Going For It: Applying The Bellman Equation To Football
Berkeley economist David Romer wrote an interesting paper in 2005 entitled Do Firms Maximize? Evidence From Professional Football in which he posits that football coaches are far too reliant on kicking on 4th down. Romer believes that teams should be going for it far more often in order to maximize their team’s chances of winning. Statistical modeling of his theories has shown an increase of just over one victory per NFL season. Some coaches appear to be early adopters of some of these philosophies, but none have been remotely as aggressive as they should be to optimize their statistical chances of victory. How do we know?
The Bellman equation says so:
Ei Di(gt) Vi = Pgt + Bgt Ei Di(gt+1) Vi - egt
Romer’s general guidelines can be summarized as follows:
A team facing fourth-and-goal within five yards of the end zone is better off trying for a touchdown.
At midfield, go for any fourth down within five yards of a first down.
Even on its own 10-yard-line — 90 yards from the end zone — a team within three yards of a first down is marginally better off, on average, going for it.
Revolutionary. It would certainly change the game of football and it might offer a significant advantage to early adopters. Adapting team and fan psychology to your new strategy would be crucial.
If you scroll down to Trend #3, you’ll see an interesting article on Pulaski Academy in Arkansas where head coach Kevin Kelley decided to stop punting in 2005 after reading Romer’s paper. To good effect. Coach Kelley hates punting
How would this fare at the college level? Aside from the psychological aspects to be considered for both your team and the opponent, I’d venture that the net wins to be gained would be even higher than professional football since college offenses generally exhibit much higher levels of production vis a vis the average college defense. I think this is groundbreaking stuff and though I don’t endorse Romer’s guidelines totally, I do think he’s far closer to empirical reality than most modern coaching folkways and mores. This is a first step in applying Moneyball type statistical modeling to football; I’m excited by the possibilities.
Your thoughts?
November 13, 2007 at 2:12 pm
Seems like he left out some variables. Can you quantify momentum?
November 13, 2007 at 2:15 pm
“I’d venture that the net wins to be gained would be even higher than professional football since college offenses generally exhibit much higher levels of production vis a vis the average college defense.”
My first impression is the opposite, since the talent disparity is much less in the NFL than in the college ranks. I would think that the net wins in college ranks would be less where a more talented team would still have an advantage over the length of the game, or am I missing something in his equation that balances that out?
November 13, 2007 at 2:35 pm
HJ:
No. Momentum is a psychological state and its validity is determined by your ability to sell your team and fans on the empirical reality of what you’re doing rather than succumbing to fraternal peer pressure. Momentum, indefinable as it is, also cuts both ways.
November 13, 2007 at 2:41 pm
“Seems like he left out some variables. Can you quantify momentum?”
…Or the value of getting a struggling offense off the field during a failed drive and regrouping.
I have to disagree with this trend when you are backed up in your own territory because of these variables.
However, I can see how going for 7 instead of kicking a field goal may be beneficial, statistically (hell, we kick field goals when down by 4 from the 1…mind-bottling). When a drive sputters and a field goal is forced, it’s a huge letdown to the offense. I can definitely see how that would drain momentum.
November 13, 2007 at 2:41 pm
srr50:
You’re absolutely right in terms of net total wins for a dominant team since fractional points don’t make a difference - it’s equally a win whether it’s 49-14 or 48-14. However, given that any productive college offense generally averages more yards per play than their NFL equivalent (and there are more possessions in a game), any team with good offense may get a higher Go For It modifier vis a vis the punters. That modifier, as you point out, may not necessarily means more wins if 8 of your 12 games were lopsided anyway.
November 13, 2007 at 2:43 pm
Fuck this.
November 13, 2007 at 2:45 pm
“Can you quantify momentum?”
Mass times velocity. Or, alternatively, the time differential of kinetic energy.
November 13, 2007 at 2:46 pm
From the Easterbrook ESPN column I linked above:
“Nearly three-quarters of fourth-and-1 attempts succeed, while around one-third of possessions result in scores. Think about those fractions. Go for it four times on fourth-and-1: Odds are you will keep the ball three times, and three kept possessions each with a one-third chance of a score results in your team scoring once more than it otherwise would have. Punt the ball on all four fourth-and-1s, and you’ve given the opponents three additional possessions. (It would have gotten one possession anyway when you missed one of your fourth-and-1s.) Those three extra possessions, divided by the one-third chance to score, give the opponent an extra score.”
