Allowing a TD in tie games inside the 2 minute warning

We've all been there. We're playing Madden, it's tied, there's 1:30 left and we have no timeouts. Your opponent has the ball at your 13 yard line after a 20 yard pass play. You're dead -- they run in twice, kick the field goal, and you lose.

Something you'll see a lot of Madden players do: allow the opponent to score a touchdown in this situation. Now, the NFL isn't a video game, but this seemingly ridiculous strategy may be the optimal strategy. The reason for this article is because this situation almost played out Monday night with the Chiefs: The Chargers had the ball at the Chiefs' 16 yard line with 1:10 remaining. The Chargers lined up as if they were going to run the ball (a terribly bad strategy to begin with). Would just letting them walk into the end zone be better than trying to stuff them and hope for a missed/blocked field goal?

Let's take a look at some of the math for this situation:

In this situation, assuming the run(s) get stuffed, we can safely assume that the game-winning field goal will be attempted from roughly 33-35 yards. Over the last 2 years, field goals from 30-39 yards have been hit at a rate of about 89.8%. Because the attempt would be from the shorter side of that range, let's just round to 90% for ease of calculation. Thus, 90% of the time, by stuffing the run, you lose in regulation. The other 10% of the time, you will go to overtime (or block the field goal and return it for a touchdown and win in regulation). Let's assume, for argument's sake, that if the game goes into overtime each team will have about a 50% chance to win. So, 10% of the time, you have a 50% chance to win (the chances of winning in regulation are beyond negligible):

0.90 *(1.00) + 0.10*(0.50) = 0.95 or 95% loss probability
0.90* (0.00) + 0.10*(0.50) = .05 or 5% WP(win probability)

Bold signifies the overtime calculation, while non-bold signifies regulation.

Now, what happens if you allow the other team to score (assuming they'll just run the ball into the end zone if you let them -- but if they don't and lay down at the 2, your chances only drop from about 5% to 3%)? Assume a TD, now you're down 7 with ~ 1 minute remaining and receiving a kickoff. With no timeouts remaining, assuming an average field position of ~27 yard line (the league average), the TD probability in this situation is about 0.18 (according to advancenflstats, though I think this may be a bit high). This means that roughly ~82% of the time, you'll lose in regulation, while 18% of the time, you can score a tying TD and head to overtime. Now, assuming the same 50/50 win probability if it goes to overtime, the outcome probability looks like this:

0.82 * (0.00) + 0.18*(0.5) = 0.09 or 9% chance of winning
0.82 * (1.00) + 0.18 * (0.50) = 0.91 or 91% chance of losing

So, you can assume that by allowing them to go to freely to the end zone helps your chances of winning. Your WP goes from 0.05 to roughly 0.09. If the 18% chance to score a TD seems ambitious, realize that even if you give the trailing team only a 10% chance to score a TD, then both strategies give you an equal win probability.

If the opposing team decides to lay down at the 2 and your win probability goes from 0.05 to 0.03, you are only losing 2% WP. Whereas, by allowing a TD, you are gaining 0.04 WP, making it a +ExpectedValue play unless they lay down at the 1 more than half the time (which is probably a lofty assumption for the intelligent people with the job of NFL RB):
0.02*x (loss from shorter FG*probability of laying down) = 0.04*(1-x) (gain from letting them score)

Math is fun, but NFL coaches often totally disregard strategies that seem "crazy" even though it increases their chances to win. There's a reason why Belichick went for it on 4th and 2, and that's because he knows isn't afraid to try unconventional things that will up his win probability.