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**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.

----------------

*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.*