I think that’s an interesting and straightforward means of understanding the issue at hand…
November 13, 2007 at 2:53 pm
Scipio,
That’s interesting, but a little oversimplified. The expected value of an offensive possession varies with several factors, including field position, time remaining in the half, the time the opponent’s defense has had to rest in between drives, etc.
While the statistical arguments might favor going for it more often in a limited set of circumstances, a flat policy against punting seems like a ham-handed solution. It can’t honestly be advisable to go for it every time your team is facing a 4th and 25 from the 1 yard line. The chances of converting would be fairly low (maybe 20% tops), while the chance of scoring a touchdown with a first and goal at the 1 yard line is pretty high (depending on the team, maybe 70-99%) compared to the chance of scoring a touchdown from around the 30-40 (assuming an average punt and coverage).
November 13, 2007 at 3:03 pm
“It can’t honestly be advisable to go for it every time your team is facing a 4th and 25 from the 1 yard line.”
All the variables listed by Scip were 4th and five yards or less.
November 13, 2007 at 3:12 pm
Put another way, if you punt you have 100% chance of NOT converting nor of scoring.
November 13, 2007 at 3:18 pm
srr50,
Correct. But the Easterbrook column discussed at length a high school team that has basically stopped punting altogether. Predicting the success of that policy requires an analysis that is sufficiently complicated that the 4th-and-1 example Scipio cited in his last post becomes misleading.
Obviously, the decision whether to punt is guided by a pretty complex surface in a multi-dimensional situational space. My takeaway from this thread is that there is good reason to believe that common football knowledge (of the type so lauded by mindless nitwits like echeese) provides too conservative a set of guidelines in regards to punting. But the optimal solution is almost certainly not “always go for it.” There are some circumstances in which the punting remains the best option.
November 13, 2007 at 3:23 pm
Uh…I was told there wouldn’t be any math on this website…
November 13, 2007 at 3:25 pm
Put another way, if you punt you have 100% chance of NOT converting nor of scoring.
Well, a significantly reduced chance, anyway. Depending on the circumstances, of course. There are a few ways that punting can turn out in your favor. First, the returner could muff the punt and the kicking team could recover it. This is pretty low probability. Second, your defense could create a turnover and possibly improve your team’s field position. Third, your defense could limit the yardage gained by the opponent and a poor punt by the opponent or good return could improve the field position relative to when you punted in the first place.
There are probably others. But the fact that these possibilities exist mean that punting doesn’t quite give you a 0% chance of improving your team’s lot.
November 13, 2007 at 3:30 pm
Brick:
It actually makes much more sense to be as aggressive as Pulaski is if you consider that the kicking game (both field goals and punts) in high school is far more high risk/low reward. Pulaski also runs a dominant spread offense while possessing a weak defense. My guess is that they don’t find themselves in 4th and long very often - particularly when they call plays as if they had 4 downs to work with.
4th and 25 is extreme, but you could make very strong arguments for Pulaski to go for it on 4th and 10.
NiceyNice:
I think you nailed it. In fact, it has even become standard football rhetoric that a long interception “is just like a punt.” That’s true. In fact, that very precisely defines what a punt is - a downfield turnover. Doesn’t sound quite as appealing now…
November 13, 2007 at 5:14 pm
Mike Leach probably gave Romer his inspiration:
To a team that gains as many yards as Texas Tech, the usual boring, penny-ante yard-eating tactics - punts, penalties - are trivial. Field position is simply a thing to improve. Cody Hodges, who has spent the last four years marveling at Leach’s in-game refusal to accept that his offense might have to be so conservative as to punt, says, “There’s been lots of times I’m on the sidelines, and I’m like, ‘Oh, my God, we’re going for it!’ We went for it on fourth and 5 on our own 23 - in the first quarter. We went for it once on fourth and 18 - and we were ahead.” E. J. Whitley, an offensive lineman, says: “If you’re on this offense, you expect to score. Most offenses on fourth down are coming off the field. On fourth down we expect a play to be called. Because we haven’t scored yet.”
http://www.nytimes.com/2005/12/04/magazine/04coach.html?pagewanted=all
November 13, 2007 at 5:33 pm
You need to ask Barry Switzer about this.
