Today being Thanksgiving, it's pretty reasonable to assume (if you live in the U.S.) that you will likely be sitting down to a large meal involving lots of turkey, stuffing, and cranberry sauce. It is also pretty likely, that somewhere in the house, football games will be on the television. Which brings us to one of the quintessential questions in football: It's 4th down, your team is on the opposing team's 30 yard line and they have only one yard to go to get a first down. Should they go for it? Most people would probably say no... that they should try for a field goal and at least get the 3 points. But most people would be wrong. At least according to a study in the Journal of Political Economy (pdf), which suggests that the payoff for "going for it" is more than twice as much compared to trying for a field goal.
To completely over simplify the study, economist David Romer determined the payoff of each decision (either to kick or to go for it) by looking at thousands of NFL plays and calculating the average costs and benefits of each decision depending on where the team was positioned on the field. In this case, the benefit would be the likelihood of scoring a touchdown (valued at roughly 7 points*) or of scoring a field goal (valued at 3 points). So, if teams that decide to kick when they are on their opponents' 30 yard line make the field goal an average of 33% of the time, then the benefit of kicking is assigned a point value of 1 (since a field goal is worth 3 points, and 33 percent of 3 points is 1 point). Since teams that only have one yard to go when they are on the 30 yard line convert for a first down 64% of the time, and teams that are inside the 30 yard line score a touchdown about 40% of the time, the benefit of going for a first down is assigned a value of 1.8 (0.64 x 0.40 = 0.24, or a 24% chance of scoring a touchdown by going for it on 4th and 1, and 0.24 x 7 points = 1.8 points). This means that "going for it" should result in scoring almost twice as many points than kicking, and this difference becomes even more exaggerated when we consider the costs. In this case, a failed attempt either way results in giving the other team the ball on their own 30 yard line. Scoring the field goal, or eventually a touchdown, will result in a kickoff which, on average, results in the other team getting the ball on their 27 yard line. This means that, no matter what your team does in this situation, they are going to ultimately give the ball to the other team at about the 30 yard line, which gives the other team a chance at scoring relating to an average value of 0.62. Thus -0.62 points is the cost associated with either kicking or going for it. This means that the net benefit of going for the first down is 1.18 points, while the net benefit of kicking the field goal is 0.38 points. So, while there is less of a chance that "going for it" will result in a touchdown, compared to a greater chance that your kicker will make the 40yard field goal, on average you will get more than THREE TIMES as much benefit if your team goes for it every time they have 4th and 1 on the 30 rather than kicking it every time you were in this situation.
So, why do coaches so rarely go for the first down? It may be that, in these instances, coaches (and likely fans) fail to take into account the differences in values of touchdowns and field goals, seeing all types of "score" as being roughly equal in value even though touchdowns are more than twice as valuable as field goals. Additionally, the costs are also poorly estimated. For example, turning the ball over by a loss of downs seems to carry greater cost than turning it over in the form of a kickoff after scoring, even though the point values of these in our hypothetical are roughly the same. These ideas tie into the explanation that Romer offers in the paper, suggesting that coaches may be succumbing to "loss aversion" type thinking. Loss aversion is a phenomenon in psychology where people generally avoid a more rewarding choice when the lesser of the two options seems more like a sure thing. For example, when offered a fifty percent chance of a $100,000 prize or a 100 percent chance for a $30,000 prize, most people choose the $30,000 prize because it is a sure thing. However, the value of the first option has an average value of $50,000 compared to only $30,000 for the second option. So, if you are only offered this choice once, it might make sense to take the sure thing, BUT, if, as in the game of football, you will face a choice like this many different times in a game or over the course of a season, it makes more sense to choose the option that has the higher average outcome. And, of course, most people do realize which decision is the better one if you take away the "sure thing" aspect of one of the options. For example, if you ask the same group of people whether or not they would go for a 5% chance at $100,000 or a 10% chance at $30,000, more people choose the 5% chance at $100,000, even though the relative difference between winning the prizes is the same as in the previous situation. (If you want to know more about loss aversion, I suggest this post over at Jonah Lehrer's blog.)
In the case of football, most coaches (and fans) see a field goal as much more of a "sure thing" because the probability of a successful try is higher than the probability of ultimately scoring a touchdown, particularly if you are less than 30 yards away from the goal line. BUT, this neglects to take into account the point differences between a field goal (3) and a touchdown (7), and the costs associated with the opposing team's resulting field position. To provide another example, Romer presents the situation of having a 4th down and goal on the 2 yard line, where a field goal really is a sure thing, but the chances of scoring a touchdown are about 3 in 7. Here, the benefit is about the same (an average of 3 points per field goal attempt, and an average of 3 points per touchdown attempt). However, the cost for both of these is NOT the same. Since the field goal is all but guaranteed, that means that on average, the opposing team will then get the ball on their 27 yard line after the kickoff. Whereas, if you go for the touchdown, there is a 4/7 chance that you will fail and leave the other team with the ball on their 2 yard line. When you calculate the cost and benefit of each of those field positions, the kickoff results in a positive chance of 0.62 points for the opposing team, thus costing your team 0.62 points (on average), leaving a resulting net benefit for a field goal try at 2.38. Leaving the other team on their own 2 yard line however, puts them at a serious disadvantage where they are more likely to turn the ball back over or get sacked in the endzone giving your team 2 points for a safety. The average value of this position is therefore actually negative (-1.5), and thus, the benefit of going for the touchdown is (3/7 x 7 =) 3, and the "cost" is (4/7 x -1.5 = -0.857), which, subtracting the cost from the benefit, yields a net benefit of 3.857 going for the touchdown (versus the net benefit of only 2.38 if you kick the field goal).
AND this calculus doesn't just apply to being on the thirty yard line, or the 2 yard line. According to Romer's extrapolation, no matter where you are on the field, except for maybe behind your own 15 yard line, it makes more sense to go for the first down on 4th and short (less than 2-3 yards to go) than it does to punt or to kick a field goal. Romer estimates that teams who adopt a strategy of "going for it" in these situations would be 5% more likely to win each game, and would at least win one extra game per season than they normally would for 3 out of every 4 seasons. So, maybe Bill Belichick, who has a reputation for "going for it" on 4th down, has done his math, or maybe he has intuitively stumbled on to something. Either way, his career coaching record of almost twice as many wins as losses, 3 Super Bowl rings and 4 AFC championships, support the idea that going for the first down conversion on 4th and short might just give your team the winning edge. Just something to keep in mind while you are watching the games later this afternoon. Happy Thanksgiving!
*Since point after touchdown ("extra point") kicks are successful 98.5% of the time, the actual value of a touchdown used in the study was 6.985, not 7, but for us, to keep the math simple, I'll just use the full 7.
And if you want to learn more about what various fields of research can tell you about your favorite sports, check out some of my earlier posts:
How the flash lag illusion may have cost Armando Galarraga his perfect game in baseball.
How magnetic necklaces and hologram bracelets may actually affect athletic performance.
How watching your favorite team win (or lose) might affect your testosterone levels.
Do you run a higher risk of getting a concussion playing boys" football or girls' soccer?
Where the students sit in the stadium could enhance your school's home field advantage.
Romer, D. (2006). Do Firms Maximize? Evidence from Professional Football Journal of Political Economy, 114 (2), 340-365 DOI: 10.1086/501171