Strategies on Fourth Down, From a Mathematical Point of View

The New York Times, "Economic Scene" , September 12, 2002

Football season has begun, and in stadiums, sports bars and living rooms across the land fans are asking the same question: "It's fourth down and what does the Bellman equation say?"

Or at least one fan is asking.

That's the title of a recent working paper by David Romer, an economist at the University of California at Berkeley, for the National Bureau of Economic Research. (The paper is available here.)

The paper uses the mathematical technique of dynamic programming -- the Bellman equation, named for Richard Bellman, a mathematician who died in 1984, is a central tool of the technique -- to analyze the value of different football strategies on fourth down. Given its field position, should a team punt, kick a field goal or go for the first down?

Consider a team that is just 2 yards away from a touchdown. Listening to a game on the radio, Professor Romer heard the commentators say the team should obviously take the easy 3 points from a field goal. "It wasn't obvious to me that it was the right decision," he said.

First he thought about the odds. If the team has a better than three-in-seven chance of scoring a touchdown -- and the all but certain extra point -- going for the touchdown has a higher expected value than kicking.

For an average National Football League team, the probability of a touchdown from 2 yards out is about 40 percent, making an attempt worth 2.8 points, or less than a field goal.

But there's more to the question than the probability of a score. "If you go for the touchdown and you don't get it, you've lost the 3 points but you've gotten something back in return," Professor Romer notes. "You've left the other team in this terrible field position," a daunting 98 yards from its goal.

A field goal, by contrast, leads to a kickoff, which on average in the N.F.L. gets the receiving team to the 27-yard line, a much better position. "Thus we need to know how much better it is to leave the opponent with the ball on its 2-yard line than on its 27," he writes.

That's where dynamic programming comes in.

The technique is a way to calculate the value of actions that have effects far into the future. Economists use it to look at things like a household's choice between saving and spending: how much you save rather than consume this year affects what you have to spend and save next year, and what you do then affects the following year, and so on into the future.

For football, points are at stake instead of money, but the mathematical principles are the same.

Using data from about 700 regular-season N.F.L. games, Professor Romer estimated the value in expected future points of getting a first down at each yard line on the field. To keep the analysis relatively simple, he looked just at the first quarter.

By these calculations, the value of a first-and-10 on a team's own 1-yard line, a terrible position 99 yards from a touchdown, is minus 1.6 points. Moving up the field, the value rapidly increases, hitting zero at the 15-yard line. From there on, each gain of 18 yards is worth about a point in the final score.

The bottom line: Teams tend to overestimate the value of field position, compared with keeping the ball, on their end of the field and underestimate it, compared with the value of a field goal, on the other team's end. When those estimates are corrected, the risks of going for it look a lot lower.

"The analysis implies that teams should be quite aggressive," Professor Romer writes. "A team facing fourth-and-goal is better off on average trying for a touchdown as long as it is within 5 yards of the end zone. At midfield, being within 5 yards of a first down makes going for it on average desirable. Even on its 10-yard line -- 90 yards from a score -- a team within 3 yards of a first down is better off on average going for it."

But, he said, "teams almost always kick on fourth down early in the game."

This result presents a puzzle for economists. Since football is highly competitive and the rewards for winning are great, teams should be doing what is optimal to win.

"We believe, we teach our students repeatedly, we base our models on the assumption that there are very strong pressures for firms to maximize, for firms to do the best thing," Professor Romer said. "We don't think they know calculus or dynamic programming. We think intelligence, ability in other dimensions, intuition, trial and error, imitation are going to lead, in a competitive marketplace," to the best strategies.

One possibility is that teams are not trying to win above all. "The costs of losing as a result of a failed gamble could be greater than the costs of losing from playing it safe," Professor Romer writes.

Or coaches may not realize they could do better by taking more chances. After all, away from the economists' blackboards, real companies constantly discover new and better ways to operate, some of which are made possible by better data and computing power.

Recently, for instance, retailers have begun using mathematical models to figure out when to put clothes on sale, something that used to be done by intuition and rules of thumb. Profits have increased substantially in some cases.

Football could be similar. If so, Professor Romer's paper could begin to change fourth-down strategies. But he'll have to overcome professional skepticism.

"If a football coach called me and said, 'I have a new way to deal with the Social Security problem,' " he said, "my first reaction would be not, 'Oh boy, someone solved this difficult problem.' My first reaction would be, 'They're not qualified to talk about this. Who are they to think that they've solved this?' "