This is a nice, short profile of David Romer, an economist and lifelong sports fan who briefly turned his attention to football some years ago.
In 2002, Romer wrote the first serious academic paper asking the question “When should football teams go for it on 4th down?”, applying rigorous analytical from economics and mathematics.
Here’s the simple summary: a touchdown in football is (usually) worth 7 points, and a field goal is worth 3 points. A team will often face the situation that, on 4th down, they can either kick a field goal with a relatively high probability of success (say 80-90%), or they can go for it on 4th down (which has something closer to a 40-50% success rate) and continue to try for the touchdown (not a guarantee).
Romer’s conclusion was basically that teams should go for it on 4th down far more often than they do. This is essentially an expected value argument: if, by going for it, you get 7 points about 40% of the time, that’s an average of 2.8 points per attempt; if, by kicking the field goal, you get 3 points about 80% of the time, that’s an average of 2.4 points per attempt. So in the long run, going for it will produce more points.
However, the fact is that teams rarely go for it on 4th down, usually only trying this strategy in desperate times. So what account for the difference between the theoretical conclusion and the practice of professionals?