WITH an annual salary last reported at $7m a year, Pete Carroll can certainly afford a small household staff. Now that the Seattle Seahawks’ coach arguably cost his team a National Football League (NFL) championship, by ordering a questionable passing play as the clock wound down in last night’s Super Bowl, he might want to consider investing in a bodyguard and food taster. After all, numerous American football fans in the Pacific Northwest are likely to have designs on his life following what a local newspaper has declared “the worst play call in Super Bowl history.” To bring readers among the 98.5% of humanity that did not tune in up to speed (around 115m people watched the game, meaning that some 6.9 billion did not), the final minutes unfolded as follows. As the fourth quarter began, Seattle led the New England Patriots 24-14. New England promptly scored touchdowns on back-to-back drives, giving them a 28-24 edge. The Seahawks had two minutes left in the game to answer. Russell Wilson, Seattle’s quarterback, immediately launched a 31-yard pass to bring his club to midfield, and secured another first down with an 11-yard completion.

Then, with little over a minute to go, the Seahawks reached the brink of a riveting come-from-behind victory, thanks to one of the most remarkable plays in the sport’s history (shown here in slow-motion video). Mr Wilson launched a deep pass down the right-hand sideline. Both Jermaine Kearse, the intended receiver, and Malcom Butler (pictured, at left), the defender covering him, leapt into the air. Neither came down with the ball, and after it was deflected into the air, it bounced off their hands and bodies five times. Finally Mr Kearse corralled it. Replays showed that the ball never touched the ground, putting the Seahawks five yards away from successfully defending their title from last year as Super Bowl champions.

On the following play, Mr Carroll had Mr Wilson hand off the ball to Marshawn Lynch, Seattle’s superb running back. He advanced four yards, putting the Seahawks inside the Patriots’ one-yard line. That put Seattle in the driver’s seat: even though they were still trailing, they had three chances (the second, third and fourth downs) to advance the ball less than 36 inches (91 cm) and gain a three-point lead. Teams in that situation go on to win just under 85% of the time.

This case, however, would be the exception. After letting another 40 seconds burn off the clock, to ensure that New England would not have time to answer with a score of their own, Mr Carroll chose to pass instead of run. Mr Wilson took the snap, and saw Ricardo Lockette (pictured, at right) appear wide open with a clear path to the end zone. The quarterback fired the ball in his direction. But Mr Butler was about to have his revenge for being beaten by Mr Kearse two plays earlier. Reading Mr Wilson perfectly—he later said that he had tried and failed to defend a similar route during the Patriots’ practice sessions—he broke straight for the spot Mr Wilson was targeting and intercepted the pass (see this detailed replay), ending the Seahawks’ season.

Even before the game clock ran out, football pundits across the country shook their heads at Mr Carroll’s decision to pass. Mr Lynch is arguably the best running back in football. Moreover, thanks to his ability to eke out a few extra inches while defenders try to bring him down, he particularly excels in short-yardage situations. This season, Seattle secured a first down or touchdown 81% of the time they attempted a run with less than two yards to go on third or fourth down. That was the second-best mark in the league, well above the 65% overall average. Meanwhile, New England’s defence was unusually porous against such plays: they allowed first downs or touchdowns on 81% of such attempts, ranking dead last in the NFL. On paper, Mr Lynch seemed like a lock to force his way into the end zone for a game-winning touchdown.

The fact that Mr Carroll took the ball out of his hands, gambling instead on a pass play, has already been described as “awful, “inexplicable” and “the worst Super Bowl play of all time”. Mr Carroll’s fumbling attempt at an explanation—“we were throwing the ball really to kind of waste that play”—only made matters worse. Even the Seahawks’ own players second-guessed their boss. “I’m speechless…I will never understand that,” said Bruce Irvin, a linebacker. “I think we were all surprised,” echoed Doug Baldwin, a receiver. “I don’t know, man. I’m just trying to make up an explanation.”

But does the fact that Mr Carroll, just minutes after a devastating reversal of fortune, could not express a coherent justification for his call mean that the play could not be justified? A brief game theory refresher may be in order. There is no play that cannot be stopped if the defence knows it is coming. If the Seahawks were to sign a blood oath promising to have Mr Lynch run the ball, the Patriots could simply throw all 11 defenders at him and stop him in his tracks. In order for a run by Mr Lynch to be effective, the opposing team must believe there is some chance, however small, that the offence will do something else. For such a threat to be credible, Seattle must randomly call a different play every so often. Just how often depends on how much better they are at rushing than at passing in short-yardage scenarios.

In one of the rare demonstrations of mathematically optimal play in the NFL, the league’s coaches as a group have found their way to a textbook equilibrium. During the 2014 season, offences on an opponent’s one-yard line ran two-thirds of the time (there were 212 rushes and 106 passes). The running plays went for touchdowns 57.5% of the time...and the passing plays went for touchdowns 57.5% of the time. This is exactly what theory would predict: if either rushing or passing offered a superior success rate, teams would shift their play mix towards the better option until the balance was restored. Because the Seahawks’ running game is so powerful, Seattle’s optimal run frequency is higher than that of the league as a whole: it might be 75% or even 80%. But there’s a lot of room between those figures and 100%. Even if the Seahawks should run 80% of the time with a single yard to go, that still means that about half the time they would call at least one pass during a three-play sequence (as Mr Carroll expected to have, starting with his second-down play).