November 13, 2007 at 5:41 pm
In other words, coaches won’t do what’s perceived by the majority to be dumb.
November 13, 2007 at 6:22 pm
I have a copy of a book by Darrell Royal where he repeatedly praises the virtues of the quick kick on 3rd down. Of course, it was written in 1963 when most games seemed to end with a score of 14-10, so it was probably smart to play field position in a defensive struggle.
In a game with 2 strong defenses, it makes sense to punt more as the extra 30-40 yards the opposing offense has to go significantly reduces the likelihood of them scoring (not to mention that the chance of you making the 4th down is also significantly reduced). With modern offenses, 30 to 40 yards does not matter quite as much. I think that coaches have just blindly copied the conventional wisdom that made sense in the 50’s but is sadly outdated.
November 13, 2007 at 7:04 pm
I would argue that there is no such thing as a statistical imperative applicable to all situations regardless of other variables.
Amount of time left in the game. The score. The relative strength of the offenses and defenses. The weather conditions. Tendencies. How your players are performing on that particular day. How much you’ve bet on the game.
That’s why I freakin’ hate the charts coaches have detailing when to go for 1 or 2 on the XP. Or rather, I don’t hate the charts, I just hate the blind adherence to them.
November 13, 2007 at 9:51 pm
By the same token, I’ve long thought that teams should go for two a lot more than they do. I think the main problem is that most coaches think about the next possession rather than the gross outcome.
In terms of punting… I think that something that is missing is an altogether different approach to the punt play. If you are going to punt, it seems to me like you would be better off teaching your QB to pooch punt and having punts of 25-35 yards with no return (and a much greater chance of a turnover from inadvertent touching) than to punt it 40 yards to a special teams that is set up to try for a block or return. Additionally, you can look for a favorable defense pre-snap then decide whether you want to go for it or pooch it. The traditional punt completely gives up the offensive initiative and it really doesn’t have to.
November 13, 2007 at 10:10 pm
Disregarding the strengths and weaknesses of individual teams, it’s not all that far-fetched to imagine a continuum of threshold 4th-and-distances at which it makes sense not to punt (or kick) based on field location:
Own 1: 4th-and-an-inch
Own 20: 4th-and-a-foot
Own 50: 4th-and-a-yard
Own 20: 4th-and-2-yards
Own 5: 4th-and-goal
or something like that. It is absolutely a risk/reward tradeoff, and most coaches are probably ill-equipped to imagine the possibilities.
November 13, 2007 at 10:10 pm
Err, those last two should be the opponent’s 20 and 5, respectively.
November 13, 2007 at 11:54 pm
JimmyJazz, in the history of football, nobody has ever been faced with 4th-and-an-inch from their own 1 yard line. However, I get what you mean and agree with you.
November 14, 2007 at 4:26 am
A solid foundation of thinking here. I agree with the basic premise. Jimmyjazz has a good way of expressing it. Seems clear the relative strength of the offenses and defenses are crucial to accurately assessing the probabilities. The “table” of decisions on distance and field position would be different for different teams and their opponents.
This starts to be too much work, or too nerdy work, for a coach to undertake, practically speaking. Not so at the pro level, of course.
Adding to the variables, would be one or two about the specific day in question, like the weather, home or away, todays’s execution level vs norm, etc.
Basically, no problem to have a spreadsheet updated every week via BCS type calcs and the game day variables, and updated during the game for the execution variable.
At any rate, everyone punts far too much. This thread has been an eye opener for me, I feel like the dumbass I am.
November 14, 2007 at 4:51 am
Some good points. I think a lot of the “book” on going for it was developed 50 years ago when teams averaged 13 points per game and field position was king. Now teams average over 20 points per game.
A coach should combine this calculation with a simple option tree analysis- i.e. “What is the reward if I make it and what is the cost if I don’t?” For example, leading by four points going into the half (or end of game), with 80 seconds left to play, seems to be a situation that calls for a field goal or punt, depending upon field position.
It’s interesting that we have a D-1A coach who happily foregoes the 4th down kick- Mike Leach, and he’s had success with that approach. I assumed that his success was due to his wide open offense, and the need to maximize plays to get it rolling, but it may be that he just has the odds figured out better than others.
November 14, 2007 at 6:50 am
“JimmyJazz, in the history of football, nobody has ever been faced with 4th-and-an-inch from their own 1 yard line. However, I get what you mean and agree with you.”
