## Go for two?

You're the head coach of a football team, and your team trails late by 14 points before you score a touchdown to cut the lead to eight. Should you call for a two-point conversion attempt? Or do you send the kicking unit out for an extra point? Conventional wisdom (and certainly observed coaching behavior) suggests that the best plan is to kick the extra point. But does this really give your team the best chance to win?

## Statistical Modeling

Since the only way your team wins is if they score another touchdown and
prevent your opponent from scoring, the one vs. two point conversion decision is
only relevant in that circumstance. So we'll assume that you'll score
again and that the other guys won't. (If this doesn't happen, you'll lose
no matter what your strategy.) We'll assume that the probability of
successfully converting an extra point is p_{1}, the probability of
successfully converting a two-point attempt is p_{2}
and the probability of winning in overtime is ½. Now, let's examine your chance
of winning for a few different strategies:

Go for two now:

There are three ways to win the game if you go for two now:

- Convert the two-point attempt now and kick an extra point after the next touchdown.
- Convert the two-point attempt now, miss the extra point after the next touchdown and win in overtime.
- Miss the two-point conversion now, convert a two-point attempt after the next touchdown and win in overtime.
Assuming independence

^{1}we can calculate the probability of each of these outcomes.

- p
_{2}× p_{1}.- p
_{2}× (1-p_{1}) × ½.- (1-p
_{2}) × p_{2}× ½.

Go for one now and two later:There are two ways to win the game under this strategy.

- Kick the extra point and convert the subsequent two-point attempt.
- Miss the extra point, convert the subsequent two-point conversion and win in overtime.
The probabilities associated with these outcomes are:

- p
_{1}× p_{2}.- (1-p
_{1}) × p_{2}× ½.It is obvious that if the two choices are "Go for two now" or "Go for two later" the former is a better choice. Denote these probabilities p

_{ now}and p_{ later}:p

_{ now}= [p_{1}p_{2}+ ½(1-p_{1})p_{2}+ ½(1-p_{2})p_{2}] =

[p_{ later}+ ½(1-p_{2})p_{2}] ≥ p_{ later}This result is intuitive. By going for two earlier, we know whether or not the attempt is successful and can plan the subsequent attempt accordingly. By procrastinating, we lose the opportunity to overcome a failed two-point attempt.

Go for one now and one later (if the first is successful):There are two ways to win the game under this strategy.

- Convert both extra points and win in overtime.
- Miss the first extra point, convert the subsequent two-point conversion and win in overtime.
The probabilities associated with these outcomes are:

- p
_{1}× p_{1}× ½.- (1-p
_{1}) × p_{2}× ½.This strategy results in a lower probability of winning (compared to going for two immediately) if

p

_{1}p_{2 }+ ½(1-p_{2})p_{2}> ½p_{1}p_{1}Even if the extra point were a certainty (i.e., there is no chance of missing it), the "Go for one" strategy is only a good decision if the probability of a successful two-point attempt is less than 38.2%. (Historically, the two-point success rate is about 44%.)

1. Essentially, independence states that the results of one event have no influence on the results of another.

## Further Analysis

If it increases your chances of winning, why do coaches seldom (if ever)
employ the "Go for two now" strategy^{2}?
One possible answer is that this strategy is unconventional. Losing a game
by following conventional wisdom never got a coach fired. Failing while
attempting something innovative is dangerous.

Another possible explanation is that the coach really isn't trying to
maximize his chances of winning. Rather he's attempting to prolong his
defeat. For example, there's a (1-p_{2})(1-p_{2})% probability
of losing in regulation with the go for two strategy, while the probability of
losing in regulation using the conservative extra-point strategy is only about
6%. Therefore, this is the classic example of a coach playing not to lose
rather than playing to win.

2. Perhaps the strangest conversion strategy was employed by then-Cowboys head coach Dave Campo in 2001 during a Thanksgiving game against the Denver Broncos. Midway through the fourth quarter and trailing by 16, Dallas scored a touchdown after which Campo initially decided to go for two. Oddly, however, Campo changed his mind, electing to kick the extra point and ensure the Cowboys would need at least two additional offensive possessions. Dallas eventually lost 26-24. Campo later explained his curious decision "To make two 2-pointers, back-to-back, the percentages were not with us. I thought it was a better chance to take the point and get the onside kick. No, it wasn't a bad decision." (We beg to differ.)

## References

Joseph A. Gallian (1981) An Optimal Football Strategy, The Two-Year College Mathematics Journal,
**12** (5) 330-331.

Richard C. Porter (1967) Extra-Point Strategy in Football, The American Statistician, **21** (5) 14-15.