you assume a non-zero chance to force FG - a tie is 50% chance to win, while allowing a touchdown is, at best, in the 30's.

Something that I wonder about in this context is clock management in the 3-4 minutes left time frame. Does a team on defense that is up by 6 or less prefer to see the opposition on first and goal at the 7 to first and 10 on the 35?

2. Brian Anderson says:

Of course, the coach of the offensive team should know this and instruct his team to NOT score when it's apparent the defense is letting them score.

Very cool analysis!

3. Alan says:

We're a long way from this state, but I wonder: if coaches really start consistently playing evidence-based football, won't this be seen as a problem? Those Chinese badminton players were punished for letting opponents win in the Olympics, right? Although if they just task officials with awarding penalties for not putting up one's best effort, it becomes such a judgement call, and players will be taught to do a kabuki dance of apparently unimpeachable "effort" that isn't effort at all. You could get comical situations where both sides have to pretend to be trying, but the real struggle is in the opposite direction than what they are portraying. It could be amusing for a while but ultimately be ugly and bad for the game. Not sure how to fix it though!

Something related that I've wondered about is when a team is down by 5 or 6 and driving in the last two minutes. So often I've seen teams I root for score the TD in this situation, but do so with still about a minute left. My instinctive feeling is "uh oh, I'm glad they scored but I wish they had taken a little longer to do it". I wonder if it's possible to model this situation mathematically or if it is too complex. Let's say for instance you're down by 6 and your opponent gets hit with a pass interference penalty in the end zone. So now you've got a first-and-goal at the one yard line, but there's a minute or two left. Your opponent has no timeouts. Couldn't it be worth the risk of not scoring at all to take a knee once, or even twice, before you try to punch it in? There are of course endless variations on this situation with different downs and distance, timeout status, and for that matter the strength of each team's offense and defense (goal line O/D going one way, and passing O/D the other way).

4. Keith Goldner says:

Great stuff, Brian. Any way you could add this to the win probability calculator? Maybe just at the bottom of the results saying "Allow the TD!!" if in that particular situation?

5. Mark M says:

"In what other sport would it be better to allow your opponent to achieve a major score?"

You can also ask the reverse for the offense. In what other sport would you be in a worse situation by achieving a major score? I guess the answer would be any time-bounded game in which you score points in multiples and the person with 'possession' scores the majority of the time.

I'm thinking of some kind of time-based darts match. Let's suppose you each alternately throw three darts each, the winner is the one with the most points after 10 minutes and you have to take all your throws within 30 seconds. If a player comes to the oche down by one point with 29 seconds left, he'll wait til the last moment before throwing. You'd much rather take all the time to score two points than score 100 pts and use only 10 seconds. If you're the player off throw, if there was some mechanism by which you could award your opponent 100 pts to get the throw back, you would use it.

6. Anonymous says:

Very much like the analysis, but find the probability curve starting near 0% at 50s to be unrealistic.
Maybe I've watched too many Lion's games, some on offense, (depressingly) more on defense, where 50s seemed more than enough time to score from the 20.

7. Anonymous says:

"In what other sport would it be better to allow your opponent to achieve a major score?"
Basketball is the first that comes to mind. But I guess only in cases where both offenses score at a high percentage.

8. Anonymous says:

"This is why football is uniquely compelling. In what other sport would it be better to allow your opponent to achieve a major score? "
Auctions when the price gets above the value, or when the value relative to other future value is low enough where it reduces the capital to bid on other more favorable auctions.

Or as some statisticians call it "winner's curse". You only can win the bid when no one else wants it, so you will be forced to bid high and "over pay" if the other teams are rational. This of course happens in every sport in terms of free agency but is less of an issue when there is no salery cap and a team has a huge payroll. Which of course then brings us to the concept of "moneyball" and how to do more with less with advance statistics and we come full circle.

9. Anonymous says:

I am a bit confused by the conclusion. So when there is 51 seconds left or less, it is never acceptable for defense to "let opponent score"? this seems completely wrong and the exact opposite of reality. The opponent will simply run out the clock and win!

Or are your chances so low of winning at that point that your odds are better of getting a fumble or turnover? or missed kick?! Down 2 or 3 I might buy that, but only down 1?! A TD and two point conversion can tie it and although odds are against you, it's a almost sure thing for opponent to just take a knee a few times and kick the fieldgoal. What am I missing? Am I reading the graph wrong?

10. Anonymous says:

ftr, in basketball just about every game does that, forcing the offense to go to the foul line and get points, in order to get possession back.

in baseball, the point is moot. It is also moot in hockey, lacrosse, etc, where scores are equal (1 point).

11. Anonymous says:

since offenses obviously know this as well, and will likely try to stop the ball at the 1 yd line and kneel, it seems like in this situation that the ideal play by the defense would be to go with an 11 man blitz, and try to sack the handoff etc.

do everything to push them back, and to knock the ball out, and if you don't, then give up the TD.

Of course, the major "flaw" in this analysis is that the defense does not have the power to give up an intentional TD. They can offer it, but they cannot force it.

Just like the Giants did (or tried to do). It shows that the offense was aware of this situation before, and most certainly everyone is aware of it now since it happened in the SB.

12. Alex says:

Depending on what you mean by 'major score', some sacrifices in chess would probably qualify.

13. Unknown says:

Agree with Mark M, this would likely only happen in a game where you score in very different multiples. Anyone know if analogous situations occur in rugby or Aussie rules?

Leading by three in basketball, some coaches foul before the 3pt shot, but that's not the same since you're allowing a chance at fewer points, not more. Sometimes it's better to miss the free throw, but again, not quite the same.

