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Football and the Fourth-Down Dilemma: Go For It?

If you are an avid football fan like me, Sunday means only one thing: Football. You sit down to watch your favorite team play and cheer as the offense drives down the field deep into the red zone. Now they are near the goal line, moments away from scoring. The running back runs into a wall of defenders only to be stopped short of the goal line. It’s fourth down now, and there are only a few feet of grass between the spot of the ball and a touchdown. The coach, however, elects to kick a field goal instead. Only 3 points are registered on the scoreboard and every fan is disappointed at the anti-climactic ending to an exciting drive.
This scenario is all too common in the world of football, and it grinds the gears of every football fan that has to witness it. Why doesn’t the coach just go for the touchdown? Why does the coach choose to kick a field goal almost every time? I came across an article by an N.F.L statistical analyst who had similar questions. He asked why teams punt the ball away every 4th down, effectively giving up a quarter of their opportunities for only 35 yards of field position. He pointed out how boxed in our thinking is, where we’ve reached a point that, when its 4th down, we unquestionably think punting is the best option. The analyst cited studies and confirmed that it pays to be more aggressive and go for it on fourth down more often than not. A team with this aggressive strategy will score more points and should win more games as a result. If you’re interested, you can read more about this article here:
http://fifthdown.blogs.nytimes.com/2009/09/17/a-new-study-on-fourth-downs-go-for-it/
After reading this article, I had to wonder why this same thinking can’t be applied to the situation above. Doesn’t it also pay off to go for it on 4th down when you’re near the goal-line? Or is it better to be safe and go ahead with the 3 points from a field goal? In an attempt to answer this question, I turned to a little basic game theory. Assuming that the time left on the clock is not yet a factor, we can simplify the scenario to look something like this:

Payoff Matrix

Here we can see that the offensive coach essentially has 3 options: call a pass play, call a run play, or kick a field goal. Likewise, the defensive coach can opt to defend a pass play, defend a run play, or defend a field goal. We are making a couple other assumptions here. We are assuming that the two teams are ideal in the sense that if the offense chooses to pass and the defense chooses to defend the pass, the offense will fail to score and the defense will be successful. If the offense chooses to run and the defense chooses to defend the pass, then the offense will always score since the defense picked the wrong strategy. We are also assuming that it is obvious to the defense when the offensive coach chooses FG, and so the defensive coach will always pick the FG defense. If the offense chooses to either pass or run and is successful, the payoff is 7 points on the scoreboard, but if the offense fails then their payoff is 0 and the defense has a payoff of 7 since that’s how many points they prevented.

So in this game scenario, what is the best strategy for the offensive coach? He can choose to play it safe for a payoff of 3, or he can be aggressive and try for a payoff of 7 but risk ending up with nothing. Since the defensive coach probably has no idea what the offense will do, he will have to take a guess, meaning there is roughly a 50% chance he will defend the pass or a 50% chance he will defend the run. What does this mean for the offensive coach? No matter what he chooses, PASS or RUN, there is a 50% chance the defense guessed wrong and so there is a 50% chance the offense will score. So in the long run, the average payoff after several of these game scenarios will be 3.5, which is greater than the payoff of 3 the offensive coach would receive if he elects to kick a field goal.
While this theoretical game scenario is overly simplified and idealistic, it is obvious that aggressive offenses will score more points in the long run and win more games as a result. This does not mean offensive coaches should always elect to go for it, since there will always be certain situations where a field goal is more appropriate, but rather proves that it is theoretically better to be aggressive more often than not. Perhaps, as the N.F.L statistical analyst from the article pointed out, the reason coaches opt for field goals isn’t strategy-based at all. Rather, it is based on what economists call Prospect Theory, where almost all people tend to fear losses more than equivalent gains. The fear of losing out to the defense and getting a payoff of 0 can make owners, teammates, and fans unhappy. So the coach fears the loss disproportionately, and this clouds his judgment. No coach wants to be in the hot seat and risk his job security after making one bad call. To him, kicking the field goal in that situation is safer and more acceptable.

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