There’s nothing like a good argument over 2-point conversions to get the NFL interested in its math homework. Week 10 was a big week for 2-pointers, but we’re also in the middle of a seasons-long upheaval in the underlying logic that goes into some of the more complex extra-point decisions, meaning that the wisdom of what to do in specific scenarios has changed, even from just a few seasons ago.

According to ESPN Stats & Information Group, there have been 1,045 two-point conversion attempts since 2001, with teams converting 501 of those tries. That’s a 47.9 percent conversion rate; given that a successful attempt yields 2 points, that means the expected value from an average 2-point try is 0.96 points.

Interestingly, that’s almost exactly what the expected value is from an extra point these days. Since the NFL moved extra-point kicks back to the 15-yard line last season, teams have a 94.4 percent success rate, which means that an extra point has an expected value of between 0.94 and 0.95 points.

This means that, all else being equal, the average team should be indifferent between going for two or kicking an extra point. Unless the game situation (i.e., late in the second half) or team composition (e.g., a bad kicker, or an offense or an opposing defense that is very good or very bad) changes the odds considerably, the decision to go for two or kick an extra point shouldn’t be controversial. In the long run, things will even out, because the expected value to the offense is essentially the same in both cases.

That’s the long run. In the short run, there will be ugly outcomes. And we saw two of those play out this weekend.

Pittsburgh and Dallas combined for a historic 0-6 day

On Sunday, the Steelers missed four 2-point conversion attempts, after scoring touchdowns to go up 6-0, 12-3, 24-23 and 30-29. The last two attempts were not controversial: Being up by 1 point in the fourth quarter is a clear “go for two” situation. The Steelers’ first two failed attempts occurred in the first quarter, but there’s good justification for those attempts: Pittsburgh has a great offense, centered around a top quarterback (Ben Roethlisberger), top running back (Le’Veon Bell) and top wide receiver (Antonio Brown). In 2014, 2015 and the first nine weeks of 2016, the Steelers converted 14 of 17 tries! In other words, Pittsburgh is not the average offense, so there’s good reason to think that over time, Pittsburgh will score more points by always going for two. It just didn’t work out that way on Sunday.

The Cowboys unsuccessfully went for two after both of the team’s touchdowns in the final two minutes, with Dallas up by 5 after each score. Those were also no-brainer decisions — you gain much less, in terms of win probability, by extending a 5-point lead to 6 points late in a game than the additional gain you’d have by extending the lead to 7 points.

Before Sunday’s game, according to Pro-Football-Reference.com, the NFL record for combined failed 2-point tries by both teams in a game was four, set four times.

But with extra points no longer functioning as gimmes — and potentially being disasters — we should expect to see more of these outlier performances from time to time. Since the rule change last year, teams have roughly doubled their rate of going for two in the first 10 weeks of the season, from 4 percent to 5 percent in previous years to over 8 percent this season and last:

The Seahawks went for the kill, but did it make sense?

With 4:24 remaining, Seattle scored a touchdown that gave them a 31-24 lead over New England. At that point, the Seahawks chose to go for two, the idea being that a 9-point lead would nearly clinch the game by turning it into a two-score game. They failed.

There is some logic to the attempt: If Seattled opted to go for one instead of two and made it, it would have an 8-point lead. That, in turn, would mean that whether the Patriots could tie the game with a single possession would depend on how they did on a 2-point attempt. So the Seahawks would be trading the risk of their own offense going for a two-point try for the risk of the Patriots offense going for two. As discussed above, teams should generally be indifferent to whether they go for two or their opponent goes for two, as the conversion rate is right around 50 percent. But given that both the Patriots and Seahawks have strong offenses, it’s easy to see why the Seahawks would feel that they would have a better chance of converting a 2-point attempt than of stopping a Patriots 2-point attempt. (However, it’s worth noting that since 2014, the Patriots are 0 for 1 on 2-point attempts, while the Seahawks are now 1 for 7.)

Four years ago, writing about a nearly identical situation, I said that going for two is not the correct call because there were more than four minutes left in the game. Even if we assume a 50/50 conversion rate on 2-point attempts, you don’t gain enough by pushing the lead to 9 points to risk leaving it at 7 points.

Why? At the time, the thinking went something like this: In some ways, being up by 7 is twice as risky as leading by 8, because when you’re up by 8 the trailing team needs to convert the two-point try to tie the game, and that only happens about 50 percent of the time. There’s no way the additional benefit of pushing the lead from 8 points to 9 points could exceed that. And it’s not like being up by 9 clinches the game.

According to Pro-Football-Reference.com, Seattle would have had a win probability of 89.9 percent if it had hit the extra point to take an 8-point lead, assuming that the Patriots would have the ball at their own 25 with 4:20 remaining. Alternatively, if Seattle had converted on a 2-point try, bumping the lead to 9, its win probability would have been 3.3 points higher, at 93.2 percent. But if the team had gone for 2 and missed, having “only” a 7-point lead would have put Seattle’s win probability at 85.5 percent, representing a more significant change in win probability.

That’s because the Patriots, with more than four minutes remaining and down by 9, would have gone into hurry-up mode and had the opportunity to try to gain a second possession. If there were only a minute or so remaining, then the difference between being down 7 or 8 would be essentially equivalent to the difference between being down 8 or 9 and the identity of a team’s offense (or opponent’s defense) should dictate that decision. But with four minutes remaining, that logic doesn’t apply.

That was the logic four years ago. However, things have changed a bit. That’s because the extra point has dropped from a near certainty to a 94 percent or 95 percent proposition. And that change moves the math in favor of going for two, in two ways:

Up by 7 after scoring the touchdown, Seattle would have a 5 percent or 6 percent chance of missing the extra point.

If Seattle goes for two and misses, New England would still need to successfully hit the extra point after scoring a touchdown.

So Seattle head coach Pete Carroll’s decision-making process in this situation needs to include those factors; the harder it is to make an extra point, the better it is for his team to avoid having to kick one and the better it is for the opposing team to have to. In other words, here are the three scenarios for teams to consider when up by 7 after scoring a touchdown late (but not too late) in the game, using the Seattle-New England scenario as a reference point:

Kick Extra Point (Go Up 8): This only happens 94 percent to 95 percent of the time. And it’s worth noting that Seattle kicker Steven Hauschka has converted on only 89 percent of his attempts since 2015, including a miss earlier in the night.

Go For Two, Miss (Now Up 7): This doubles the opposing team’s odds of winning (because a 50/50 event is now eliminated), assuming the opponent connects on the extra-point try. But that’s no sure thing, and Stephen Gostkowski of the Patriots has missed two this season (93 percent).

Go For Two, Make (Now Up 9): This doesn’t quite end the game and encourages the Patriots to play as aggressively and quickly as possible. New England had all three of its timeouts remaining, so instead of methodically driving down the field (as happened in real life, with the Patriots down only 7), the Patriots would have treated it as a true two-possession game.

So was it the correct decision? Frankly, this one is on the border. Under the old extra-point rules, unless either New England or Seattle was significantly more likely than 50/50 to convert on a 2-point try, going for two this early would have been the wrong move. Now, with extra points no longer guarantees, it’s a closer call. But in general, I would choose to kick the extra point. That encourages the opposing team to play conservatively and treat the game as a one-score game even though it would still need to convert the 2-point try just to tie the game.

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