Dynamic "Scoring" In Football
September 27, 2002
A recent working paper by David Romer, entitled "It's fourth down and what does the Bellman equation say?", uses the mathematical technique of dynamic programming to analyze the value of different football strategies on fourth down. It asks the question, "given a team's field position, should it punt, kick a field goal or go for it on fourth down?"
Dynamic programming 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 money and the affects those choices have from year to year. Romer applied those same mathematical principles to football. Instead of money, it's points.
Romer used data from over 700 regular-season NFL games to estimate the value of expected future points of getting a first down at each yard line on the field.
By these calculations:
- The value of a first-and-10 on a team's own 1-yard line (99 yards away from a score) is minus 1.6 points
- Moving up the field, the value increases, hitting zero at the 15-yard line.
- From there, each gain of 18 yards is worth about a point in the final score.
Romer concludes that teams should be quite aggressive, but that they almost always kick on fourth down.
The study suggests that teams tend to overestimate the value of field position, compared with keeping the ball, on their end of the field -- but underestimate it, compared with the value of a field goal, on the other team's end of the field.
In 1,575 fourth-down situations where Romer concluded that coaches would have been better off going for it, they did so seven times.
Source: Virginia Postrel, "Strategies on Fourth Down, From a Mathematical Point of View," The New York Times, September 12, 2002; Kevin Sherrington, "Coach, it's OK to go for it on fourth down," Dallas Morning News, September 2, 2002; David Romer, "It's Fourth Down and What Does the Bellman Equation Say? A Dynamic-Programming Analysis of Football Strategy," NBER Working Paper No. 9024, June 2002, National Bureau of Economic Research.
For NBER abstract
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