On Dec. 23, the 8-6 Dallas Cowboys trailed the New Orleans Saints 31-17 late in the fourth quarter. Dallas’ path to victory seemed obvious: score two quick touchdowns and kick two extra points, then win the game in overtime. The first half of the plan worked perfectly. The Cowboys scored a touchdown and added the PAT with 3:35 left, then got another TD and PAT with 15 seconds to go. In overtime, however, the Saints forced Tony Romo’s crew to punt, then drove for the winning field goal. That’s how it goes in the NFL—sometimes you do everything right and the luck doesn’t go your way.
Well, that’s likely how the Cowboys saw it. But in reality, Dallas fouled up the end game, succumbing to a fallacy that statisticians have recognized for going on five decades but NFL coaches have yet to comprehend. In a game they were desperate to win, the playoff-hungry Cowboys bowed to gridiron conventional wisdom and played for a tie. When his team scored that late touchdown to cut the lead to eight, head coach Jason Garrett should’ve made a call that seems radical only because nobody’s smart enough to do it. He should have gone for two.
I first learned about the wisdom of this strategy a few days ago, when a Slate reader named Walter Sun emailed me some calculations. Sun, a 38-year-old development manager from Seattle, assumed that—as per this article by Advanced NFL Stats’ Brian Burke—NFL teams have a 47.9 percent success rate on two-point conversions and make extra points 100 percent of the time. (It’s actually more like 98 or 99 percent in recent years, but Sun’s guesstimate simplifies the math and doesn’t change the ultimate result.)
If two-pointers are a 48 percent proposition, then the expected value of going for two is 0.48 * 2 = 0.96 points. That means that in the long run, the close-to-a-sure-thing extra point is the better percentage move.
But as this chart shows, it’s sometimes smart to go for two. Many of these cases are intuitive. If you’re down eight with three seconds to go, you need a touchdown and a two-point conversion to tie the game. And if you’re trailing by 11 with a minute left, it’s wise to get a touchdown plus two to reduce the margin to a field goal.
Going for two when you’re down by two touchdowns is less obvious but no less logical. Let’s stipulate that our hypothetical team—we’ll call them the Cowboys—does score the required touchdowns in the waning minutes while keeping the opposition off the scoreboard, because otherwise this exercise is pointless. In that case, the touchdown + extra point + touchdown + extra point approach will lead to a victory 50 percent of the time—the game will be tied at the end of regulation, and we’ll assume the Cowboys and their opponent have an equal chance of winning in overtime (and we’ll ignore the possibility of a rare-for-the-NFL tie game).
The touchdown + two-point conversion approach plays out like this:
- 52.1 percent of the time, the Cowboys miss the two-point conversion. In that instance, they’re trailing by eight points. After the second touchdown, they’ll have a 47.9 percent chance of converting the two-pointer to force overtime, and will then win in OT half the time. The Cowboys’ win probability in this case: 0.521 * 0.479 * 0.5 = 12.5 percent.
- 47.9 percent of the time, the Cowboys make the two-point conversion. In that case, they’ll trail by six. After the second touchdown, they kick the extra point and win the game. The win probability in this case: 0.479 * 1 = 47.9 percent.
Add those two possible paths together, and the total win probability for the alternate-universe Cowboys is 12.5 + 47.9 = 60.4 percent.
The difference between the win probability for the right decision (60.4 percent) and the wrong one (50 percent) is significant but not enormous. Kicking the extra point in this situation isn’t nearly as dumb, then, as Ron Rivera’s much-derided late-game punt in Week 4. But the choice is still clear: Going for two takes a 50-50 proposition and makes it a winning strategy.
(For what it’s worth, touchdown + extra point + touchdown + two-point conversion is the worst route of all. That’s because it gives our Cowboys no path to an overtime-producing tie and no opportunity to recover from a botched two-pointer. The win probability here is the same as the probability of making a single two-point conversion: 47.9 percent.)
Walter Sun, it turns out, isn’t the first guy to figure out that football coaches are doing this wrong. University of Michigan economist Richard C. Porter knew it in 1967. In “Extra-Point Strategy in Football,” an article published in the American Statistician, Porter worked out that a football team need only have a 38 percent success rate on two-pointers to make going for two the right approach in a late-game, needs-two-touchdowns scenario. (Porter was writing about college football in the pre-overtime, regulation-tie era. The 38 percent finding was based on an assumption that “the utility of a tie” was 0.5 for the trailing team. That’s functionally equivalent to assuming that said team would win half the time in a theoretical overtime period.)