In last night’s Bears/Packers game, Coach Marc Trestman (of the Bears) decided to go for it on 4th and inches at the Bears’ 32 yard line during in the fourth quarter with 7:50 left and when the Bears were up 4. Normally, teams decide to punt in this situation, which reflects a hyper-conservative decision-making approach adopted by most football coaches. The Bears got the first down, and the ensuing drive led to a field goal, putting the Bears up by 7 with 0:50 left in the game.

In hindsight, it was obviously a great call. But decisions aren’t made with hindsight – both good and bad outcomes are possible with different likelihoods.

An article by Chris Chase at USA Today [Link] argued that while going for it on 4th down was a bad decision because the bad outweighed the good. There isn’t much analytical reasoning in the article. I prefer base decisions on number crunching rather than feeling and intuition, so here is my attempt to argue that going for it on 4th down was a good decision.

### The basic idea of football decision-making

There are a number of models that estimate the expected number of points a team would get based on their position on the field. To determine the best decision, you can:

- look at the set of possible outcomes associated with each decision,
- find the probability and expected number of points associated with each of these outcomes,
- then take the expected value associated with each outcome, and
- choose the outcome with the most expected points.

Let’s say going for it on 4th down has success probability p. Historical data suggests that p=0.8 or so. If unsuccessful, the Packers would take the ball over on the Bears’ 32 yard line with a conditional expected value of about -3.265 points. This value is negative because we are taking the Bears’ point of view. If successful, the Bears would be around their own 35 yard line with a conditional expected value of 0.839. When considering both outcomes (success and failure), we can an expected value associated with going for it on fourth down: 0.839 p – 3.265(1-p).

Let’s look at the alternative: punting. The average punt nets a team about 39 yards. This would put the ball on the Packers’ 29 yard line with an associated expected number of points of -0.51. However, this isn’t the right way to approach the problem. Since the expected number of points associated with a yard line is non-linear, we can’t average the field position first and then look up the expected number of points. Instead, we should consider several outcomes associated with field positions: Let’s assume that the Packers will get the ball back on their own 15, 25, 35, and 45 yard lines with probabilities 0.28, 0.25, 0.25, and 0.22 and with expected points 0.64, -0.24, -0.92, and -1.54, respectively. This averages out to the ball on the Packers’ 29 yard line with -0.45 points (on average).

Now we can compare the options of going for it (left hand side) and punting (right hand side):

Solving this inequality tells us that the Bears should go for it on fourth down if they have a success probability of at least 68.6%.

These values are from Wayne Winston’s book *Mathletics*.

### But time was running out!

The method I outlined above tends to work really well except that it ignores the actual point differential between the teams (which is often important, e.g., when deciding to go for one or two after a touchdown), the amount of time left on the clock, and the number of timeouts. It’s worth doing a different analysis during extreme situations. With 7:50 left on the clock, the situation wasn’t too extreme, but the Packers’ 3 remaining timeouts and 4 point score differential are worth discussing. Going for it on 4th down allowed the Bears to score a field goal and eat up an additional seven minutes off the clock, which was almost the perfect outcome. Let’s consider a range of outcomes.

Very close to the end of the game, it’s best to evaluate decisions based on the probability of winning instead of the expected number of points. Note that you find the probability of winning as the expected value of an indicator variable, so it uses the same method with different numbers. Making this distinction is important, since if you are down by 4 points, going for a field goal may maximize your average points but would guarantee that you’d lose the game.

One way to address these issues is to look at how many possessions the Packers will have if the Bears punt or go for it on fourth down. Let’s say that the Packers would get one possessions if the Bears punt. They would need to score a touchdown on their single possession to win. Let’s say that the Packers would get two possessions if the Bears punt. The Packers could win by scoring two field goals or one touchdown, unless the Bears score on their possession in between the Packers’ possessions. If the Bears score an additional field goal, that would put the Bears up 7, and the Packers would need at least one touchdown to tie (assuming a PAT), and an additional score of any kind to win. If the Bears score an additional touchdown, that would put the Bears up 10-12, and the Packers need two touchdowns to win and could possibly tie or win with a field goal and a touchdown (assuming a PAT or 2-point conversion was successful). The combination and sequence of events need to be evaluated and measured.

Without crunching numbers, we can see that punting would likely increase the Packers’ chance of winning because it would give them 2 chances to score (unless the Packers’ defense is so poor that they think the Bears would be almost certain to score again given another chance).

This is just one idea for analyzing the decision of whether to go for it on fourth down. Certainly, more details can be taken into account so long as there is data to support the modeling approach to support the decision.

Brian Burke blogged about this as I was finishing up my post [Link]. He used the win potential instead of the expected number of points (which I recommend but don’t calculate). This yielded the Bears’ break-even success probability of 71%, which is close to what I found. In any case, this more or less supported the decision to go for it on fourth and inches (although not going for it would also be reasonable in this case since the probability of successfully getting a fourth down is only slightly higher than the threshold) but maybe this analysis wouldn’t have supported the decision to go for it if it were fourth and 1.

### More on fourth down decision-making:

- Brian Burke has a post on fourth down decision making. He also blogged about this very same call.
- Brian Burke defended a Patriots’ unsuccessful 4th down conversion in the NY Times.
- I once blogged about the Patriots’ decision to go for it on fourth down (slides & slidecast available).
- Going for it on fourth down is discussed in detail in Scorecasting by L. Jon Wertheimand Tobias Moskowitz, a book that I highly recommend if you are a sports nerd or number cruncher.

What sports play have you over analyzed?

November 6th, 2013 at 11:37 am

Vince Knight left some feedback on Google+. He blogged about decision-making in rugby using game theory. Here is a link to his post: http://drvinceknight.blogspot.com/2011/12/to-box-kick-or-not-to-box-kick.html?m=1

November 8th, 2013 at 6:00 pm

This article by Hurley focuses on why the fourth down decision should be made before the first down play is called. Hurley, W.J., “Optimal Sequential Decisions and the Content of the Fourth-and-Goal Conference,” Interfaces, Volume 28, Number 6, pages 19-22, 1998. Of course the coach wouldn’t know exactly how much time or yardage remained, but the team would know what is coming.

November 10th, 2013 at 2:33 pm

Thanks Jeffrey! I will check that out.