Late in the game, when the score is close, it's pretty obvious what the trailing team needs to do. They need to score. But what about the team ahead? Should they try to score and pad their lead? Should they pass the ball instead of running it, risking interceptions and stopping the clock? Or should they rely almost exclusively on the run, limiting risk and letting the clock run?

Combining two concepts of utility in football--expected points (EP) and win probability (WP), I'll analyze team strategy in the 4th quarter. Ultimately, we'll find out if NFL coaches are making the right strategy decisions in the end-game. If my analysis is correct, it would have meaningful implications for NFL strategy in close games.

Recap

Before I get to the meat of my theory, I should review the concepts on which it's based. EP is the average number of points scored next at each field position. For example, at an opponent's 10 yd-line, a team can usually expect about 5 points on average. But on their own 10, a team would expect -0.5 points because the opponent is actually more likely to score next. For the NFL as a whole, here is what the EP curve looks like for a 1st down at each position on the field.

WP is the probability a team will win a game given the current score, time remaining, and any other relevant variables including field position, down and distance to go, and time outs remaining. Here is a graph of WP for selected score differentials. So far I have not made adjustments for field position or other variables, but we can already notice some very interesting things. To read the graph, each curve represents the WP of the team with the ball for a given point differential. For example, a team with the ball and a 7-point lead at halftime (30 min remaining) has a 75% chance of winning.

Observations

In the graph above, notice how the lines are somewhat steady until some point late in the 3rd quarter/early in the 4th quarter when they suddenly diverge. As time dwindles, teams with small leads become far more likely to win, and teams that are behind see their chances diminish rapidly.

Something big happens right around the 4th quarter mark that changes the nature of the game. This seems to be the point where teams stop playing a conventional contest of point maximization, and they start maneuvering for the end game. Teams that are ahead would take fewer risks while teams that are behind would take greater risks. Both teams would start managing the clock, either jockeying for that extra possession or running down the clock.

Also notice the purple line--the WP for teams with the ball and trailing by -1 point. Throughout much of the 2nd half, a team with the ball, down by 1 point, will probably win the game. I was a little surprised by that at first, but I shouldn't have been. The EP curve above shows that possession of the ball at anywhere past a team's own 30 yd-line is, on average, worth more than a point.

Effectively, this means that throughout much of the 4th quarter, teams on defense with a one point lead are already losing. They just don't know it yet.

Further, and more surprising, is the overlap in WP for teams with a +1 point lead and a -1 point deficit. In the first several minutes of the 4th quarter, a team with the ball that's down by 1 is actually more likely to win than a team with the ball and is up by 1!

What? How Can That Be?

Why would a team be better off being down by a point than ahead by a point? I think it has to do with mindset and risk tolerance. Teams that are down increase their risk tolerance and teams that are up decrease their risk tolerance. My theory is that, in general, almost all offenses usually play below the optimum risk level. They should typically be playing more aggressively. But teams may actually start to optimize when behind in the 4th quarter. Teams with small leads would restrict their risk tolerance even more, tilting further from the optimum risk-reward balance.

Consequently, teams with a very small lead need to count on the high likelihood that the opponent will score, and strategize accordingly. They should play as if they're behind, before they actually do fall behind. When the trailing team scores (and they probably will), it may be too late to become more aggressive and try to reclaim the lead.

If I'm right, we should see the EP curves for teams with small leads look different from teams with small deficits. The chart below illustrates the 4th quarter EP for teams with a +1 point lead (green), tied (red), and a -1 point deficit (blue). The data set has been narrowed considerably compared to the overall NFL expected point curve, so the curves are a little noisier, but the trends are clear enough to draw conclusions.

The chart excludes the last 5 minutes of the 4th quarter. The final minutes are excluded because teams with leads of any size could simply run out the clock and leave little or no time for an opponent to score. I wanted to be sure to exclude this final possession from the analysis which would bias the results. (I'm not cherry-picking the data, as including those final minutes actually makes the differences more stark.)

There is one point at which teams with a +1 point lead become more efficient. That's inside the 20 yard line. Teams behind by a point, knowing they're within field goal range, tighten up. They effectively have a 2-point lead at this point and reduce their risk levels accordingly. Teams ahead by a point appear to keep grinding toward the end zone when inside the 20. Keeping the drive going keeps the ball out of the hands of the opponent and burns time off the clock. So the added risk of the continuing the push for the end zone may be balanced by the advantage of keeping the ball out of the opponent's hands. By this point in the game, you don't want to score too quickly.

Teams that are tied even seem to be playing very timidly. Their EP profile is very similar to teams with a +1 point lead. If tied teams or teams with a lead simply played normally, or even slightly less conservatively, they'd win more often.

My advice to NFL coaches would therefore be to become more aggressive with a small lead in the 4th quarter. In fact, whatever it is teams do when down by 1 in the 4th quarter, do that all game long, even with a small lead. Go for it on 4th down more often, go no-huddle, and air it out. You'll score more and prevent your opponent from scoring more often. Ultimately, you'll win more often.

