In one of the more intriguing match-ups of the weekend, Tony Romo and the Cowboys received the ball in a tie game just over midfield with 0:47 seconds remaining. After two incompletions and one pass to Jason Witten, the Cowboys faced a 4th-and-4 at the Steelers 43-yard line. With 0:32 seconds to go and one timeout left, the Cowboys trot out their field goal team for a miraculous 60-yard game-winning kick. Both teams would call timeout and Jason Garrett would ultimately decide to punt instead, essentially assuring overtime. Let's think about this decision generally before looking at the frequency specifics.

If you punt, you are conceding to play overtime, giving your team a 50/50 shot to win the game. If you kick a field goal from that range, the probability that it goes in is at best the same as the probability that you miss and your opponent can score before the end of regulation, giving you a 50/50 chance to win optimistically. If you go for it and convert, you will have a much higher probability of making the field goal even with time constraints -- if you fail, you are pretty much in the same situation as a missed field goal.

So, with that in mind, here are the numbers:

4th-and-4 converts historically at 53% resulting in an expected 61% for the Cowboys to win the game if they go for it (84% on a successful conversion vs 36% on a failure). A punt results in a 49% chance of winning (likely due to the possibility of a block/botched punt, a return for TD or extremely unlikely case of the returning team scoring after the punt). 60-yard field goals convert at about 15% and a miss would mean a 70% chance for the Steelers to win with approximately 0:25 seconds left on the clock. That makes the break-even point between deciding to punt or go for it a mere 27%. In other words, unless the Cowboys think they can only convert one out of four times or less on 4th-and-4, they should go for it.

Keep in mind, these probabilities are only league-average baselines, so there are definite margins for error. But, the difference between 53% and 27% is extremely significant. One other note is that this analysis assumes an average amount of timeouts left for both teams. In our case, Dallas was out of timeouts and Pittsburgh had one remaining -- most likely about average given the situation. Since it was a tie game and timeouts have a much greater affect on the probability of a team scoring when they are on defense, rather than offense, the timeouts left assumption seems reasonable.

The Cowboys also had a 4th-and-1 from their own 21-yard line right at the two-minute warning. Obviously going for it here is a much riskier proposition -- meaning a much more difficult decision for a coach in the conservative NFL, especially one on the hot seat under Jerry Jones. The numbers say to go for it (as they almost always do on 4th-and-1): E[WP Go-for-it] = 49% vs E[WP Punt] = 30%. This makes the break-even point 33% compared to league-wide 74% conversion on 4th-and-1. The primary reason is that if you punt the ball back, the Steelers will be in a position to drive down field and take the last possession of the game for a game-winning field goal. Similarly, if the Cowboys convert on 4th-and-1, they would have the same situation. Both teams did have three timeouts at this point, though, which would increase the Steelers chance of winning were the Cowboys to convert and would increase the Cowboys chances of winning on a punt -- since there would be more time left on the clock to respond to a Steelers scoring drive. As such, the 74% to 33% conversion rate to break-even rate discrepancy would likely shrink. Garrett, as expected, elected to punt and Brian Moorman hit a 59-yard boomer.

And whats so sad is because they won it gets lost in the shuffle and the pundits will say it "worked". Where if they lost and you bring the above up its called a "second guess". Culture of conservatism in the NFL and only getting worse.

Even without the math, any casual NFL fan knows kicking the field goal is the wrong move. The chances of making it are well below 50% and when you inevitably miss it you've set up the Steelers for quick field goal drive.

How these NFL coaches make millions of dollars is beyond me

Watching the game, I remember saying attempting the field goal is the worst of all the decisions... low chance of hitting it, leave defense in position to score. I say you either punt and force OT or you go for it for the win. I didn't realize punting wasn't that much better.

BTW I think if Garret spends some time reading up on this blog, it might help save his job.

