Extra point vs. two-point conversion

The defense

Simple strategies

One simple, uniformed strategy the defense could employ would be to simply defend against the 2-point conversion all the time. What would the offense’s expected score be?

\[ \begin{aligned} E[S_{CONV}] & = p_{XP} \cdot (1.0) + 2 \cdot (1 - p_{XP}) \cdot (0.4) \\ & = (0.2) \cdot p_{XP} + 0.8 \\ \end{aligned} \]

The other simple, uninformed strategy would be to always defend against the extra point all the time. The expected score there would be:

\[ \begin{aligned} E[S_{XP}] & = p_{XP} \cdot (0.9) + 2 \cdot (1 - p_{XP}) \cdot (0.6) \\ & = (-0.3) \cdot p_{XP} + (1.2) \\ \end{aligned} \]

We can graph both simple strategies and the offense’s expected score as a function of their selected probability of going for the extra point \(p_{XP}\):

General strategy

Since the defense actually knows the offense’s selected probability of going for the extra point, they can select the strategy that minimizes the offense’s total scoring. They can defend against the two-point conversion until they know that that the offense is very likely to go for the extra point.

What’s the threshold? We can set the two scoring equations equal, and that will give us the threshold at which we should change tack: \(p_{XP} = 0.8\).

\[ \begin{aligned} E[S_{CONV}] & = E[S_{XP}] \\ (0.2) \cdot p_{XP} + 0.8 & = (-0.3) \cdot p_{XP} + (1.2) \\ (0.5) \cdot p_{XP} & = 0.4 \\ p_{XP} & = 0.8 \end{aligned} \]

So, the defense should defend against the two-point conversion until it knows the offense is going to go for the extra point with at least a 80% probability.

The offense

General strategy

The offense knows that the defense knows the offense’s strategy. (Ugh, that’s a mouthful.) But using the same graph above, the maximum expected score under the defense’s approach will also be at \(p_{XP} = 0.8\). Substituting into either equation gives us an expected score of \(E[S] = 0.96\).

So, the offense should consistently send in a probability of 0.8, expecting to score 0.96 points per attempt.