A deficit of 14 points in the NFL often acts as a critical breaking point. Trailing by further would be a very clear indication of a blowout whilst scoring a touchdown would make the game close and competitive. Fans hope that their team will score the touchdown, cutting the deficit to 7 points, just one more touchdown away from tying the game.
However, what the average viewer rarely would consider is the possibility of going for two after the first touchdown is scored. The theory behind this idea is that if you convert the first conversion then you don’t have to go for two on the second TD to win; you can instead kick an extra point to win. If you do not convert the first conversion, however, it is still possible to tie the game up with a two-point conversion after the next touchdown. Basically, you only have to convert one two-point conversion after the first touchdown in order to gain a lead after two touchdowns, whilst you would have to miss two consecutive two-point conversions in order to still be behind. This creative idea sounds good in theory, but does it mathematically hold up?
Let’s look at the decision and outcome tree of going for it, or just kicking the extra point after the first touchdown, based on the assumption that if the trailing team is going to have to make up the deficit, it will force defensive stops and score touchdowns. Although this seems to be a large assumption to make, I will cover that thinking like this does hold up in practice, especially towards the end of a game.
Written algebraically the decision after the first touchdown looks like this:
Go For It After First TD: (Value of a Win)(First Conversion Made)(Extra Point Made) + (Value of Overtime)(First Conversion Missed)(Second Conversion Made) + (Value of Overtime)(First Conversion Made)(Extra Point Missed) + (Value of Loss)(First Conversion Missed)(Second Conversion Missed)
Kick Extra Point: (Value of Overtime)(Extra Point Made)(Extra Point Made) + (Value of Overtime)(Extra Point Missed)(Conversion Made) + (Value of Loss)(Extra Point Missed)(Conversion Missed) + (Value of Loss)(Extra Point Made)(Extra Point Missed)
What we are doing is multiplying the possibility of the various end results by the value that those end results would produce in order to figure out the expected value of each decision. Whichever choice has the highest expected value will logically be the better decision for a coach to make.
Now let’s replace those variable names with actual numbers.
- In an average game, a team should value a win as 1, a loss as 0, and the game going into overtime as .5.
- From the 2000-2009 NFL seasons, the two point conversion rate was 47.9%.
- The Extra-point conversion rate in the NFL is 99.3%
Solved:
Go For It After First TD: 1(.479)(.993) + .5(.521)(.479) + .5(.479)(.007) + 0(.521)(.521) = .6021
Kick Extra Point: .5(.993)(.993) + .5(.007)(.479) + 0(.007)(.521) + 0(.993)(.007) = .4947
Because the expected value (in terms of wins) of going for two after the first touchdown is higher than the expected value of kicking the extra point, it would make more sense to go for two after scoring the first.
However, there is substantial debate about expected rate of two point conversion. Although the percentage has been 47.9% from 2000-2009, many coaches commonly believe it to be as low as 40%, especially at levels of football lower than the NFL. Additionally, coaches can also base their probability of conversion on their current confidence in their players and the play call they have for the conversion (the most stark example would be Chris Peterson’s confidence in the Statue of Liberty play for Boise State in the 2007 Fiesta Bowl Overtime against Oklahoma, it is very likely he estimated the conversion probability at much higher than 47.9%). Therefore, it is important to determine the conversion percentage at which it becomes more logical to go for two after the first touchdown, as opposed to traditionally kicking the extra points. This is known as the breakeven point. To find this point, we set the two decision equations equal to each other, and let the conversion rate be variable x.
1(x)(.993) + .5(1-x)(x) + .5(x)(.007) + 0(1-x)(1-x) = .5(.993)(.993) + .5(.007)(x) + 0(.007)(.521) + 0(.993)(.007)
We end up with the quadratic equation .5x^{2 }– 1.493x + .493 = 0
Solving for x, we get the conversion probabilities of 2.608 and .378, of which obviously only .378 makes sense. 37.8% is the breakeven point.
Therefore, we can determine that if the expected probability of conversion is above 37.8%, that it makes more sense to go for two, whilst if it is lower than that, it makes more sense to just kick the extra point.
Let’s go back to the assumption that a team will score those two touchdowns as the way they close the deficit. In a late game situation with limited time on the clock the trailing team might as well play as if they are assuming their defense will get a stop when the other team has the ball. After all, as the other team scores, eventually the clock will rule out all chances of victory for the trailing team. The clock also necessitates the assumption that there are a limited amount of possessions left. This situation forces the trailing team to score two touchdowns, rather than one or more field goals. We can additionally ignore the scenarios where the opposition increases their lead, because it does not change the complexion of this game theory. If the opposition scores a field goal, resulting in a 17 point lead, the game is three possessions no matter what, and the two touchdown scenario remains in a late game situation. The team behind will have to score two touchdowns at some point to make up the deficit, provided there is not enough time on the clock for numerous possessions resulting in field goals.