"One other note is that this analysis assumes an average amount of timeouts left for both teams. In our case, Dallas was out of timeouts and Pittsburgh had one remaining -- most likely about average given the situation."

What about the selection bias? For the past data indicating going for it on 4th and 4, from opponents 43 with 0:32 remaining, produces an 84% WP, don't you think it it likely the teams that went for it in Dallas' situation had more than 0 timeouts remaining?

I find the 84% WP on a successful conversion hard to believe, given no timeouts remaining.

And now I see people are killing Pats for going for it late on 4rth and one from their 12. This is like baseball was when Bill James first came to the scene- just total denial from those in the game, reporting on the game, and watching the game. Just dont want to believe you can apply math and probabilities

"I find the 84% WP on a successful conversion hard to believe, given no timeouts remaining"

seems reasonable: If they convert, they get at least one more play, spike the ball, and have a last second GW FG try from maybe 50 yards. Roughly 50/50 to hit the FG, and 50/50 if they miss.

"seems reasonable: If they convert, they get at least one more play, spike the ball, and have a last second GW FG try from maybe 50 yards. Roughly 50/50 to hit the FG, and 50/50 if they miss."

Like you said, roughly 50/50, which is not very close to 84/16.

"seems reasonable: If they convert, they get at least one more play, spike the ball, and have a last second GW FG try from maybe 50 yards. Roughly 50/50 to hit the FG, and 50/50 if they miss."

NEVERMIND! Ok, I forget the score is tied, I see your point.

The choice was a certain 50% WP chance to win, or a 50/50 chance to drop to 36% WP.

It seems to me that the choice to take the 50% win certainty is the best decision. These coaches do not get to have 1000 similar situations where their long term strategy will produce a mild advantage, they need to avoid losing this season or they will get fired.

How about the Giants going for it 3 times on fourth-and-short? From a dispassionate analysis that takes each play as a random event, this makes sense. But their offense had struggled all game, and some points (any points) might have improved their mood and make their subsequent attempt more likely to work. I'd like to see these decisions being viewed as somewhat interdependent. Controlling for season-long chances of making a fourth-and-one, do prior attempts within the same game affect subsequent attempts' success significantly.

This is so rational and straightforward for anyone with a modest understanding of logic and probability and statistics. That so many coaches continually make irrational decisions especially at the end of the game when the possible outcomes are well defined and few, boggles my mind. So much planning and preparation goes into these games and it appears that virtually none of them (except maybe Belicek) are even curious about using objective data to make critical decisions. I mean, Jason Garrett has an Ivy League education. Amazing.

Joe L - as a very initial step at answering your question. I had a look through the data I have, found all games where a team had more than one 4th and 1 conversion attempt.

The results were that on the 100 occasions where the previous attempt failed, 64 of the following attempts succeeded (64% - obviously). On the 232 occasions where the previous attempt succeeded, 149 succeeded again (64.2%). I have to admit, I thought I'd see a bigger gap as teams that failed on the first attempt would be more likely to be worse, therefore more likely to fail on the second attempt.

Still, the whole "mood" argument doesn't seem to hold much water. Not too surprising. No-one, as far as I've ever known, has been able to demonstrate the existence of 'momentum' in football.

Did you check the math on the Patriots 4th and 1 from their own 11 with 2;24 left? I bet that is a real close decision though you have to tip the scales slightly to go for it with Brady at QB.

The Patriots decision was not a close one, they were correct to go for it. 60% conversion rate on 4th-and-2 -- expected win probability of 13% versus 2% of a punt. That makes the break-even point a 5% conversion rate (1/12 of the estimated conversion rate).

Thanks, Ian. That's really interesting -- no difference whatsoever between success rates based on a previous attempt. I'd have never guessed that.

I know that you folks don't believe in momentum. But I do think that players get frustrated and that it clouds their judgment, which can lead to bad plays (and fights that lead to ejections). I find it interesting that the world's best pro golf players will attribute their success to being able to forget their last shot (ie., eliminating momentum) -- which, therefore, suggests that momentum exists for the vast majority of golf players. Why wouldn't the same apply to pro football players?

