Was Bill Belichick’s decision’s to go for a fourth and two conversion at his own 28 with 2 minutes left in the game against the Colts a boneheaded move?

There’s already been a lot of Monday morning quarterbacking by media talking heads (here, here, here, and here) saying yes. The stat gurus at Advanced NFL Stats say no, prompting this response from a commenter:

“Well, he went for it and it didn’t work. Then his team lost a game it was winning by six points with two minutes left. We don’t need any more proof then that to know it was a dumb decision, no matter what any stat geeks claim. This isn’t calculus calls. This is the NFL.”

The Nudge blog thinks there was nothing crazy or boneheaded about Belichick’s decision. Sure, it was a close call, but all the Monday morning quarterbacks are suffering from a condition known as the hindsight bias – because something happened means that something was always destined to happen.

Where were all these critics when Belichick successfully went for it on fourth and 1 from his own 24 against the Falcons last month?

Here is the Advanced NFL Stats analysis that we find convincing. Warning: if you are not a football fan, stop reading here.

With 2:00 left and the Colts with only one timeout, a successful conversion wins the game for all practical purposes. A 4th and 2 conversion would be successful 60% of the time. Historically, in a situation with 2:00 left and needing a TD to either win or tie, teams get the TD 53% of the time from that field position. The total WP (winning percentage) for the 4th down conversion attempt would therefore be:

(0.60 * 1) + (0.40 * (1-0.53)) = 0.79 WP

A punt from the 28 typically nets 38 yards, starting the Colts at their own 34. Teams historically get the TD 30% of the time in that situation. So the punt gives the Pats about a 0.70 WP.

Statistically, the better decision would be to go for it, and by a good amount. However, these numbers are baselines for the league as a whole. You’d have to expect the Colts had a better than a 30% chance of scoring from their 34, and an accordingly higher chance to score from the Pats’ 28. But any adjustment in their likelihood of scoring from either field position increases the advantage of going for it. You can play with the numbers any way you like, but it’s pretty hard to come up with a realistic combination of numbers that make punting the better option. At best, you could make it a wash.

For those who are curious, .60 is the probability of succeeding on fourth down and .53 is the probability the Colts will score a touchdown given a Patriot’s punt plus any increase in the probability of a Colts’ touchdown given that the Patriots don’t convert on fourth down.

Here’s another way to look at the decision with an additional probability assumption.

Let y = the probability the Colts score a TD, given that the Pats punt. The Advanced NFL stats equation calls y the winning percentage.

Let x = the increase in the probability that the Colts score a TD, given that the Pats do not convert.

0.6 + 0.4 (1-(y + x)) > (1-y).

Rearranging terms, x < 1.5 y.

Suppose the Colts probability of scoring would have been 0 if they took the ball over on their own goal line, and 1 if they take over on the Pats’ goal line. This basically means the Colts would definitely not score a touchdown if they had to go the length of the field, and would definitely score if they got to start at the Patriot’s 1-yard line. Suppose also that this probability increases linearly with field position. That means x is the net yardage of the punt (divided by 100), and y = .70 – x. (.7 is the probability of scoring when taking over on the opponents’ 30.)

Plugging terms into the algebraic condition x <1.5 y produces x < .42. So Belichick should punt if the expected net yardage of the punt is more than 42 yards. Guess what New England punter Chris Hanson’s lifetime indoor punting average is? 42.9 yards. His 2009 average is 39.6 yards.

Belichick’s critics treat the Colts’ touchdown as an inevitability. But the probability of a Colts’ touchdown was not 1. The New England defense could have held them to a field goal and still won the game.

Hat tip to Thomas Hubbard to whom all hate mail should be sent.

**Addendum:** For more on fourth down conversion calculations by Berkeley economist David Romer check out his paper, “Do Firms Maximize, Evidence from Professional Football.”