Another great article as always. This is one of the things that always seems to be true but hadn't really been proved out before. I think the defense is as big of a factor in this anomoly as the offense. Defenses become a more conservative and vanilla with the lead at the same time the offense is becoming more aggressive. Back to your previous article where you discussed the optimal run/pass %'s if you knew the offense was going to be more aggressive and pass more the defense should be more aggressive as well but they do the opposite thus making the offensive strategy even more succesful. If it is true that according to your other article that the offensive strategy didn't matter if the defense adjusts then it is up to the defense to recognize what the offense is doing and adjust their game plan when behind accordingly.

it kills me when teams lead by 3 or less and run the ball on 1st and 2nd down late in the game, haven't tried to calculate its effective or anything, but to the naked eye that just seems like the most predictable playcalling in the game,

seems like whenever the playcaller mixes it up with a playaction or just straight pass they get a few first downs and kill the clock

just my take on it

I've always thought that championship-winning teams are much more likely to play the way you suggest: keeping the pressure on and not becoming too conservative. As a longtime Chargers fan I always thought that Marty Schottenheimer's biggest weakness was playing too conservatively in the end of close games.

I'd love to see this analysis extended to see if the way teams approach these situations correlates with their overall success. Do championship teams better resist the temptation to become conservative?

Trailing teams in any sport (even by tiny margins) tend to outperform relative to expectation (that is, relative to pre-game expected performance). This can be seen dramatically by the fact that 2nd half lines in sports betting are never simply a fraction of the whole-game line. They are consistently adjusted to reflect the (correct) expectation that trailing teams will outperform.

How much of this is due to strategy changes vs. emotional factors is unclear.

Interesting article. To help a non-statistician make sense of the graph 'Win Probability by Lead' when looking at the +1 and -1 lines (crux of the article), how can BOTH lines be above the 50% probability of winning the game between 20 and 5 minutes remaining in the game? Both teams can't have a better than 50% chance of winning -- shouldn't the two lines be mirror images of each other (symmetrical) above and below the 50% line?

Thanks. It's because the graph is for the win probability for the team

with possession.What are your thoughts on scoring when down 14 points in the final 5 minutes. Go for 2 or 1?

Slam dunk--You go for 2. Assume a 0.45 chance of 2-pt conversion and .99 chance of 1-pt XP.

Convert 2-pt then get XP =

0.45 * 0.99 = 0.45...ahead by 1

Fail on first 2-pt, convert on second =

0.55 * 0.45 = 0.25...tied

Fail on both 2-pt =

0.55 * 0.55 = 0.30...down by 2

In total, by going for the 2-pt conversion, you have a 70% chance of being tied or ahead, and only a 30% chance of being behind.

That's much better than the 50% chance you get by kicking both XPs, and taking your chances in OT.

Agreed, but do you think more than 1 NFL coach ( maybe Belichek) would go for 2 here?

I'm not sure, but I'd love to see the press conference after the game in which a coach tries to explain the logic to the media!

While I agree with your conclusion on the 2-pt analysis, I don't agree with how you got there. Comparing the 70% chance of being ahead or tied to the 50% of being ahead or tied after overtime is not the best comparison. What should really be compared are the probabilities of winning alone. Kicking two extra points gives a 99% chance of being tied or ahead which would appear to be even better than the 70%. The probability of winning should be isolated to provide the proper comparison.

By going for two, the probability of being tied is .25 and the probability of winning in regulation is .45, as you had noted, while the probability of going to overtime is .25 and given overtime, the probability of winning is roughly .5 (I'm assuming that the probability of being tied after overtime is quite small). So the total probability of winning is:

.45 + (.25 x .50) = .575

Since this is greater than the probability of winning when two kicking extra points (roughly .50), then going for two while down by 14 is the optimal solution.

*kicking two, not two kicking. Sorry about that :)

I think that the fact that -1 is above +1 with 10 minutes left needs to be discussed more. Most of the analysis on this site is done by saying "do what maximizes the win probability" which is certainly valid. However, win probability with 10 minutes left says that the best thing to do, if it was available, would be to give the other team 2 points for free, and then continue playing. Obviously this is not the right decision (even if it was possible), however this tells that one of two things is wrong:

1. Not every decision should be made to maximize WP.

2. WP is not (entirely) valid as computed here.

These are the only two possibilities.

Fascinating article. I always had a gut feeling that being behind by 1 or 2 early in the 4th quarter was actually a good thing. It's cool to see the data actually bear this out.

Does this trailing paradox hold at point differentials of +/- 2? If not, why not?

David Kravitz: I don't think your statement regarding two possibilities is correct. The point is that when you are up by 1 you should play the way teams play when they are down by 1. That will improve WP.