Perhaps a team's ability to resist the frustration and clouded judgment (that might appear to be momentum for the other team) is, in fact, reflected in their team efficiency numbers... Thus accounted for in the model.

Joe L,

I haven't looked closely at golf but I believe I remember a study that shows that momentum is a myth there as well. Have you thought about basketball? Basketball is a sport that should easily be able to demonstrate or not demonstrate the presence of momentum. There are whole lots of games and many possessions per game. Each possession is roughly the same. They don't start at different places on the court like football or play different terrain like golf. In basketball, it can be clearly shown that there is negative momentum. What do I mean by negative momentum? I mean that if two teams are equal in terms of talent level and one team has a 10 point lead, it is far more likely that the game is about to be tied than that they are about to go up by 20.

There is no momentum in basketball either. Especially when it comes to "hot" shooting. Like clutch hitting it doesnt really exist. Read a few studies on will see if i can locate.

As a thought experiment, can you imagine how much people would talk about momentum if the team that scored then received the following kickoff?

Obviously just a thought experiment, but I'll bite -- in that case, Ian, I think "momentum" would be a real thing in the form of defensive exhaustion. I haven't looked into it, although I have been meaning to, but I believe long drives have a greater affect on the exhaustion of the defense than offense since the offense is dictating the pace of play. Would definitely be interesting to look at.

Howard,

The studies you refer to deal with free throw shooting. Removing shot length and defender pressure makes it a lot easier to see whether momentum actually exists. In this sterile environment, there is no momentum positive or negative. In actual games, the widespread belief in momentum is one of the main reasons why there is actually negative momentum. Defenders give extra attention to "hot" shooters. Hot shooters then proceed to take ill-advised low percentage shots simply because they think they have momentum.

Take a look at halftime betting lines. Say the Lakers are a 10 point favorite in the game. If momentum was completely non-existent, the Lakers should expect to be a 5 point favorite in the 2H line regardless of what happens in the first half. That isn't what actually happens. If they have a huge lead, they can expect to be a 2H underdog. On the other hand, if they are behind, they can expect a greater than 5 point line.

If negative momentum didn't exist, you would see far more blowouts and far fewer close games. In some ways, its beneficial to the viewing experience. On the other hand, you can't help but wonder how good a team could be if you could just convince them to play as hard as they do when they are 10 points behind when they are 10 points up.

Wasn't even fishing there Keith, but fun to get a bite. I'm always interested in a testable hypothesis.

Perhaps to clear the debate up a bit, a definition of what momentum is might help. In my mind, momentum would be something that affects WP outside of the game state. In mathematical terms, a normal WP model is a memoryless Markov chain. A model in which momentum exists would not be memoryless.

I don't think momentum applies to this discussion on 4th down and 1s. There must be an enormous time lag between the two situations.

One would expect to see both positive and negative correlations though, if you play a lousy defense, you'd expect 2 successes more often than random chance. When you play a great D, you'd expect 2 failures more often. Average them all together, and it washes out (though the effect is still in there)

Jeff, actually the 1980's study details both free throws and field goals. Was called the The Hot Hand in Basketball: On the Misperception of Random Sequences. John Wertheim's Scorecasting book also examines this in detail. Very good book by the way even has a section on golf

Jeff, I would suspect that the answer may not be "negative momentum" as much as conscious decisions (often wrongheaded ones) to play conservatively and burn clock while ahead. In football teams that built a lead by passing hunker down and run to avoid turnovers and clock stoppages but end up hurting their chances to move the ball and make long drives. In basketball teams with leads sometimes just waste time on offense, then have to take hurried shots late in the shot clock. Teams that play best with high-variance, fast-paced strategies can really struggle to play with the lead because of the impulse to slow down.