Jesus, that’s embarassing. Thanks for catching it.
Drunk or stupid, you make the call.
November 14, 2007 at 9:55 am
My thought is that you should only punt when your opponent is releived that you are punting instead of going for it.
During the Tech game I am always releived when Tech punts.
November 14, 2007 at 10:18 am
Gregg Easterbrook did most of his research by consulting the Left Behind books.
November 14, 2007 at 10:32 am
I’ve read Romer’s paper and it is very sound as far as variables are concerned. As a high-level research effort he does as well as can be expected. Based on play-by-play data one can assign an expected value of the following scoring plays from any situation.
So, on average, if your team has a first-and-10 on the opponent’s one-yard line you have an expected value of 5.55 points. The difference between this and 7 points is due to failing to score, kicking field goals, and giving up points on the ensuing possession (value of a kickoff is -0.6 points). Similarly, the value of a first-and-10 on the opponent’s 41-yard line is 2.4 points, which is equal to a field goal (remember the -0.6 from the kickoff).
It becomes a straightforward calculation based on punt field position improvement, etc. to determine when you should kick and when you shouldn’t. Basically, will it improve your expected value or not?
Obviously things like end-game scenarios aren’t included in this. When down by two points and facing fourth-and-seven on the opponent’s 15-yard line with one second remaining, clearly a 95% chance at three points is better than a 50% chance at a touchdown regardless of the EVs.
And, also as stated, team morale, etc. aren’t calculated. Getting the team’s buy-in is imperative to implementing something like this.
November 14, 2007 at 10:57 am
Obviously things like end-game scenarios aren’t included in this. When down by two points and facing fourth-and-seven on the opponent’s 15-yard line with one second remaining, clearly a 95% chance at three points is better than a 50% chance at a touchdown regardless of the EVs.
Good point. In that situation, the only relevant expected value relates to the binary values “win” (1) or “lose” (0). So, the field goal situation gives you a 0.95, while the touchdown situation gives you a 0.5. The obvious choice would be to kick the field goal.
November 14, 2007 at 11:50 am
My two nephews played QB at Pulaski (all before the “no punt decision”) and one is in the national high school record book (several times) for yards per game, completion percentage, touchdowns, and a few others — some are still national records.
Surprised to see their coach adopted that position completely, but there is no doubt they can throw the ball in any situation.
November 14, 2007 at 1:37 pm
what I meant to say that you should go for it if your opponent is releived that you are going to punt.
I never want Tech to go for it.
November 14, 2007 at 1:49 pm
Most coaches coach not to get fired rather than to win. When you do something unconventional, it had better work. Think about Mackovic if he’d failed on 4th-and-inches from his own 30 against Nebraska. If they get the ball back and punch it in, we wouldn’t have had Mackovic for the 1997 season. (Which would have been bad because…ok, I’m stuck.)
Les Miles is an exception, but he’s insane. The angry Pirate is an exception, but his employer knows they can’t do any better. The other 115 D-1 coaches would rather play it safe rather than play the statistical odds to their advantage.
November 14, 2007 at 7:16 pm
Some great dialogue thus far.
My thinking is in line with LH Scott’s…the traditional punt is not logical. A 4th and 3 from the 50 should mean your first team offense stays on the field and your QB takes it in shotgun. He has three options: Run, pass, kick.
Imagine: the QB takes the snap. The defense has things covered. He rolls right. No receiver breaks open. He starts to get pressure as he rolls…and he calmly punts the ball on a line drive straight down the field to be downed after it dribbles forward inside the opponent’s 10. Oh, and a defender better not try to block my gunner/wide receiver until he’s sure the ball was actually punted - pass interference you know. You want to give me another converted 4th down?
The run/pass/kick option means you’ve just created another tool to mitigate the “risk” of going for it on 4th down.
Anyone think Vince Young might be pretty good at this?
November 14, 2007 at 8:33 pm
Scipio,
Maybe we should just do away with helmets, ban the forward pass and start calling the huddle a “scrum.”
November 14, 2007 at 9:27 pm
on the punt/pass/kick, isn’t there some rule like illegal man down field that would cause a problem? or just assume all lineman stay behind the LOS till you pooch it away?
