Unlike some of our other strategy essays, the complexity of this problem precludes constructing a concise yet representative decision tree.  So we'll answer this problem numerically using the SportsQuant football simulator.  For this analysis, we consider games in which the defensive team trails by one point.  The defense employs one of two possible strategies:

1. Play hard and hope to get the ball back.

2. Concede the touchdown and get the ball immediately.

We estimated the probability that the defensive team wins using 10,000 simulated conclusions for games inside the two-minute warning.  Furthermore, we've given the offense the ball first-and-goal at the 10-yard line (a field position from which allowing a touchdown wouldn't necessarily look bad).  The figure below shows the difference between probabilities of winning by conceding the touchdown and by playing conventional defense.  When this difference exceeds zero, the defense improves its chance of winning by surrendering an uncontested touchdown.  When this difference is less than zero, conventional defense is the best approach.

The figure shows that as the time remaining decreases (left to right), allowing the offense to score becomes more appealing.  This is to be expected since such a strategy is clearly self-defeating if there is any reasonable amount of time remaining.  Only when there is time for a small number of plays is this strategy appealing.  Furthermore, the figure shows that this strategy is more appealing with fewer timeouts remaining than with many.  Again, this would be expected since a team with three timeouts could force a kick with very little time elapsing from the game clock.

The analysis shows that conceding a touchdown becomes more appealing to a team trailing by a point on defense as the game winds down.  The time at which to surrender depends on the number of timeouts remaining.  When the defense has 0, 1 or 2 timeouts, conceding a touchdown becomes the optimal strategy with a little more than a minute left (between 65 and 70 seconds).  However, when the defense has its full complement of timeouts, this strategy is only useful with fewer than 15 seconds remaining.  (Our analysis suggests 00:12.)  The dramatic difference is because the team with three timeouts can stop the clock immediately after the first three downs.  So they will either allow a touchdown (unintentionally) or force the offense to attempt a field goal -- either scenario using only a few seconds.  On the other hand, teams with two timeouts or fewer can expect at least 30 seconds to run off the clock, provided the offense runs the ball and stays in bounds.

Discussion

It's interesting that a team can improve its chances of winning by allowing its opponent to score deliberately.  But our analysis shows that this is the case for a football team trailing by a single point on defense.  (For more on how point-scoring and game-winning are occasionally at odds with each other, see our discussion on the marginal value of an additional point scored.)

These simulations assumed that the offensive team had a first-and-goal from the 10-yard line.  Down, distance and field position influence the particular times at which the optimal strategy changes.  However, the general pattern remains -- when the time remaining decreases enough, the defense is best served allowing a touchdown.  It would be interesting to see this strategy employed.  For instance, if instead of first-and-goal, the offense had a first-and-10 at mid-field, such a defensive strategy could reasonably save much more time and may be optimal with as much as five minutes remaining.  The difficulty (aside from the psychological aspect of surrender) would be in disguising the true intent.  A defense would need to allow a long touchdown run without looking like they did it on purpose (lest fans become angry, or the opponents deduce their plans).

It should be noted, though, that while clever strategy can improve the defense's chances of winning, it's not a cure-all.  Either way, things are bleak, and the defensive team is unlikely to come back.  The offense has a high probability of running out the clock or scoring again (or both).  Conversely, even if the defense gains possession, a scoring drive in with under 2:00 remaining is far from given.