Why Baltimore was correct to kick an extra point for an eight-point lead against Washington

With 4:47 remaining in their game vs Washington, the Baltimore Ravens scored a touchdown that increased their lead from 21-20 to 28-20.  You probably know what happened next: the Redskins drove down the field, overcoming the loss of RG3 along the way, with Kirk Cousins hitting Pierre Garcon for a touchdown with 32 seconds remaining and then successfully executing a QB draw for the tying two-point conversion; Washington then went on to win the game in overtime.  In the aftermath of the game, some analysts, such as Grantland's Bill Barnwell and espn.com's Gregg Easterbrook (aka TMQ) suggested that Baltimore ought to have attempted a two-point conversion after their touchdown, which if successful would have given them a nearly insurmountable nine-point lead.  They both suggest that the two-point try offered great upside with little risk, pointing out that even if it failed, Washington would still almost certainly only end up tying the game with a touchdown.

But closer examination of the issue reveals that this is a rare case where Barnwell and Easterbrook (normally very sharp football thinkers) are wrong, and the conventional wisdom manifested in John Harbaugh's decision is right.  In fact, Barnwell and Easterbrook have fallen prey to the very same error that underlies many of the legitimately foolish strands of football's conventional wisdom: defining the issue in terms of vaguely-specified risk/reward criteria, instead of a rigorous analysis of the win probabilities involved.  The math involved in this particular case is actually too simple to be of much inherent interest, but it still serves as a valuable reminder of how important it is to think through these questions in quantitative terms - we see that even top-caliber analysts can be led astray when they rely entirely on their intuition. 

Here, the key point is that by kicking an extra point, Baltimore made it considerably more difficult for Washington to tie the game with a touchdown, as they now needed to convert a two-point try, rather than simply kicking an extra point.  The fact that Washington would have faced a much easier road to a tied game following a failed two-point attempt by Baltimore may not trigger any visceral perception of risk, but it still endows the extra point with a fair amount of upside in relative to a failed two-point try.  In order to estimate the success probability at which Baltimore's two-point try would have been a break-even proposition, we need to weigh this upside against the greater reward offered by a successful two-point conversion, which would result in a nine-point lead. 

We will use P(B) to denote the probability of Baltimore successfully executing a two-point conversion, P(W) to refer to the probability of Washington executing a successful two-point conversion.  (For simplicity, we assume that probability of extra point success is 100% - this variable largely cancels out on both sides of the equation, and the logic of the conclusion can easily be restated in order to allow this assumption to be relaxed.)  Now, restricting ourselves to the space of potential outcomes where Washington outscores Baltimore by exactly one touchdown over the remainder of the game (since these are the only feasible outcomes where Baltimore's post-TD decision is directly reflected in the final result) we see that Baltimore faces the following dilemma:

Option 1: Kick an extra point.  Now with probability P(W), the game goes to overtime, and with probability 1-P(W) Baltimore wins in regulation.
Option 2: Attempt a two-point conversion.  Now with probability P(B), Baltimore wins the game in regulation, and with probability 1-P(B), the game goes to overtime.

So to first approximation, Baltimore's two-point try breaks even if P(B) = 1-P(W); i.e. if their probability of success is as high as Washington's probability of failure.  While we cannot say for certain that this hurdle was not reached in the game, we note that it requires that at least one team to have a two-point success probability of over 50%* - in which case they ought to be adopting the two-point conversion as their general post-touchdown strategy.  This conclusion can be summarized by pointing out that statistically, a team is better off needing to defend a two-point conversion than they are needing to make one of their own, unless one or both teams would be successful more than half the time. 

And in fact, the two-point conversion attempt has yet another factor in its disfavor.  As indicated above, the game situations where Washington outscores Baltimore by exactly one touchdown over the remainder of the game are the only ones in which Baltimore's post-TD decision has direct bearing on the final score.  But the probability of the various rest-of-game outcomes is not an entirely exogenous factor to their decision; in particular, a successful two-point conversion, by clarifying Washington's scoring requirements for the remainder of the game, would make it slightly more likely that Washington would outscore them by one TD plus another score of some variety, thereby winning the game outright themselves.  This is still a very low-probability event - scoring twice in the final five minutes is always a tall order - but it still means that a slight downward adjustment to the equity of the two-point try is in order.  Therefore, even if the nominal break-even point of P(B) = 1-P(W) were reached, we conclude that the two-point attempt would still be an incorrect decision. 

* Here is we can bring extra-point probability back into the mix: the actual threshold is not 50%, but P(XP)/2.  But the general consideration remains the same: if a team's two-point probability were in excess of this value, they ought to attempt a two-point conversion after virtually every touchdown.