Here’s the situation: Your team leads 21-20 with 2 minutes left in the game, has just scored a touchdown to go up 27-20, and your head coach kicks an extra point to take a 28-20 lead.  Seemingly every coach kicks this extra point in all similar game situations that we’ve witnessed – it’s a no-brainer, right?

Back in my video game-playing days – it’s been a few years, but I’d bet that I’ve played football video games for over 1,000 hours of my life – I used to always go for 2 points in this situation in an effort to build an insurmountable 9-point lead.  My logic was this: in practice, at least in the NFL, no team that scores a touchdown to put them down 1 point in a game-ending situation goes for 2 points to win the game.  (I believe this has happened less than 10 times in the (brief) history of the 2-point play in the NFL, which represents a negligibly small percentage of similar game-end situations).  Therefore, the difference between a 7- or 9-point lead to me was far greater than the difference between a 7- or an 8-point lead; at 7 or 8, the other team has a chance to tie in regulation (but not win, given my assumption), but at 9 the game is effectively over.

In the intervening years I’ve gone along my merry way just assuming that all coaches were making suboptimal decisions with respect to this situation.  Now in my first year as a PhD student in a business program, my brain is starting to work a little bit differently.  Thinking of this situation earlier this week, I developed a simple model to help infer whether either strategy here is dominant.

Assumptions

• Your team is Team A, the opponent is Team B

• Team A has just scored a TD with 2 minutes left in regulation to take a 27-20 lead; PAT/conversion pending

• There is only one meaningful possession remaining in regulation, for Team B, starting with Team A’s kickoff to Team B

• We assign a probability of β to represent the likelihood that Team B scores a TD on their possession (0 ≤ β ≤ 1)

• The probability of either team successfully converting a 2-point conversion is 44% (I believe this is the NCAA historical average conversion rate)

• The probability of either team making an extra point is 100%
  

• If Team B scores a TD on their possession to reduce Team A’s lead to 1 point, they will kick the extra point 100% of the time*

• If the game goes to overtime, both Team A and Team B have an equal 50% chance of winning the game

* - I expect this to be the most controversial assumption, as in college there is always some consideration with respect for going for 2 in this situation (e.g. the Michigan-Michigan State game this year).  I submit that this is a very matchup-specific assumption at the college level – a heavy underdog is more likely to take their chances on a conversion attempt than on overtime – but as noted above, the assumption should be uncontroversial for the NFL, where going for 2 and the win is a nonfactor.

Probability of Winning – Go for 2

There is a 44% likelihood of making the conversion, which makes the score 29-20 and results in a win likelihood of 1 given our assumptions (i.e. one possession remaining in the game).  If the conversion attempt is missed (56% likelihood), we consider that Team B will score a TD with β probability.  If they score, this results in a 50/50 chance to win in overtime; so, in this state, Team A will win with (1 – β/2) probability.  Therefore, the Total Win likelihood is (.44)(1) + (.56)(1- β/2), which reduces to: 1 - 0.28 β.

Probability of Winning – Kick extra point

There is a 100% likelihood of making the extra point, giving Team A a 28-20 lead.  In order to lose the game at this state, the following has to happen: (1) Team B scores – β probability; (2) Team B makes a 2-point conversion (44% likelihood); (3) Team B wins the game in overtime (50%).  The total loss likelihood is therefore 0.22 β, meaning that the Total Win likelihood is: 1 – 0.22 β.

So what?

Umm…Brian’s bolded alter-ego, is that you?

No.  Brian’s bolder alter-ego has long, curly hair; I’m bald.  Get it?

Yes.

Well then.

Well then.  What’s next is that we start playing with β.

Sounds kinky.

It’s not.  We can now calculate the β at which these two decisions provide an equal probability of winning, which is clear from looking at the formulas:  only when the other team has a 0% likelihood of scoring a TD are these two strategies equal.

How, exactly, does this help us?

What this tells us is that, given these assumptions, we have a dominant strategy.  If we set β equal to 1 – that is, there is a 100% likelihood that Team B will score a TD on their drive – we find that going for the 2-point conversion in this situation provides for a 72% probability of winning, whereas kicking the extra point provides for a 78% chance of winning.  Lowering the β into a more realistic region – for convenience, say 0.5 (i.e. 50%) – we find that that going for the 2-point conversion provides for a 86% chance of winning, while kicking the extra point provides for an 89% chance of winning.

It’s important to not dismiss this difference out of hand and treat the strategies as equal – if you told a coach that a particular decision would increase the chance his team loses from 11% to 14%, I’m quite certain that the difference would be meaningful to him.  And it’s also important to keep in mind that these are just fun game theory assumptions that would need to be modified for each specific scenario; for example, I might have had a play on Madden that I knew would work on a 2-point conversion 80% of the time given the poor game AI.  In that situation, my decision to go for two was probably rational.

Which leads me to the following conclusion: given that real game situations will have realities that diverge quite a bit from the basic assumptions in this model, over the course of thousands of games there must have been individual circumstances where teams were at least as well off attempting a 2-point conversion in this situation as kicking the PAT.  In fact, it seems likely that there would have been at least a few instances where they would have been better off attempting a 2-point conversion – say, in the college football fringes where PATs are not ~100% propositions and where weak kickoffs will lead to greater βs .  However, in my football-watching experience, I can’t recall ever hearing a discussion as to whether a team should go for the deuce in this situation.

Anything else?

I discussed my “model” with an experienced PhD student, and his feedback was invaluable.  One major issue that he raised was that there is a covariance between (1) the likelihood of Team B successfully converting a 2-point conversion and (2) β, the likelihood of Team B scoring a TD on their final possession.  The point being, the strength of Team A’s defense (and Team B’s offense) will cause these values to be related.

There are also economic concepts of utility and risk-aversion which are being ignored here.  And of course, the emotional and psychological implications that any given Result A will have on each team, thereby potentially influencing the outcomes of Result B.

So, minor quibbles with assumptions aside, through a very, very simple economic model I’ve provided evidence to help answer a question that’s been bugging me for some time.  Unfortunately, the results are inconclusive – while I find no fault in the general strategy of kicking the extra point in this situation (indeed, a dominant strategy in this model), I have to believe that the ingrained nature of this decision and the strict adherence to the conversion chart has caused a few coaches to make suboptimal decisions.  In any event, hopefully this creates some fun discussion, and hopefully a future look at a different question will provide a more conclusive and illuminating result.

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Thanks to my friend Andy for catching an embarrassing error with my initial model and thereby proving the immense value of editors.  If there are any less-than-minor quibbles with the assumptions or any other issues with the model, please let me know – constructive feedback is welcomed.