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Two-Point Conversions and Rational Coaching

        Analytics is becoming an increasingly hot topic in the football world, perhaps most noticeably with decision-making scenarios that were previously seen as a given. Punt if you’re out of field goal range? Kick the extra point if you’re down by eight points after a touchdown? Both these status-quo situations are being challenged analytically by incredibly smart people in the industry, and it’s bringing to light the power applied probabilities can have on the game of football.

        Today, I want to dive deeply into a concept I consistently see people misinterpret regarding the two-point conversion and its trade-off to simply kicking an extra point. The problem, as I see it, lies in the interpretation of Expected Value. Several discussions on the value of the two-point conversion, both on Twitter and through mediums like television and radio, have led people to the following conclusion; if the Expected Value of the two-point conversion is greater than the Expected Value of the extra point, then teams should, in theory, always go for two. I’m here today to show you why this seemingly obvious statement is wrong.

Expected Value & Binary Outcomes

        To begin, we must focus on what expected value represents in the context of the post-touchdown plays. In these situations, the outcomes are binary. A team either succeeds at their attempt – either for two or for one – or they fail. This concept of binomial outcomes is incredibly important for later analysis. The formula for computing the expected value of a binomial model in this situation is very straightforward and easy to conceptually understand: E = v * p.

        Here, v represents the value of the attempt (either two or one point), and p represents the probability of success. Let’s say, for example, that the probability of success for a two-point conversion is 48% and the congruent probability for an extra point attempt is 95%. The expected value, henceforth, of the two-point conversion is 0.96 whereas the expected value for the extra point is 0.95. The two-point conversion attempt has a higher expected value, which could lead to the conclusion that it is the better option. However, we are missing one huge piece of the puzzle.

Coefficient of Variation

        To best exemplify this missing piece, let’s look to finance and its applications for better context. In portfolio theory, the use of the Coefficient of Variation (CV) is of the utmost importance when deciding between two investment opportunities. The formula for the Coefficient of Variation is as follows: CV = sd / E. Here, sd is the opportunity’s standard deviation and E is it’s expected value. The determining factor for selecting between two investment opportunities is whichever one has the lower CV. It stands that sd and CV have a positive relationship, meaning as one goes up, so does the other. The opposite is true for E and CV: as one goes down, the other goes up. Ultimately, this means we would prefer to have high expected value and low standard deviation.

Financial Examples

        Using a real-world example, we can see this in action. Suppose I offer you a choice: an expected $1,000 in a year or an expected $100 in a year, and both could possibly differ by $100. The choice is clear, you’d, of course, prefer the one with the higher expected value, aka the $1,000.

        The intuition for why higher expected values are better is rather straightforward. The intuition for why a lower standard deviation is better, however, is a bit less clear on the surface. Suppose I offer you a choice. You can pick option A, which is a guaranteed $200 every month, or option B, which at the end of every month I flip a coin. If it’s heads, you get $400. If it’s tails, you get $0. The expected values of both options are the same: $200. So why should one pick the guaranteed option? Over an infinite time period, the two options should give the same amount.

        However, we inherently work and live in a finite environment. For instance, I need $500 at the end of the third month to pay rent or otherwise get evicted. I could theoretically have $1,200 at the end of the third month with option B, and that would be awesome. However, there’s a chance I don’t have even $500 to pay my rent. There’s value in the guaranteed option in that it is inherently predictable and has less far less downside in the short(ish) term.

        There are of course situations where one should pick option B. Let’s say, for example, I need $800 in three months to pay rent. Option A does not even give me a chance at this money, so I would necessarily go with option B. This is synonymous with being down 16 in an NFL football game. However, in the absence of a short-term goal that is unattainable with the lower variance option, a “rational investor” would always pick the lower variance option assuming the same expected value.

Variance in Binomial Outcomes

        We’ve discussed at length thus far why a lower variance in outcomes is valuable in deciding between two options that have a similar expected value. But what does all of this have to do with the two-point vs extra point attempt discussion? We now know what the missing piece of our analysis is: standard deviation. Coaches act just like financial investors. They value both the expected value and the variability in outcomes. It holds true, then, that coaches look to minimize the coefficient of variation as well.

        The binomial nature of post-touchdown plays makes this part of the analysis much, much easier. For a binomial outcome, the standard deviation of outcomes is as follows: sd = sqrt(n*p*(1-p)). Here, n represents the number of attempts and p represents the probability of success. The intuition of n is important but negligible in this analysis, as we’ll see later on that n ultimately gets canceled out of the equations.

Putting it all together

        Now, we have both parts of the CV equation. We know the formula for binomial expected value and standard deviation and understand why both are important in determining if coaches should be going for two or kicking the extra point. Given this knowledge and mathematical tools, we can actually graph the CV function for both situations! Below is the CV (on a logarithmic scale for visual purposes) for both types of plays.

