With 10:52 left in overtime against the Saints, the Falcons faced a 4th and inches at their own 29. Head coach Mike Smith decided to for it. Was it a smart call?
Calculating Win Probability (WP) in OT is surprisingly simple compared to regulation time. Except for the final few minutes when there is real possibility of a tie, time is not a factor and the score is always tied.
A punt would be the conventional call. A typical punt from the 29 nets 38 yards, giving the Saints a first down at their own 33, worth 0.58 WP (a 58% chance of winning). This makes intuitive sense, because teams that win the coin toss are in a similar situation and win just under 60% of the time. The Falcons would therefore have a 0.42 WP following a punt.
If the Falcons successfully convert the first down, they'd have a fresh set of downs at (at least) their own 30, good for a 0.57 WP. Notice that if they succeed the situation is nearly the symmetrical opposite of the punt. Atlanta would have the ball at slightly worse field position as New Orleans would following a punt.
If the Falcons fail to convert, they'd hand the ball over to the Saints at their own 29, a nearly fatal situation, worth only 0.18 WP. I realize this is surprisingly high, but it reflects actual overtime game results. Often, coaches will conservatively run three times into a brick wall to set up a 40-plus-yard field goal, which is far from a slam-dunk.
Conversions on 4th and 1 are typically successful 74% of the time. But this includes all 4th "and 1" situations, everything from a yard-and-a-half to go to an inch to go. For now, let's stipulate that it's 74%.
The total WP for the conversion attempt is:
0.74 * 0.57 + (1 - 0.74) * 0.18 = 0.47
The go for it option is worth, on net, a 0.47 WP. That's better than the 0.42 WP of the punt option, at least according to league-average percentages. I'm certain the fact that Drew Brees and the rest of the Saints offense is not league-average helped sway Mike Smith's mental calculus.
If you do the math the other way around, the break-even conversion probability would need to be 62% for the conversion attempt to be worthwhile. And if we don't buy the 0.18 WP following a failed conversion attempt, it would still have been worth going for it. Even If failing to convert meant an instant loss, the possibility of success would be slightly better than punting (0.74 * 0.57 WP = 0.43 WP).
Smart call. It just didn't work out. To Sean Payton's credit, he got aggressive and didn't settle for the long field goal attempt. The Saints got aggressive, moved into chip-shot range, and won the game.