November 14, 2007 at 9:55 pm
Exactly Scipio. We’ve seen time and time again that plays where the defense has to deal with conflict between run/pass responsibilities are when the offense has the chance for big plays. Vince thrives in that no man’s land. If you throw special teams responsibilities in the mix then you get a whole secondary of really confused players. How about a defensive end who is coming in for the sack before he realizes that now he’s about to be flagged for roughing the kicker?
November 15, 2007 at 6:51 am
I don’t think that the QB can be roughed like a kicker if he makes any move to run or pass
November 15, 2007 at 7:12 am
punter is protected whenever he is in the act of punting.
November 15, 2007 at 8:06 am
I don’t think that’s true. You can’t run a play and then just punt before the ball carrier is about to get tackled and claim roughing.
November 15, 2007 at 8:16 am
Nerds.
November 15, 2007 at 8:25 am
“I don’t think that’s true. You can’t run a play and then just punt before the ball carrier is about to get tackled and claim roughing.”
I don’t know the answer, but I have to think with HJ. Logically, it has to be tied to some kind of punt formation. Otherwise, why not always run the punter around in a fake punk run and have him kick it just before he’s tackled and get the penalty? It has to be tied to some kind of initial indicator of punt. Like a reverse fair catch, or something.
November 15, 2007 at 8:46 am
It’s a subjective call, obviously. But it’s not tied to formation. Once the kicker establishes the act of kicking or punting (usually this is established with the plant foot), they are supposed to be protected. Now, granted, if the guy is running around and kicking on the run, the ref is likely to give a lot more leeway to the tackler. In my mind it’s not even a question of the exact conditions in which it’s called but the fact that it will cause doubt on the part of the defender… unless of course we are talking about Robert Killebrew. He would tackle the ref, eat the yellow flag, take a crap on the punter, then roll onto his back so Mack could scratch him on the spot that makes his leg twitch.
November 15, 2007 at 8:51 am
This is the NFL rules
A member of the receiving team may not run into or rough a kicker who kicks from behind his line unless contact is:
(a) Incidental to and after he had touched ball in flight.
(b) Caused by kicker’s own motions.
(c) Occurs during a quick kick, or a kick made after a run behind the line, or after kicker recovers a loose ball on the ground. Ball is loose when kicker muffs snap or snap hits ground.
(d) Defender is blocked into kicker.
The penalty for running into the kicker is 5 yards. For roughing the kicker: 15 yards, an automatic first down and disqualification if flagrant.
In this case it would be a quick kick
November 15, 2007 at 9:05 am
Alright, after reading the NCAA and NFL rules for the penalty… I am just wrong
Anyhow, let it not distract from the broader point that offenses could be dictating on 4th down rather that playing an extra down of defense.
November 15, 2007 at 11:31 am
Your point is still valid about Vince and the conflict that the defense faces
November 16, 2007 at 8:51 am
It’s an interesting argument, but an academic one. For the same reasons managers and GMs make provably inferior short- and long-term choices, football coaches will do the same.
Namely, we live in an anti-intellectual country, and this is a strategy that will, on some days, fail spectacularly. And when you go to the media after the game and try to explain your reasoning, it’ll never, ever get through to the public. Not only will the media be looking to tear down your strategy, but the public at large will be seeking for you to fail.
A football coach would be run out of town just as fast as Paul DePodesta was run out of LA. There are too many Bill Platschkes and dumb fans in the world. They’d never accept the games when it didn’t work.
November 16, 2007 at 8:53 am
Just to follow up: that’s why the conventional wisdom is embraced so readily. You don’t take nearly as much heat when you don’t win.
If you do something outside the norm and you lose? You’re the goat.
November 16, 2007 at 10:22 am
Peter:
Agreed. However, today’s academic argument is often tomorrow’s conventional wisdom. The mechanism that allows progress in an anti-empirical world (I’d argue that this is as much a human phenomenon as American) is competition.
On base percentage and slugging percentage was the vocabulary of a small society of baseball nerds who were roundly mocked by baseball traditionalists and their media mouthpieces for 25 years - now it’s the standard of evaluation for a hitter by any credible GM or fan. Now it’s the metaphysical purists who are mocked for their ignorance and mysticism.
The A’s are probably a good example of how this sort of paradigm shift could occur in college football: a small market team with nothing to lose and absent major market media pressures overachieves for several years in a row - humiliating several big market/big money teams in the process. Perenially underachieving giant (Boston) rolls the dice and embraces these ideas (bulwarked by their own huge salary cap and resources) and wins two titles in three years. Others take note. Precedent. The cautious are now emboldened. Copycatting ensues…poor Billy Beane, he’s about to be operating in semi-efficient markets with 1/4 the stack.
