We’re in week 16 of the NFL season and playoffs are close on the horizon. We’re also in college bowl game season. With that in mind, let’s consider a key decision that faces coaches multiple times a game: should they go for it when faced with fourth down? On fourth down, a coach has three possible calls:

• Punt
• Attempt a field goal
• Go for it

The choice among the three is affected by factors such as the score, how many yards are needed for a first down, and where they are on the field.

Most coaches choose to punt or go for a field goal unless they are 4th and 1 inside their opponent’s 45-yard line. Math suggests that coaches should instead choose to go for it on fourth down way more than they currently do.

The theory that coaches should more often go for it on fourth down was championed by Nobel Prize-winning economist Paul Roemer. In 2005 he ran the math of fourth-down attempts and determined that coaches would best be suited by following the Bellman Equation which is a “dynamic programming equation associated with discrete-time optimization problems.” Here’s the equation:

Ei Di(gt) Vi = Pgt + Bgt Ei Di(gt+1) Vi – get

Here are the rules of thumb Roemer produced using the Bellman Equation:

• A team facing fourth-and-goal within five yards of the end zone is better off, on average, trying for a touchdown.
• At midfield, on average, there is an argument to 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.

The New York Times ran more scenarios and produced this helpful chart:

On the left is what the math says about going for it on fourth down and on the right is what coaches do.