Do NFL coaches understand utility theory? Possibly better than we think!
This Sunday, America will stand still as the Pittsburgh Steelers and the Green Bay Packers meet in Super Bowl XLV. For players, this is the pinnacle of their career, whilst for coaches it is a chance to cement greatness. For statisticians, American football is an intriguing sport. There has been an abundance of research on the ratio of running to throwing the ball; on the statistics that contribute to winning; measuring the contribution of individual players, and a host of other areas. This article is going to focus on a very tiny part of the game – 4th down.
For those unfamiliar with American football rules, the offensive team must gain 10 yards per 4 downs (or plays). If they reach 10 yards or more, they receive an additional 4 downs. If they are on their 4th down and they have additional yards to achieve, they have one of three options. They can attempt to make the 10 yards; knowing that if they fail to reach it, the opposition receive the ball where they are. They can attempt a field goal; which gains them 3 points. Or they can punt the ball and force the defensive team deep into their own territory.
On Sunday, each team will face numerous 4th down situations. And research by David Romer of the University of Berkeley  (and extended excellently by Brian Burke at Advanced NFL Stats) using expected utility theory suggests that coaches will typically be risk averse when faced with these situations. Expected utility theory, broadly speaking, allows us to choose between different uncertain options. For American football, we can use this method to calculate the expected number of points a team expects to achieve depending on where we are on the field at a 1st down. For example, if a team is on its own 1 yard line, then it expects to gain -1.6 points, i.e. it is likely that the opposition will score next. Alternatively, being on the opponent’s 10 yard line gives the offensive team an expected 5.0 points.
By having this measure of expected utility, we can begin to compare the different options for punting, kicking a field goal, and “going for it” on 4th down. Let’s take an example to demonstrate this. You are a coach and your team faces a 4th and 4 on your opponent’s 40 yard line. It's early in the game and you are currently tied with the opposition. You believe your kicker is a 50/50 to score a field goal from this distance, and historical data suggests that the probability of successfully gaining a 1st down is 0.52. You have 3 options – which would you choose?
For punting, historical data suggests it would land around the opponent’s 12 yard line giving us an expected point’s score of 0.2. This is the classical decision taken.For kicking a field goal, you would expect the kicker to be successful 50% of the time. When he is successful you receive 2.3 expected points - not 3 as the resulting kickoff will give the opposition 0.7 expected points! If you are unsuccessful, the opposition receives the ball around the 47 yard line and has an expected points score of -2.1.
A simple calculation of E(Points) = p(A).E(A) + p(B)E(B) where p(A) is the probability of event A and E(A) is the expectation of event A. This gives us 0.5 x 2.3 + 0.5 x (-2.1) = 0.1. At this point, choosing to punt is the optimal decision - but it is close!
However, what about going for it on 4th and 4? Most armchair fans will call this madness, but what do the statistics tell us? If we are unsuccessful and the opposition receives the ball on their 40 yard line, we have an expected points score of -1.5. But if we are successful and gain the 1st down, we now have at least 2.5 expected points. By doing the same calculation as above, we get an expected points of 0.6 – greater than both punting and field goal options!
As we can see, going for it gives the highest expected utility, and is the one we should choose over the long run – so why do coaches not go for it more often? Well within this simple model we are only capturing the expected return for the team – not the coach making the decision. Going for it and being unsuccessful could lead to the coach losing his job – no matter what the statistics say. Maybe they know more about utility theory than we give them credit for!