Stop Shy of the First Down

Figure 1

Drive value versus where the carrier is downed

Consider a first-down play. The curve below plots the expected points for the remainder of the drive as a function of where the ball-carrier is downed, under a simple "advance the chains" continuation. It is a single, consistent measure across the whole range — and it has an interior maximum located before the line to gain.

Drag the marker along the field. The corresponding point on the curve moves with it; note that crossing the line to gain lowers, rather than raises, the drive's expected points.

Figure 2

Field position in expected points

The common unit for the analysis is the expected points of a possession as a function of field position: approximately zero near one's own twenty, rising to about four near the opponent's. As a rough guide, fifteen yards correspond to one point — and, equivalently, a possession is worth on the order of forty yards of field position. That exchange rate is what the next figure puts to work.

Drag along the curve to read off the value at any field position.

Figure 3 · The central point

What would a possession have to be worth?

Rather than assume a value for a possession, invert the question. If a ball-carrier is downed a yard short (2nd-and-1) and rushes it out, the chains are advanced with probability ≈ 0.94, and the ball is lost only ≈ 0.06 of the time; when converted, the play gains about five yards. Reaching for the first down instead avoids that small risk but forgoes those extra yards.

Setting the two equal gives the implied value of a possession at which reaching for the first down would just break even: P(convert)/P(lose) × (avg yards − 1) ≈ 66 yards. A possession would have to be worth roughly sixty-six yards — far beyond any plausible value — for the stretch to make sense.

Drag your own estimate of a possession's value along the line; vary the conversion rates to move the break-even threshold.

Figure 4

Comparing 2nd-and-1 with 1st-and-10

The same comparison, simulated directly in expected points. From one point on the field: take the first down (1st-and-10), or be downed a yard short (2nd-and-1) and advance the chains. Simulating the continuation many times yields the mean expected points of each; the conversion probabilities and field position can be varied.

Run the simulation, then adjust the parameters and re-run.

Figure 5

Run versus pass on 3rd-and-1

A related question: on third-and-one, the run converts more often, but the pass gains substantially more when it succeeds. The decision tree below evaluates each call in expected points, accounting for a punt on a failed conversion. The preferred call depends on field position.

Switch the call and vary the field position to compare.

Figure 6 · An aside

The distribution of yards gained on first down

A curiosity, offered for interest rather than as evidence for the argument. Across 33,951 first-and-ten rushes (2009–2013), gains of exactly ten yards — the line to gain — are markedly rarer (383) than gains of nine (1,058) or eleven (634). The cause is not obvious: where the ball is spotted, how defenders converge on the marker, and ball-carrier behaviour could all contribute. Make of it what you will.

The highlighted bar sits exactly on the line to gain. Data: microprediction/nflMarkov.