Bill Belichick is no bonehead. He just understands probability.

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.”


8 Responses to “Bill Belichick is no bonehead. He just understands probability.”

  1. the people who bash Bill Belichick’s 4th-down decision at things kevin hates Says:

    […] The Nudge blog notes that while media opposition to the decision has been widespread, it’s largely a case of hindsight bias. This hindsight bias—focusing on results instead of probabilities—is the source of a great many ills in today’s society. People only consider what actually happened, instead of considering what decision was best given the information known at the time and the relative probabilities of success for the various available choices. It comes down to a concept called “expected value.” I’ve been planning for a while to write a blog post on how much I hate the fact that so few people understand the concept of expected value. I promise to get around to it eventually, but for now, here’s a short example, because I don’t care for any of the first ten links that popped up on google. […]

  2. Jack Says:

    The primary problem with this analysis is that it fails to take into account the very complex factors that average out in the big database, for example, that good teams play better than the average in the clutch, or the fact that the Patriots had already intercepted Manning twice in the game.

    Second, it considers the game in a vacuum. How does Belicheck’s perceived inconfidence in his defense affect the next few games or the remainder of the season? How will the fan base respond to being robbed of a dramatic defensive stand at the end of the game? Etc.

    I don’t think he’s a bonehead, at all. I do think that he failed to diversify his investments–he ultimately put all of his eggs in one basket, boiling down winning or losing the game to a single play.

  3. michael webster Says:

    Does it matter to your a) calculation or b) to your assessment of the decision that New England is currently 5/17 on 4th down conversions?

  4. Thomas Hubbard Says:

    Sometimes it does pay to put all of your eggs in one basket. (See, for example, doubling down in blackjack, where you forego the opportunity of additional cards.)

    What matters here is the expected probability of winning — you need to take actions that maximize this, and sometimes this means putting it all on one play. This is not like investing, where individuals are not maximizing expected returns, but rather are maximizing more complicated objective functions.

  5. Tim Williams Says:

    Are your calculations taking into account that the Colts, having no choice but to do so, would have gone for it on any fourth downs they might have faced on their ensuing drive (from any point on the field?) If going for it was the Patriots’ best chance, presumably it would have improved the Colts’ chance of scoring, as well. Not sure that this changes the balance of the odds, it just isn’t clear to me that it was taken into account above.

  6. Ted Says:

    You wrote “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.”

    You say if the NET yardage is greater than 42 and then use the NON-NET average into your calculation. Not sure what is net average indoors is, but is career net average is 34.8. Usually net average is around 5 yards below average.

  7. gobears Says:

    As for the “all your eggs in one basket” argument, I would have loved to make bets with y’alls the moment after the Pat’s attempt failed, given that you put the odds for a Colts victory at 1:0.

  8. Mike Gill Says:

    I just read Mr. Hubbard’s thoughtful and in depth analysis of Bill Belichick’s decision to go for it on 4th and 2 in the now famous Indy game.
    Thank you Mr Hubbard.
    It was fascinating.
    I do have some questions at the end of this that I hope you can clear up for me.

    Here’s the equation I have questions about:
    (0.60 * 1) + (0.40 * (1-0.53)) = 0.79 WP

    The equation is sound arithmetically.
    The math is right because it is describing something with a 79% probability of occurring.
    But I don’t think it accurately models what is occurring in this Punt versus Go For It situation ;o)

    I think when the equation is expressed with dimensions it’s easier to understand the assertion that it does not model what is occurring.
    Here’s the equation with dimensions:
    (60% chance pats convert) + (40% chance pats do not convert*(47% chance colts do not score)) =
    (60% chance pats convert) + (19% chance colts do not score) = 79% chance pats win

    That second term is 0 while the Patriots attempt to convert.
    The event the second term is describing can never really occur.
    There is no real event in which the Colts have a 19% chance to not score.

    The real possible scenarios are:
    1.0 60% chance pats convert and win
    2.0 If pats don’t convert, 47% chance pats win, 53% chance colts win
    3.0 If pats punt, 70% chance pats win, 30% chance colts win

    The equation is stating the Pats opted for 79 chances out of 100 to win the game.
    But I believe this is inaccurate because:
    When the pats did not get the first 60 chances, the next 19 were no longer available as possibilities.
    The scenario changed completely as described in 2.0.
    The only other possible scenario was 3.0 above.

    Do you agree with my conclusion?
    If not, could you please correct the flaws in the analysis that brought me to the conclusion?

    Thanks again for the article.

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