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# Pro Football Reference Blog

## David Romer’s paper II

Posted by Doug on May 11, 2006

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.

1. Would you give the other team half a point for the right to receive the kickoff in both halves? A full point?
2. 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?
3. 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?
4. 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?

5. 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:

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.