By Max Mulitz
This post is arguably a month late, but good game management is always timely and I don’t think anyone on the internet really got at the problem from a game theory perspective.
On the one hand, in the article linked above, Pete Carroll notes that being up 7 gave his team about a 79% chance of winning, he further suggested that making the field goal to go up 8 would give his team an 85/86% chance of winning and that making the two point conversion would raise his teams chances to 90-92%. These numbers are more or less aligned with Brian Burkes old Win Probability model from Advanced NFL Stats.
On the other hand, win probability estimates can get pretty noisy in the endgame, especially when you are talking about crossing over meaningful thresholds of points. For instance, weather a team is trailing by 4 or 5 points with 0:30 left makes no practical difference to the teams chance of winning, but the one point difference between leading by 6 and 7 has an enormous impact on win probability. For this reason, relatively time remaining, it makes sense to model a game in terms of number of score differential rather than point differential.
It’s time for some game theory. I’m going to take the view that with less than 5 minutes remaining, number of score differential is more important than points.
The first question is if an 8 point lead constitutes a one or two score game. If you believe leading by 7 points and 8 points are each one score games and a 9 point lead is a two score game, then going for two up 7 is risk free proposition! Of course this is absurd because a team is much more likely to fail on a two point conversion than they are on a PAT.
The chance that an 8 point game is a two score game is simply the probability that the opponent will convert their two point conversion if they score. So a team with a 25% chance of converting a two point conversion is in a Two Score game 75% of the time when they are down 8, but a team with a 75% chance of converting only has a 25% chance of trailing by two scores. In reality, the vast majority of teams are probably between 40-60% to convert a two point conversion in a given game, with the mean right around 50%.
The chance that a 9 point game is a two score game is obviously 100%, since there is no three point conversion.
The idea that a 7 point game is always a one score game is a parochial misconception. In fact, with the new PAT rules, teams only make about 94% of their PATs. So when trailing by 7 there is actually a 6% chance you trail by two scores.
Therefore, the chance of leading by Two Scores when going for Two when leading by 7 is equal to the chance you fail to convert multiplied by .06 (the chance your opponent scores but misses the PAT) plus the chance you do convert to go up multiplied by 1 (as a 9 point lead is always a Two Score game), as show below.
Two Score Game Probability (Go For It)=Probability you fail on your two point attempt and your opponent misses their PAT to tie + probability you succeed on your two point attempt.
Two Score Game Probability (Go For It)= .06*P(Fail) + 1*P(Success)
Two Score Game Probability (Go For It)= .06*(1-P(Success))+P(Success)
Two Score Game Probability (Go For It)= .94*P(Success) + .06.
On the other hand, if you attempt a PAT to go up 8, the chance you are in a two score game is .06*the chance you miss your PAT (i.e. you miss your PAT and are still up 7, then the opponent misses theirs and you still lead by 1) plus the chance the opponent makes a two point conversion after you make your PAT. If we assume a teams own chance of making the PAT is 94%, we can model this situation the following way.
Two Score Game Probability (Attempt PAT)= Probability both teams miss their PAT attempt + Probability you make your PAT and the opponent fails on their Two Point Conversion.
Two Score Game Probability (Attempt PAT)= .06*.06 + .94*P(Opponent Fails)
Two Score Game Probability (Attempt PAT)==.036 + .94 & P(Opponent Fails)
A team should Go For It if their probability of leading by two scores is at least as high as it would be if they chose to kick the PAT. That is, they should go for it if
Two Score Lead Probability (Attempt PAT) > Two Score Lead Probability (Go For It) which is equivalent to saying to go for it if
.94*Probability of success + .06>.036 + .94*Probably opponent fails
This can be rewritten as:
.94*Probability of success -.94*Probability Opponent Fails>.024
Which can be further simplified to
Probability of Success -Probability Opponent Fails>2.6%
So if a teams probability of converting a Two Point Conversion is a little higher than their chance of stopping their opponent on a Two-Point Attempt then Going For Two up 7 is justified.
Of course, some kickers are better or worse than others. I made both kickers league average so I could easily show the difference in success rates when going for two, but in an applied setting it would be easy to change these numbers slightly on a situation by situation basis.
It seems, the Go For Two vs. Kick a PAT decision late in a 7 point game is close enough to essentially be a toss up, so there’s no clear right or wrong answer in a given situation. Essentially, the decision a team must make is if they think they are more likely to convert their own Two Point Conversion attempt or if they are more likely to stop their opponent from converting a Two Point Conversion. At some point I will post about at what happens if we expect our opponent to go for Two when they are trailing by 7, though the math is a little more complicared and the conclusion (it’s a judgment that comes down to a teams relative offensive and defensive strength) stays the same.