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For more from Chase and Jason, check out their work at Football Perspective and The Big Lead.

## David Romer’s paper II

This is a continuation of yesterday's post, so you might want to read that one real quick if you haven't.

Assume you are coaching an average team against another average team and assume it's very early in the game.

- Would you give the other team half a point for the right to receive the kickoff in both halves? A full point?
- Would you rather have the ball, first-and-10, on your own 14 yard line, or would you rather your opponent have the ball, first-and-10, on their 14 yard line?
- If you had first-and-10 at your opponent's 42 yard line, would you trade it, on the spot, for a guaranteed field goal?
- Would you rather (A) be leading by three points and have the ball, first-and-10, at midfield, or (B) be leading by seven points, but your opponent has the ball, first-and-10, at midfield?
- If you had first-and-10 at your own one yard line and you knew with 100% certainty that you could execute a 56-yard quick kick (with no return), would you do it?

If you want to answer these questions, then you need a way to translate situations into point values. Is 50 yards worth three points? Four points? How many points is possession of the ball worth? And do the answers to these questions depend upon where the ball is? The cornerstone of Romer's paper is putting a point value on every situation, and I'll explain later exactly how he does it. If you are trying to maximize your point differential, which, early in the game, is essentially equivalent to maximizing your probability of winning, then the paper suggests that each of the above decisions is between two equally attractive options. If you found them to be difficult decisions, then your intuition matches up with Romer's model.

Here is a picture that summarizes his model:

Starting at the left, note that if the yard line (on the *x*-axis) is 1, the associated point value (on the *y*-axis) is -1.6. That says: if it's a tie game and you have first-and-10 at your own one yard line, then it's really not a tie game. You're morally trailing by 1.6 points. At about your 15 yard line, the point value is zero. That says that you should be indifferent between having the ball at your own 15 and having your opponent have the ball at his 15. At midfield, the point value is about +2.

What doesn't show up on the chart is the kickoff situation. If your team is lining up to kick off, you are in a -.6 point situation. This jives with the fact that +.6 is the value of having first-and-10 on your own 27, which is roughly the average starting field position after a kickoff. The fact that a kickoff is worth -.6 points is crucial, because scores are always followed by kickoffs. So a field goal isn't really worth 3 points. It's worth only 2.4 points. Likewise, a touchdown is worth 6.4 points.

Let's look at those questions again:

**Would you give the other team half a point for the right to receive the kickoff in both halves? A full point?**- as discussed above, the right to receive a kickoff is worth about .6 points, so either of those trades would be a close call.**Would you rather have the ball, first-and-10, on your own 14 yard line, or would you rather your opponent have the ball, first-and-10, on their 14 yard line?**- as discussed above, the 14 or 15 yard line is the place where the point value is zero, so this is an even trade either way.**If you had first-and-10 at your opponent's 42 yard line, would you trade it, on the spot, for a guaranteed field goal?**- the point value of a first-and-10 at your opponent's 42 is around +2.4, the same as the value of a field goal. Again, even trade.**Would you rather (A) be leading by three points and have the ball, first-and-10, at midfield, or (B) be leading by seven points, but your opponent has the ball, first-and-10, at midfield?**- a first-and-10 at midfield is worth about +2 points, so the difference between having the ball and not having the ball is 4 points. Even trade.**If you had first-and-10 at your own one yard line and you knew with 100% certainty that you could execute a 56-yard quick kick (with no return), would you do it?**- first-and-10 at the 1 is worth -1.6. First-and-10 at your own 44 is worth about +1.6. So if your opponent has first-and-10 at his 44, that's worth about -1.6 to you. Same as having the ball at your own 1. Even trade.

Before I talk further about it, I need to point out that the model is built from play-by-play data taken from the first quarters of games only. It therefore does not take into account end-of-half or end-of-game situations where a particular number of points are crucial. For example, consider the third question above. That question starts to look a lot different if there is one minute left in the game and you're trailing by 2. Or by 4. Or leading by 6. Romer's model serves as a guide only for teams that are trying to maximize points and that are not worried about end-of-half or end-of-game maneuvering. For that reason, you should assume that all the strategy questions above and below are taking place in the early stages of a generic game.

A very similar system of equating situations with point values was described (more than twenty years ago, it's worth noting) in *The Hidden Game of Football*, by Pete Palmer, John Thorn, and Bob Carroll. When I read that, it drastically changed the way I watch football. I'm a bit of a freak, I'll give you that, but even for a normal and well-adjusted football fan, this chart has all sorts of interesting implications.

First let's talk about what we're supposed to be talking about: fourth down strategy. Suppose you have fourth-and-one at your own 20 yard line. No one ever even considers going for it in this situation because everyone focuses on what happens if you fail. Indeed, if you don't get the first, you have given your opponent the gift of a +3.6 point situation. That's bad. But what people fail to see is that the punt --- let's assume it nets 40 yards --- puts them in a good situation too: +1.4 points. So you're not gambling 3.6 points, you're gambling 2.2 points.