Consider the growth of the spread offense in Texas High School Football. Not exactly a place replete with visionaries. But getting your ass kicked over and over by a bunch of mediocre athletes will get a coach’s attention…
What do you think would foster the uptake of some of these ideas?
November 16, 2007 at 10:52 am
Legitimate success by a legitimate title contender. Just like baseball.
November 17, 2007 at 11:49 pm
“particularly when they call plays as if they had 4 downs to work with.”
The effect on play calling is a big reason why I also support this line of thinking. You see it a little bit with Tech. 3rd and 8 or more is a killer for most offenses, but Tech just throws a quick crossing pattern for 7 yards and gets in a manageable 4th down situation. That crossing pattern probably has a 75-80% chance of being successful, while the odds of hitting a deep or intermediate throw against a defense expecting a pass are much lower.
One other thing I would like to see more of is the “throw it across the field” lateral on kick returns like the Titans and SMU used in those famous finishes. I have no statistical evidence to back that up, other than the fact that kickoff returns for TDs are very, very rare unless you have a Devin Hester type at returner. I just think it would work really well if you do it when the coverage team isn’t expecting a trick return.
Plus if you show you will run this return during normal game situations, then the other team would have to change its coverage to account for it. That would probably mean that less defenders would converge on the initial ball carrier, which would then make your normal kick returns more successful.
November 22, 2007 at 4:46 pm
No, you’re right, Scipio. It’s a slow process, but today you’ll hear mainstream journalists talk about VORP pand OPS and win shares and all the rest. The change is much slower on the field in the management offices, of course.
The other thing that’s worth mentioning is that it’s much more likely to be something we see in college than the pros. So who knows?
I do think, though, that the conservatism at big programs runs from the huge fanbases to the athletic directors; I think you’re right that it would take a medium-size school to go on a real kind of tear before change would trickle down (up?) to the big programs.
Great food for thought, though.
November 22, 2007 at 4:56 pm
One other thought: thinking back to Vince Young’s college days (which is always fun) - can you imagine any situation where it shouldn’t be four down territory? I mean, seriously - even a conservative coach should consider every possession to be “four-down territory,” no matter where on the field.
Thinking about that, maybe it wouldn’t be a medium-size school that brought the revolution. Maybe someone like Urban Meyer decides “I can get 10 yards every time with Tebow - or at least often enough to justify going for it every time.” Maybe it would take a big name at a big program going on a special kind of tear to make others realize 1) it CAN work and 2) you might get left behind if other big boys are gonna do it.
Again - all fun to think about…
December 6, 2007 at 9:18 pm
Any Texas fan can imagine a lot of situations where going for it on 4th down could produce a nigh-invincible offensive juggernaut. Does anyone think that there’s was a defense in the NCAA that could hold ricky williams or earl campbell to under 2.5 YPC if you just handed him the ball every play? With even a token passing threat, that seems like a no-brainer offense. Maybe throw in a pass 10% of the time to keep the defense honest, and cram it down their throats the rest of the time.
August 25, 2008 at 12:25 pm
“Aside from the psychological aspects to be considered for both your team and the opponent, I’d venture that the net wins to be gained would be even higher than professional football since college offenses generally exhibit much higher levels of production vis a vis the average college defense.”
The better offenses work both ways. Since the article advocates going for it on your 10 yard line, it might be worse off in college since failing there is a lot more likely to give up a TD.
“Nearly three-quarters of fourth-and-1 attempts succeed, while around one-third of possessions result in scores. Think about those fractions. Go for it four times on fourth-and-1: Odds are you will keep the ball three times, and three kept possessions each with a one-third chance of a score results in your team scoring once more than it otherwise would have. Punt the ball on all four fourth-and-1s, and you’ve given the opponents three additional possessions. (It would have gotten one possession anyway when you missed one of your fourth-and-1s.) Those three extra possessions, divided by the one-third chance to score, give the opponent an extra score.”
That can’t be right - you aren’t giving your opponents extra possessions by punting unless you would have run out the clock if you had made the first down. Punting gives the other team a possession with worse chances of scoring on you than failing on 4th down does.