The Last Two Minutes in Cincinnati

Evan McPherson
Evan McPherson
Photo: USA Today Sports Images

NFL Week 17 - There was enough drama and controversial decision making during the final two minutes of the Bengals-Chiefs game to dominate the Week 17 headlines.  Although they came up short on three prior attempts from the Chiefs’ 1-yard line, the Bengals stayed the course and still went for it on fourth down with the score tied and less than one minute remaining in the game.  After offsetting penalties on the first fourth-and-goal attempt, the Bengals once again went for it and this time benefited from an illegal use of hands penalty on the Chiefs’ L’Jarius Sneed.  Now the Bengals were able to run the clock down for a game-winning field goal.  Many questions immediately followed around both the decision not to kick on the fourth-and-1 and whether the Chiefs would have benefited from intentionally letting the Bengals score on first down after the two-minute warning.

First, let’s examine the fourth-and-1 decision that induced the fatal penalty on the Chiefs.  A custom simulation, with full consideration to turnovers, penalties, and overtime scenarios, suggests this was a significant error by the Bengals.  A field goal attempt would have produced 8.7% more wins on average.  While the model plays out hundreds of thousands of simulated games to reveal the value of kicking the field goal, it can be helpful to construct a simple decision tree to see what might be going on here.

First some assumptions derived from NFL historical data and model analysis:

  • Bengals successfully score a touchdown and kick a PAT 57% of the time
  • Bengals successfully kick a field goal 98% of the time
  • Chiefs can win 15% of games when trailing by 3 with no timeouts and 45 seconds from 25-yard line
  • Chiefs can win 4% of games when trailing by 7 with no timeouts and 45 seconds from 25-yard line
  • Chiefs are a 52% favorite in overtime
  • Chiefs can win 55% of games when tied with no timeouts and 45 seconds from their own 25-yard line
  • Chiefs can win 48% of games when tied with no timeouts and 45 seconds from 1-yard line (note the ball position and clock make this a little worse than their chances in OT)
  • Ties count as half wins

Bengals go for win

  • (.57)(.96) + (.43)(.52) = 77.1% GWC

Bengals kick field goal

  • (.98)(.85) + (.02)(.45) = 84.2% GWC

Note: The figures from the decision tree are generalized and do not account for all the variables and scenarios picked up in the simulation. However, it still provides strong support for the field goal attempt.

The question of whether the Chiefs should have allowed the Bengals to score a touchdown at the two-minute warning is a bit more complicated.  First let’s set some assumptions.

  • The Bengals will try to score a touchdown on fourth-and-goal from the 1-yard line.
  • The Bengals will kick the extra point to go up 38-31 after the touchdown.
  • The Chiefs will only voluntarily allow the Bengals to score a touchdown on first down at 2:00, and the Bengals will oblige.
  • The Bengals can convert the touchdown 57% from the 1-yard line (without assistance from the Chiefs), and each subsequent attempt will be identical.
  • The Chiefs have 19.3% Game-Winning Chance (GWC), per the simulation, to win the game trailing 38-31 with 1:55 remaining and two timeouts from their own 25-yard line.

Per these assumptions, the Bengals will score a touchdown on each specific down as follows:

Down Probability of Scoring a Touchdown
on this Specific Down
Approximate Time and Timeouts
Remaining after the Score 
Chiefs’ Resulting
GWC (per simulation)
First 57% 1:55     2 TO 19.3%
Second 24.5% 1:50     1 TO 17.1%
Third 10.5% 1:45     0 TO 14.4%
Fourth 4.6% 0:58     0 TO 6.7%
(No score) 3.4% 0:50     0 TO 48.0%

In this generalized analysis, the weighted average of GWC for the Chiefs is 18.4% without voluntarily offering the touchdown and 19.3% if they do.  A custom simulation shows the Chiefs to be only 17.0% without volunteering the touchdown on first down.  While the decision is somewhat close, it appears correct to give them the touchdown at first opportunity.  Even if the Chiefs knew the Bengals would correctly kick the field goal on fourth down it is still more favorable for them to volunteer the touchdown on first down.

Comments

9 comments, Last at 08 Jan 2022, 11:56am

#1 by RickD // Jan 03, 2022 - 4:54pm

I hope everybody involved understands that any probabilities claimed here or elsewhere are the outputs of mathematical models with a countless number of parameters and estimated values, many of which could be off.  It's very hard to say with any certainty what the probability for success is for this team on this day against this opponent under these conditions.  When a statement like "A field goal attempt would have produced 8.7% more wins on average " is made, one has to be a bit skeptical.  It's very, very hard to have that level of certainty about processes that are much simpler and have fewer variables.

Why did Zac Taylor continue to go for a TD when a FG was at hand and could be easily made?  A great deal of his thinking must be that, given the down and yardage, his odds of success were ultimately reasonably high. And, on the flip side, he felt that giving the ball back to Mahomes with only a 3-point lead was not really a great idea.  