14. Unknown says:

Anonymous, with 50s left and the defense out of timeouts, the offense really only needs to run one more play. They bring out the fieldgoal unit, wait till the last second, and kick. If they're dumb enough to run a play then by all means, let them score, but we're assuming they will only run one play if they can. If there's a few seconds left, they can onside kick to run out the clock. Or go offside on the kickoff, if you're Wisconsin a few years ago.

15. Anonymous says:

Don't know about Aussie rules football, but this strategy wouldn't work in rugby. In rugby, the team that gets scored upon has to "kick-off," and has a relatively low chance of recovering the kickoff. Also, it's quite a bit harder to willfully score a try in rugby--there's no equivalent of throwing the ball deep down the field.

16. Anonymous says:

"Of course, the major "flaw" in this analysis is that the defense does not have the power to give up an intentional TD. They can offer it, but they cannot force it."

Is there a rule against a defense carrying a ball carrier into the end zone?

17. Brian Burke says:

Re: basketball. Fouling to get the ball back is not a "major score." It's not like giving up a 3-pointer. It's like giving up a FG to try to get your own TD.

Thank you for all the great criticisms and comments. A lot to digest and this analysis is only the beginning. More work to be done.

18. Dave says:

Fantastic analysis. It's cool to see studies like this push the sport forward (eventually).

19. Anonymous says:

Disagree with your disagreement regarding basketball. This 'major score' distinction is not relevant.

in football you cannot make a team kick a fg, but you can avoid tackling and thus give them a TD. If you could force them to kick the field goal on 2nd down on the 20 with 60 seconds left, you would.

It is identical to basketball where the defense trades points to get possession. In both cases, time is the enemy and is what is addressed.

the fact that it is "a major score" doesn't matter. It is only a 'major score' because that is the only one the defense can surrender. In both football and basketball, the defense gives up points for possession.

What IS different, is that in football, you give the points when you are leading. That almost never happens in basketball. And the reason for that is because of the near certainty of a short FG in football.

20. Anonymous says:

In basketball, if the scoring environment is high enough, a team up by one point, on defense with little time remaining, could have incentive to foul and risk giving up the lead to get the last shot.

21. James M says:

I can think of two slightly similar situations in rugby competitions where there are bonus points for scoring 4 tries in addition to the 4 points you get for winning a game.

If a team has a big lead and has scored three tries and there is not much time left they would be better to allow the opposiiton to score to give them a chance to score a 4th try before the end of the game.

A second scenario is where a team with a big lead that has scored its third try with little time left should not attempt a converison kick (worth 2 points but not always as easy as its american cousin the PAT) as that can take about 60 seconds. I once saw a teeam in this position (Munster vs Connacht in the Magners league on St. Stephen's day a few years ago.) Where I said to my sister that Munster should not attempt the conversion after their third try with about 2 minutes left as that was wasting time and they had a big lead although as they managed to score their 4th try in stoppage time it didint matter but it could have.

Obviously this is maximising the schance of getting a bonus point as the game is won in this scenario but it is an area where conceding points or forgoing the opportunity to score could be to a team's overall advantage

James

22. Brian Burke says:

I don't understand why the anonymous (3 comments above) disagrees with me. The comment explains a bunch of reasons why bball and football situations are *not* analogous, which was my point.

23. Anonymous says:

In the 1994 Caribbean Cup soccer championship, a match between Barbados and Grenada saw both teams intentionally score on their own goal due to a quirky rule that determined which team would advance. Google it.

24. Brian Burke says:

Oh yeah. I remember that. I think I linked to it a couple years ago.

25. Ken R. says:

Brian,

Thanks ... Really enjoy this analysis. I recall in the Giants/Pats Super Bowl, that when the Giants got the ball back in the closing moments, recognizing Eli's track record with 4th qtr drives - I was thinking that at that moment, the Pats should take some outrageous risks... All out blitzes, anything to force the issue. Not exactly giving up a TD, but recognizing that you have to kill the drive immediately or give up the TD immediately. Either way Brady gets the ball with enough time to either win the game or end the game. So, yeah, Belicek intentionally gave up the TD, but I felt he could have taken a more stealthy approach to getting the ball back sooner. Instead, he gave up the TD too late.

26. Zipple41 says:

I don't know if Belichick's decision to delay using timeout #2 was necessarily a mistake. I think he concluded (and rightly so) that if he did take that timeout, Tom Coughlin would surely have the time to tell all of his players deliberately not to score. By getting the next play to occur as soon as possible, there was a higher likelihood that the Giants would make the wrong move and score.

When he finally DID call that timeout, there was no other choice, time was too low. He had to give them time to discuss it which, sure enough, they did. Bradshaw was told specifically NOT to score, but fell into the endzone anyways for who-knows-what-reason.

27. Hugh Nightingale says:

The one thing that really bothers me in the analysis is that of the probability of the FGA.

For there seems to be no allowance for the effective range of a kicker or consideration that the decision to convert or kick within 2-3 yards of the ER will slew.

Muddying the waters even more so is the difficulty around the Def38 having to decide on 4th Down whether to attempt the conversion, punt or FGA.

On or within the Def5 the expectancy is that the FGA will be converted .995 of the time but the true probability is again slewed because of the score/field position.

The most reliable probabilities will therefore probably fall within the range Def6-35, plus or minus, and it is difficult to get a true reflection due to limited data even when a FGA is attempted close to the end of a half or OT let alone a critical one with various permutations from leading by six to trailing by three specifically near the end of Q4.

Brian - can you revisit the data assuming the FGA has to taken from up to five yards closer than the raw data presumes, moderated of course in the range Def6-10, and/or analyse critical FGA decisions inside the LFM on a rolling basis for the last 10 seasons in +6 to -3 situations.