Expected Points

The previous post looked at one way of assessing the success of a football play, namely, by measuring the increase or decrease in the probability of getting a first down. We saw that, in general, an offense needs at least 5 yards on any play to break even in terms of its probability of getting a 1st down. I’ll continue the discussion by looking at another measure of utility called point expectancy.

Every spot on the field has an abstract value in terms of points. We can begin assigning values at the end zones, where having the ball has a clear and concrete value. Possessing the ball at the opponent’s end zone is worth (nearly always) 7 points. And having the ball at your own end zone is worth -2 points.

Every other yard line has a point value too. We can measure it by averaging how many points will be scored next. For example, having a 1st down and 10 from an opponent’s 20 yard line is worth, on average, about 4.2 points. Often the offense will score a touchdown, and failing that, it is likely to be able to kick a field goal. But sometimes, the offense will fail to do either, and the opponent may be the next to score. In other cases, neither team will score immediately, and they will exchange possession until someone does score. This is something I’m used to watching as a Ravens fan.

The concept of point expectancy originated with the work of Virgil Carter, a former NFL quarterback who studied operations research in the early 1970s (while an active player). Carroll, Palmer, and Thorn adapted the concept in their 1987 book The Hidden Game of Football.

One flaw in the early applications of the concept was the assumption of linearity. Both Carter and the authors of Hidden Game planted stakes for the obvious point values at both end zones and then drew a straight line between them. We’ll see that isn’t exactly right. Additionally, things change in the 4th quarter as teams with leads become conservative and teams that are behind trade overall scoring optimization for urgency.

The graph below plots the expected points for a 1st down at each yard line. For simplicity, I’ve named each yard line in terms of its distance from an opponent’s end zone. Having the ball at one’s own 20 is “the 80 yard line” for example.


One immediate application of point expectancy is measuring the cost of a turnover. If an offense loses a fumble at the 50 yard line, the expected point value swings from +2 to -2, a difference of 4 points. Or we can measure the value of a punt. If a team punts for a net 35 yards from its own 35 yard line (“the 65”), the expected point value swings from +1 to about -1, a difference of 2 points. In this regard, we could say that a turnover (at the 50) is 'twice' as bad as a punt.

Expected points is also the methodological basis of the Romer paper on kicking vs. going for a first down. In it, the author measures the expected point value of attempting a field goal or punting vs. the expected point value of ‘going for it’ on 4th down. Romer also points out that touchdowns and field goals are actually worth 1 point less than we think. Unless a score takes place with very little time remaining in a half, the other team will receive a kickoff, worth on average about 1 expected point if they have enough time remaining to mount a scoring drive. Touchdowns and field goals are not quite as valuable as thought, at least in abstract terms.

I should point out that a turnover has different values at different parts of the field. This is something researched early on at the Football Outsiders site. For example a turnover in the red zone, say at the 10, results in a swing from +5 to about +.25, for a difference of 4.75 points expected.

We can see that the neutral point on the field is at a team’s own 15 yard line. There, it’s equally likely that either team will be the next to score.

Things become more complicated when we consider other down and distance situations. Suppose at any given yard line, a pass falls incomplete on 1st and 10. Second down and 10 represents a drop off of about 0.5 points expected. Second and 9 represents a slightly smaller drop off, until at about 2nd and 5 when the expected points are approximately equal to those for 1st and 10. This is consistent with the 1st down probability method I described in my previous posts. Third down and 10 represents a further drop off of about 0.5 points.

Another complication is that various teams have different curves. Defenses would each have their own curve as well. When using point expectancy to weigh decisions about kicking or going for a first down, each team would have to take into account its own and its opponent's unique expectancy curves. Take for example the Ravens and Colts over most of this decade. Each team is typical of opposing extremes--great defense, mediocre offense, and vice versa.

A drawback to this method of measuring utility in football is that it does not consider time remaining. In other words, it assumes that every game is indefinitely long and the object for each team is to maximize point differential. But this is not how football really works. Take the case of a team trailing by 4 points late in a game. A touchdown is essential, but a field goal would be pointless. Even on 4th and very long, it wouldn't make sense to purely maximize "expected" points by kicking. For the vast majority of situations in football, however, this method would be adequate. This brings me to the next method of measuring utility in football—win probability. I’ll discuss that in a forthcoming article.