Let's say your probability of picking up the first is *p*. If you get it, you'll be in a +.4 situation. If you don't, you'll be in a -3.6 situation. So if you decide to go, your expected situation is *p * (.4) + (1-p) * (-3.6)*. If you punt, your expected situation is -1.4. Setting those two equal and solving for *p* yields a breakeven point of about *p = .55*. In other words, if you think you have a 55% or better chance of making the first, you are better off going for it. As any football fan knows, but Romer demonstrates anyway, there is good reason to believe that teams often have a better than 55% chance of making a first down, but punt anyway. (What Romer does is actually a bit more complex, and takes into account the possibilities of blocked punts, fumbled punts, longer gains on the 4th down attempt, and essentially anything else that might happen.)

Let's examine a few other interesting implications of the model:

- What is a successful inside-the-5 punt worth? According to the model, the difference between a first down at the 20 and a first down at the five yard line is about 1.25 points.
- Take a look at the symmetry of the graph. A consequence of that symmetry is that the cost of a turnover is virtually independent of where it takes place on the field. If you turn the ball over on your own 10, you go from a -.3 situation to a -4.3 situation, so it costs you about 4 points. If turn it over at midfield, you go from about a +2 situation to a -2 situation. Again, 4 points. Likewise, a turnover at your opponent's 20 moves you from a +3.6 to a -.4.
- The chart quantifies what we all know: that yardage between the 20s is cheaper than red zone yardage. Moving 10 yards from your own 1 to your own 11 is worth the same amount of points as moving 23 yards from your 11 to your 34. I think this is part of why punting isn't that great of a deal. Unless you're backed way up, the yardage that you gain by punting is cheap yardage. The slope of that curve in the non-red zone is about 1/18, which means that 18 yards is worth a point. So most punts gain you about two points worth of yardage. You lose the ball though, which is a four-point swing, so a typical punt is a -2 point play. A failed fourth down attempt, obviously, is worse than that, but it's not
*that*much worse.

Tomorrow I will describe the particulars of how Romer arrived at the all-important chart pictured above.

This entry was posted on Thursday, May 11th, 2006 at 4:10 am and is filed under Statgeekery. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.

Hey Doug,

I do think it crucial to note that Mr. Romer's field-position-to-points conversion chart deals in EXPECTED values. A field goal is worth three points ON THE BOARD. Scoring is a way of changing the status of points from Romer expectation to guarantee, from probabilistic mathematics to deterministic mathematics. It's a subtle distinction, but noteworthy, I think.

I suspect that kickoff conversion to -.6 pts. is a league average taken over the course of one or more seasons. That expectation, I suspect, changes when restricting to particular season, particular kicking team, particular receiving team (particular returner, you can go nuts with this). In fact, the whole Romer expectation graph changes when two teams are picked to play. Given imbalances between offensive and defensive teams for particular clubs, "flipping the chart over" when the ball changes hands may be contrainidicated.

I think I just said "specific information yields better predictions than general information", which is so true it may not be worth saying.

The chart is fascinating and thought provoking. THANKS!

wthii,

Good points all. I believe I addressed most of them at some point in the post. But you are correct that I should have been more explicit in stating that we are dealing with long-run averages for hypothetical average teams playing against other hypothetical average teams, with all such teams in point-maximization mode.

As such, the chart is obviously only a starting point for your strategy calculations.

One other thing I just thought of. I don't have the paper in front of me, but I believe the exact words Romer uses in the paper are that he assumes teams are "risk neutral over points." Something like that. What that means, if I understand it correctly, is that Romer is assuming that teams value 3 expected points the same as 3 actual points.

Romer argues that, at the beginning of the game, that's a reasonable assumption. And I tend to agree with him.

Per post #3, it is a reasonable assumption at the beginning of the game, because all points scored are in the future. Deterministic mathematics plays no role -- yet. The probabilistic viewpoint is the only one which applies. During the game, however, there are points on the scoreboard, already determined, and conditional probabilities rule additional points which may or may not be earned.

It's the tension between determinism and probability which makes this nether world of "the game in real time" so exciting. Choices made during the game as dictated by situation, score and specific probabilities are key. And, as per #2, basing choices on league averages is suboptimal.

Good stuff! I look forward to tomorrow's post.

The crucial factor in the 4th down decsions is "p". Teams with great running games are going to obviously have a higher probability of picking up the first than those with poor running games - M. Holmgren and M. Shanahan are going to view a 4th and 1 a lot differently than D. Green (for 2005 anyway).