These scenarios assume a late-game situation, leading to the assumption of stops on defense and a very small amount of remaining possessions. An early 14 point deficit may not be as game theoretically cut and dry, so it is important to view the scenario with a win probability basis, based on the time remaining.
Advancednflstatistics.com has a great win probability calculator with a wide array of variables, including the point difference and time left in a game. It takes empirical data (read: actual games), makes the necessary adjustments and comes up with a win probability for every game state. We can use these win probabilities to see if the advantages of going for two after scoring the first touchdown when down by 14 hold up in this more practical approach.
We can compare the difference in win probability directly after the first touchdown to the win probability after converting, failing the conversion or kicking the field goal to see whether, in terms of win probability, if it is more “good” to be down 6 as opposed to 7, than it is “bad” to be down 8 as opposed to 7. We will compare,
Win Probability(Down by 6) – Win Probability(Down by 7) vs. Win Probability(Down by 7) – Win Probability(Down by 8)
However, we also have to factor in the probability converting. To get the percentage of conversion at which it is more advantageous to go for the two, the breakeven point, we multiply the (Win Probability(Down by 6) – Win Probability(Down by 7)) by the (Probability of Conversion) and set it equal to (Win Probability(Down by 7) – Win Probability(Down by 8)) times (Probability of Failed Conversion), with variable x as the conversion probability. Solving for x gets the breakeven point for each scenario.
Taking a look at a wide variety of times during a football game at which this scenario could take place, we can see whether the correct decision, based on the expected conversion rate, could change depending on the time remaining in the game.
At all times, I set the field position as the average field position after a kickoff, the 23 yard line, with the opponent in possession with a 1^{st}. and 10, as would happen after the trailing team scores a touchdown. While this could distort things a little bit because it ignores the drastic swing in win probability because of the chance of a return TD, it is the best we can easily do with AdvancedNFLStats.com data. Unfortunately the expected win probability is rounded to two decimal places, so there remains a degree of error in the breakeven probabilities.
Scenario |
WP(Down 6) |
WP(Down 7) |
WP(Down 8) |
Breakeven Conversion Probability |
Better to Go For Two? |
Beginning of 2^{nd}. Quarter |
.31±.005 |
.27±.005 |
.24±.005 |
42.6%±14.2% |
Uncertain^{1} |
Beginning of 3^{rd}. Quarter |
.26±.005 |
.24±.005 |
.22±.005 |
49.7%^{2}±24.8% |
Uncertain^{1} |
Beginning of 4^{th}. Quarter |
.22±.005 |
.19±.005 |
.16±.005 |
49.7%^{2}±16.6% |
Uncertain^{1} |
10 Minutes Left in 4^{th}. Quarter |
.18±.005 |
.15±.005 |
.11±.005 |
56.7%±14.2% |
NO |
8 Minutes Left in 4^{th}. Quarter |
.16±.005 |
.13±.005 |
.09±.005 |
56.7%±14.2% |
NO |
6 Minutes Left in 4^{th} Quarter |
.15±.005 |
.09±.005 |
.07±.005 |
24.8%±12.4% |
YES |
5 Minutes Left in 4^{th} Quarter |
.16±.005 |
.07±.005 |
.07±.005 |
0%±11.0% |
YES |
4 Minutes Left in 4^{th} Quarter |
.17±.005 |
.06±.005 |
.05±.005 |
8.3%±8.3% |
YES |
3 Minutes Left in 4^{th} Quarter |
.15±.005 |
.03±.005 |
.03±.005 |
0%±8.3% |
YES |
2 Minutes Left in 4^{th} Quarter |
.07±.005 |
.01±.005 |
.01±.005 |
0%±16.6% |
YES |
^{1}The error range results in there being no definitive answer for whether it is better to go for two.
^{2 }Not exactly 50% because potential extra point misses have to be factored in.
We can see here that there is a large degree of volatility on the decision to go for it based on the time left in the game. While it is very clear that in very late, end game scenarios that going for it is the better decision based on the empirical data, it appears that it is not advantageous to go for two before the late-game scenario. The reason for the dramatic shift that occurs somewhere between 8 and 6 minutes left in the 4^{th} quarter is most likely due to the fact that most of the time the trailing team will only have two possessions to work with at the 6 minute mark.
The drastic unquestionable support that the data gives to the late-game decision to go for two after the first touchdown validates our game theoretical approach in practice.
We have seen that through both a purely game theoretical perspective, as well as through an empirical approach, that when trailing by 14 points as a football game approaches its end, the right decision after scoring a touchdown is going for two.
Xavier Weisenreder
Georgetown University Class of 2016
Follow Xavier on Twitter: @BeMoreChillNext
Follow GSABR on Twitter: @GtownSports
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Excellent write-up. I definitely love this site. Continue the good work!
I think it also applies if you are down 10 then score a td. go for two if you get it a fg wins if you miss it you know you need a td and can play accordingly. Sick of watching nfl coaches do moronic things like kicking fgs with 2 mins left on opponents 3 when up by 3. Hopefully they will learn basic game theory eventually