        This illustration is simple yet powerful. As to be expected, at each probability of success, the two-point conversion has a lower CV than the extra point. This should hopefully be intuitive; if the probability of success for a two-point conversion was the same as an extra point, one would always go for two. However, we know this is not the case. The beauty of the above chart is that we can now compare the CVs for different probabilities of success for each play. The actual value of the CV is not of significance. Rather, we are interested in the comparison between the two-point and extra point values. Whichever is smaller is the one that is the better option.

        Using the same example as at the beginning of this article, let’s look at the CV is the probability of success was 95% for extra points and 48% for two-point attempts. The log(CV) at 95% for the extra point is -0.64. Comparatively, the log(CV) at 48% for the two-point conversion is -0.28. Thus, despite having a higher expected value, the two-point conversion attempt is not the preferable option. In fact, the success rate of the two-point conversion that would have an equal CV to a 95% extra point success rate is a staggering 83%! We can actually calculate and chart the success rate CV for a two-point conversation that would equal the CV for each extra point success rate.

Equivalent CVs

        Let’s start by setting the CVs for two-point and extra points equal to each other, as that’s our ultimate goal. In the following equations, let y be the success percentage for two-point conversions and x be the success percentage for extra points. All the formulas/steps before are simply taken from the above analysis, so refer to the aforementioned formulas if anything is confusing! The below reads like a flow chart. We will start at step 1) and ultimately work down to step 10).

        There we have it! Recall that y is the probability of success for two-point conversions. We now have an equation that if given a percentage of success for extra points (x), we can compute the probability of success for two-point conversions that would make the CVs equal to one another. If the CVs are equivalent, there is an inherent selection indifference. Coaches that are more risk averse would pick the extra point, but more risk-heavy coaches would pick the two-point conversion. Both options would be “rational” from a coaching perspective, neither better than the other. Below is the graph of the equation in step 10) for all probabilities x.

What does this mean for the NFL and Coaches?

        In 2018, the league success rate for extra points is about 94%, or 0.94 expected value from extra points. Comparatively, two-point conversions are being converted at one of the highest rates we have ever seen: roughly 57%, or 1.04 expected value. Given this disparity, why are NFL coaches still insisting on kicking the extra point with almost no second-guessing? As we’ve shown here, NFL coaches are acting “rational”. This idea of rational coaching is in line with the minimizing CV analysis performed in the financial world.

        Now, a 57% success rate for two-point conversions as mentioned is one of the highest rates in recent memory. This is not expected to be a sustainable number, as for 2016 and 2017 the average hovered around 44%. However, let’s assume that the league is improving on converting two-point attempts because of their increased popularity. Therefore, let’s assume the going-forward success rate for two-point conversions is 50%. Then, in order to have selection indifference, extra points must be converted at a rate of 80%. From Pro-Football-Reference data for 2016 and 2017, I charted all field goal attempts and their conversions and added a trendline.

        The current distance for an extra point attempt is 33 yards. Consistent with the graph above, we’ve seen about a 94% conversion rate for these kicks. The NFL very recently (in 2015) moved the extra point distance back a staggering 13 yards from what formerly was only a 20-yard field goal. This decision prompted coaches to consider the two-point conversion more than they did previously, but the length still does not provide rational coaches with decision indifference. A rational coach, under the current structure, would by default kick the extra point even if the expected value going for two was marginally higher. If the NFL wanted to really make coach preference matter in the post-touchdown play, they should move the extra point back nearly an additional 10-12 yards.

Summary

        Coaches are inherently rational. The expected value of a two-point conversion, over an indefinite period of time, has the possibility to have a higher expected value than an extra point. This extra expected value, however, comes at the price of increased variability. Through the binomial distribution and its characteristics, we can build out the coefficient of variation curves at each level of success. Comparing the curves for the extra point and two-point conversion, it has been clear that coaches are acting rationally. Without hesitation, under the current structure, coaches kick the extra point after a touchdown.

        Analytics is becoming more prominent in the game of football. Whether teams choose to accept its findings or not, high impact probability analysis is helping shape coaching decisions. Analytics can cause problems, however, if used incompletely. In only utilizing the expected value to determine the best possible play after a touchdown, one fails to consider the short-term variability factor. In this holistic analysis, I’ve hopefully shown why coaches behave rationally despite a small potential expected value disparity. The NFL moving the extra point back was a great move for the strategic aspect of the game. I argue, however, that they did not go far enough. The post-touchdown play should be a real decision on any given touchdown. In its current state, rational coaching suggests an extra point is simply the better play.

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