A forward thinking GM/Coach could have a statistician on staff to help compute the probabilities for a variety of situations (both offensive and defensive) that combined with league average information could be used as a helpful tool in making these decisions.

I know better than to try and tell Doug he made a math error, so instead I'll just say I don't understand this:

If you had first-and-10 at your opponent’s 42 yard line, would you trade it, on the spot, for a guaranteed field goal? - the point value of a first-and-10 at your opponent’s 42 is around +2.4, the same as the value of a field goal. Again, even trade.

Wouldn't the point value for a first-and-10 at your opponent's 42 NOT include a subsequent deduction of .6? I would think those 2.4 points are points on the scoreboard, not net points. Therefore, 3 guaranteed points on the board (with a resulting -.6 kickoff) is worth more than 2.4 expected points, with a discounted (maybe by 33%) kickoff rate. So 3 - 0.6 > 2.4 - 0.4.

Funny you should ask, Chase. Earlier in the week I promised you a SUPER DUPER cool application of the rating system I posted on Monday.

That rating system in some sense takes into account the strength of your opponents and the strength of your opponents' opponents and the strength of your opponents' opponents' opponents, and so on.

Similarly, the point values in the chart take into account points expected to be put on the scoreboard, plus the point values of the various situations that could follow from that one, plus the point values of the various situation that could follow from those situations, and so on.

In particular, the kickoff after any potential score is accounted for in the point estimates. Hopefully this will make more sense after I present the math. As I have foreshadowed, it's exactly the same as some math you've already seen. Until then, I'll offer two "proofs by intuition."

(1) Look at Romer's point value of a 1st-and-goal at your opponents' one, which is about 5.5. If that includes the value of the resulting kickoff, it would imply about a 78% chance of a TD and 22% chance of an FG. If it didn't include the kickoff, it would imply a 63% chance of TD and 37% chance of FG. Which seems more right?

(2) Go back to the question you were asking about. If the point values did not include the potential kickoff, then trading a guaranteed field goal for a first-and-ten at about the 31 would be an even trade. That doesn't seem right, does it? Doesn't that seem like an easier question to answer than first-and-10 at the 42?

Hope all that made a little sense. If not, wait for the next post.

Good stuff Doug. I actually thought of both of those, but 1) thought Romer's point value system didn't go all the way to the one yard line (the line fades out) and 2) I don't have Capy around to tell me what's the percentage of FGs made/FGs attempted. I'd guess around 80%. But I think you're proofs by intuition are pretty reasonable. Thanks Doug.

"* The chart quantifies what we all know: that yardage between the 20s is cheaper than red zone yardage. Moving 10 yards from your own 1 to your own 11 is worth the same amount of points as moving 23 yards from your 11 to your 34. I think this is part of why punting isn’t that great of a deal. Unless you’re backed way up, the yardage that you gain by punting is cheap yardage. The slope of that curve in the non-red zone is about 1/18, which means that 18 yards is worth a point. So most punts gain you about two points worth of yardage. You lose the ball though, which is a four-point swing, so a typical punt is a -2 point play. A failed fourth down attempt, obviously, is worse than that, but it’s not that much worse."

I think I see a flaw in this logic. Your measuring the punt by starting with the value of field position X, moving to field position Y and then turning the ball over. But surely the value of field position X is much lower on third down, right? So aren't you gaining a lot more value in the yards of the punt, since the "turnover" is more likely anyway?

Well, whatever I was doing, I was doing it pretty loosely and I have to admit I wasn't thinking my way through it very clearly.

I think what I was trying to say is that, on a typical 4th down between the 20s (subject to all the usual caveats), a typical punt is roughly two points worse than getting the first.

Reading it again, "a typical punt is a -2 point play" is at best a sloppy way to say that. A typical punt is a -2 point play from the last first down you had. But, yes, if it's 4th-and-12, you've already lost most of those two (expected) points already, so it's no longer a -2 from that point.

I don't know if that makes any sense or not. The short version would be: you're right.

[...] Support pro-football-reference.com « Another ranking system David Romer’s paper II » [...]

[...] As you may have guessed, this is a continuation of David Romer’s paper II which is a continuation of David Romer’s paper I. The first one is optional I suppose, but to understand this one you need to read the second. [...]

You have a small error in your analysis of question #1.

The change is between a change of 0 advantage (kicking off once and receiving once 0.6 - 0.6 = 0) and a 1.2 advantage (receiving twice 2 * 0.6 = 1.2)

You are correct, Dan. Thanks.

In 2005, the average field position following a kickoff was at the 29 yard line, not the 27 yard line. As far as these things go (given the large sample of 2365 kickoffs), that seems like a fairly large difference. I wonder why.

(I got the stats for each kicker from PFW/Stats Inc. and aggregated them.)

[...] This post draws upon the ideas contained in David Romer’s paper, discussed here, here, here, and here. [...]