When the Safe Move Makes it Less Likely to Win
On Week 15 of the 2014 NFL Season, the division leading Arizona Cardinals faced off in St. Louis vs. the Rams. In the 4th quarter 6:10 remaining, with the Cardinals' defense playing very well, St. Louis faced a 4th and 1 at the Arizona 1 yard line, down 3-12. Needing a touchdown and a field goal (and assuming no more Arizona points), the "safe" move is to "take the points," and try to get a stop. Taking the points keeps you alive, going for a touchdown and failing effectively means you lose the game as time is running out. But does staying alive actually mean you have a better chance of winning the game?
The Logic
People in favor of taking the Field Goal say "you need to take the points" or "if you fail, you've lost the game. You have to stay in the game to win it!"
The counterargument of going for the touchdown is "You're on the 1 yard line now, it's easier to score a touchdown here and then try for a field goal than it is to kick the field goal now and drive the length of the field for a touchdown!"
Which one is right?
The Assumptions
The choices coach Jeff Fisher had were to a) Kick a field goal, going down by 6 and kicking off back to the Cardinals with about 6 minutes to go, and b) Go for a touchdown, which would lead to one of 2 cases, b1) Scoring, being down by 2 and kicking off, or b2) Getting stopped, and continuing to be down by 9.
No matter what happens, the Rams would be still behind and giving the ball back to the Cardinals so the chance of them winning the game are less than 50%.
Assuming a FG% of 100% from the 1 yard line, a 19 yard field goal, that would put the rams down 6, kicking off to the Cardinals. If they did that, assuming an average kickoff return to the 25 yard line, the Advanced Football Analytics Win Calculator has the Rams' win probability at 14%.
Assuming the Rams score a TD, with the same field position, the probability balloons up to 31%, more than double that of kicking the field goal. This makes sense as it now only takes a field goal to win the game for the Rams.
Those in the field goal camp protest, "but if you miss the TD, you have now lost the game!" Using the same Win Calculator, if the Rams are still down by 9 points, but the Cardinals have the ball at their own 1 yard line, the Rams' win probability is down to a dismal 3%, not quite 0, but close enough. This doesn't take into account the Cardinals are playing with their 3rd string quarterback after backup Drew Stanton was injured earlier in the game.
The Calculations
Simplifying the chance of winning if the miss down to 0%, The Rams should go for a touchdown only if the chance of a touchdown meets this criteria:
P(Touchdown)*(0.31) + (1-P(Touchdown)*(0) > P(FG)*(0.14)
Assuming a 100% chance of kicking the FG, and simplifying P(Touchdown) as the variable x, the inequality boils down to this:
x*0.31 > 0.14
Solving for x
x > 0.45
Thus, the Rams should go for it if they have a 45% chance of scoring a touchdown.
The Fourth Down Calculator at Advanced Football Analytics uses similar numbers, making a couple of different assumptions and also crediting the Rams with a 4% chance of winning on a failed 4th down play. In recent history, 4th and 1 has an overall success rate of 64.7%, though the type of play called has a much a quite an effect. The New York Times 4th down bot says the chance of success is 55%, with teams choosing to go for it winning 19% of the time (corresponding mostly to 55%*.31, but also the 4% chance of winning even when failing).
No matter what the numbers, 55% or 64.7% are both much more than 45%. The Rams clearly should have gone for it, and it isn't even close.
To make matters worse, before looking at the outcome of the play, the Advanced Football Analytics Win Calculator has the Rams' WP at 30% at that 4th-and-1. By kicking the field goal, they actually cut their chances of winning in half!
The Non Math Arguments
Let's look at the non-math arguments one more time. The Cardinals defense had been playing very well all game, so a touchdown might have been unlikely. But even if that were the case, kicking a field goal would mean driving the length of the field again to score that touchdown, which is more unlikely than converting 1 yard on 4th down. If they did score a touchdown, they would only have to drive part of the length of the field to kick the the field goal.
Against a good defense, which is more likely to happen, getting 1 yard, or getting 70+ yards?
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