Moreover, nobody was more aware of Mr Lynch’s abilities than Bill Belichick, the Patriots’ notoriously crafty coach. Half the Patriots on the field—including Mr Butler himself—subsequently told reporters that they were expecting a rush. To maximise the odds of stopping the run, Mr Belichick stacked eight defenders right on the line of scrimmage, ensuring that it would take every muscle in Mr Lynch’s body to reach the end zone. Mr Belichick’s trade-off was to leave just three New England players to keep an eye on potential receivers, each in man-to-man coverage. And the play Mr Carroll called, a pick in which one receiver (Mr Kearse) tries to impede the path of a defender (Mr Butler, an untested and unproven rookie) assigned to cover another receiver (Mr Lockette), is most effective against man-to-man schemes. Mr Carroll’s postgame assertion that “it was not the right matchup for us to run the football” has been widely derided. But Mr Belichick was clearly daring Seattle to beat him with someone other than Mr Lynch, making a pass more likely to succeed and a run less so.

The phrase “Monday morning quarterbacking”, which originated with American football but is now used broadly, was invented for a reason: to heap appropriate scorn on critics who wait for the benefit of hindsight before rendering judgment. Had Mr Lockette secured the ball, Mr Carroll would be the toast of the town, praised for thinking outside the box and being bold enough to call a pass when everyone expected a run. And if the play had ended with a garden-variety incompletion, it still would have served its purpose, by establishing the pass as a meaningful risk and increasing the odds of Mr Lynch fighting his way to a touchdown on a subsequent down. The only reason Mr Carroll is being raked over the coals is because the play happened to end in an exceedingly improbable interception. Not one of the previous 106 passing plays that NFL teams launched from the one-yard line in 2014 was picked off. Moreover, the odds of a turnover on a pass were no greater than on a run: in the 2014 regular season, 1.5% of Mr Wilson’s pass attempts were intercepted, while Mr Lynch fumbled the ball on 1.4% of his carries. There was nothing wrong with Mr Carroll’s play call. It just didn’t work out.

Partly due to diligent preparation, but clearly with a generous dose of dumb luck, Mr Butler made a good guess and a great play. Maybe he will turn out to be as good a cornerback as Mr Carroll is a coach.

CORRECTION: One commenter has helpfully noted that this piece overstated the odds of a turnover had the Seahawks called a rush for Mr Lynch. Of the 26 fumbles in his career, seven have come as a receiver rather than on running plays. His fumble rate while rushing has thus been just 0.9%. Moreover, offences have the opportunity to recover fumbles but not interceptions, and do so about 38% of the time. Factoring in the likelihood of a recovery further cuts the chances of a turnover on a rush by Mr Lynch to around 0.6%. (A more rigorous analysis would use a projection for fumble rate rather than a historical average, by weighting recent seasons more heavily and then regressing the forecast back to the league average. But the result would probably be fairly similar.)

Furthermore, Mr Wilson had an unusually good season in 2014 when it came to avoiding interceptions. His 1.5% rate was significantly lower than both his 2.1% career mark and the 2.5% league average. With a dose of regression, in the future, defences would probably be expected to pick off around, say, 2.2% of his passes. So it was wrong to say that “the odds of a turnover on a pass were no greater than on a run”. In fact, they were probably nearly four times (2.2% versus 0.6%) higher.

Nonetheless, the most important point is that four times a small number is still a small number. Even though the cost of a turnover was clearly catastrophic—about 84 percentage points of win probability—the increased likelihood of coughing one up on a pass compared with a run was something on the order of 1.6%. That differential reduces the expected value in win probability of the pass play by just 1.3% or so, which is a rounding error given the uncertainty in all the other projections that go into a play call. The only questions that should have been in Mr Carroll’s mind were how to maximise the odds of a touchdown while minimising the amount of time left on the clock after doing so. Turnovers are sufficiently rare that the difference between an incomplete pass and an interception, or between a stuffed running play and a fumble, was barely worth factoring into the calculation.

Numerous commenters have also raised another distinction, between defending Mr Carroll’s decision to pass at all and supporting the specific passing play he called. The above post exclusively addressed the choice of pass versus run, and made no effort to evaluate the comparative merits of the different passing options in Mr Carroll’s playbook. It certainly appeared risky to throw up the middle, into the jaws of the Patriots’ goal-line defence, rather than running routes towards the outside where there was less traffic. Moreover, given that the strongest argument for calling a pass was that New England was expecting a run, the best option might have been a playaction scheme, in which the offence fakes a run before unleashing a throw. That would also have provided Mr Wilson the opportunity to run the ball in himself if he had a clear lane to the end zone, or to throw it out of play if no receivers were open.

On the other hand, it’s hard to criticise the choice too strongly given how good it looked when Mr Wilson released the ball. With better blocking, a better receiver or a better throw, it would have been a touchdown.

Dig deeper:

Applying game theory to American football (Feb 2015)

Analysing trick plays in American football (Jan 2015)

Explained: the simple principle behind the read-option (Jan 2014)