The opposite question is also interesting, and perhaps deserves even more consideration.  Why didn't the Chiefs simply open the gates to a TD, as the Patriots famously did to the Giants in Super Bowl 46? The situation in the Super Bowl was different because the Pats had a 2-point lead, so there was virtually no chance that the Giants would risk turning the ball over on downs by going for it on 4th down.  In any case, why did the Chiefs not just open the doors on 1st down?

I suspect the Chiefs felt that, if they could stop the Bengals on 1st and 2nd down, that their odds for further success would greatly improve.  Football coaches (unconcsiously?) think there is a good deal of correlation from play to play, basically because the same people are out there trying the same thing.  Mathematical models tend to assume each play is an independent event.  That's clearly untrue, but it makes the modeling considerably easier. 

I think this kind of coverage should not rely too much on probability estimates, but instead plot a graph of likely results, possibly as a heat map, based on each coach's estimates of the respective success probabilities.  After all, nobody really knows that a field goal attempt would have produced 8.7% more wins on average.  That's just a number from a model, based on some estimates.  

Points: 0

#2 by ImNewAroundThe… // Jan 03, 2022 - 6:44pm

I swear we have this comment. 

And I still don't know what it's trying to argue other than trying to produce Aaron Rodgers level of skepticism with no real intention of staying on topic.

But it is longer than past ones so I'll give ya that. 

Points: 0

#3 by nat // Jan 03, 2022 - 7:27pm

Short form, for new people with a short attention span:

1) These articles imply WAY too much certainty about the win percentages. There is no way percentages like these should be taken to point to a “correct” or “incorrect” decision, except in cases where the percentages are very heavily weighted to one decision.

2) Better analyses look at a range of assumptions about outcome probabilities of each choice, and present the data in a way that encourages critical thinking about how those assumptions should factor into the decisions.

Points: 0

#4 by ImNewAroundThe… // Jan 03, 2022 - 8:28pm

What's a range gonna do besides invite more (the same) questions. And if you think that's what it's implying...that's on you and that's why these questions will never end and are no longer fun (were they ever?)

Just say you want to live in a gray world so you can go against the strongly worded "correct" decision when you feel like it. 

Points: 0

#9 by MikeWhiteGOAT // Jan 08, 2022 - 11:56am

This article says they have an 85% chance if they kick. This implies Mahomes has around a 25-30% chance to go down and get the chiefs a field goal and 0% at a touchdown. I would argue these probabilities are way too low. .  

This article would be better if it gave the odds (of Mahomes scoring a field goal) that would make the decision a wash. It's fine to show the results of a model but it's dumb to argue that they support a conclusion when they're probably wrong

Points: 0

#5 by Wifan6562 // Jan 03, 2022 - 11:12pm

Can I sum up your response by saying you don’t like rounding to the tenths place? What is your preference for rounding? Are we talking nearest percent? 5%? 10%? Maybe categorize into unlikely, toss up, and likely and then draw comparisons? 

I think the second point is essentially that the data only says so much. I definitely agree with that. This one in particular just isn’t enough to make a clear winner / loser with a 1% difference. Assumptions are imperfect after all. Nitpicking, maybe he should have worded the conclusion a little softer. 

I am interested if there’s any data to support the theory that when a team fails on an and-1 from the 1 yard line, do they have less success on successive attempts (assuming no loss of yardage)? I don’t know if there’s enough sample size to answer this question. And I imagine there are some complex variables at play there. 

Points: 0

#8 by Pat // Jan 04, 2022 - 10:14am

I am interested if there’s any data to support the theory that when a team fails on an and-1 from the 1 yard line, do they have less success on successive attempts (assuming no loss of yardage)?

Can't figure that out properly. Think about it: you would need to compare the result of the next play after a failed attempt versus the result of the next play after a successful attempt. And, obviously, you don't have the latter.

You could of course just compare the second play to the average results of a first play, but of course failing on the first play means the typical team in that situation is worse than average.

Retrodictive studies suck.

Points: 0

#6 by JacqueShellacque // Jan 04, 2022 - 8:58am

A field goal attempt would have produced 8.7% more wins on average.

Nothing is scarier than aggregating data, calculating an average, and using that to make a decision. Would you walk across a river that's on average 4 feet deep? If the answer to what is the 'correct' decision here changes because the opposing offense is the 2 time defending conference champions, and not run by Mike Glennon, then the use of averages is meaningless. 

Points: 0

#7 by Tutenkharnage // Jan 04, 2022 - 10:12am

The figures are based on the teams that are playing, not on league average (except for small-sample-size numbers such as onside kickoff recovery rates, I suspect). The "average" here consists of all the possible outcomes. It's about expected value, that's it.

Points: 0

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