Abstract

It is relatively common for football teams to find themselves down by two touchdowns late in the game. If they score a first touchdown then coaching folklore says that the team should go for the extra point at that time. In this paper I will show that this strategy, which appears to be universally used in both the NFL and the NCAA, is incorrect, and that going for the two-point conversion after the first touchdown is nearly always significantly better. I will also show that going for the extra point after the first touchdown is only correct if either the coaches believe that they have about a two thirds chance of winning in overtime (which seems rash after a tied game when the result of the coin toss is still obviously not known) or if they believe that their chances of making a two-point conversion are far below national averages.

Introduction

On September 10 th, 2005, the University of Michigan football team was trailing by 14 points when they scored a touchdown with 3:47 left in their game against Notre Dame. Their coach decided to kick an extra point to get within seven points. Even though this strategy is followed in the NCAA and the NFL almost without exception, it is, in general, incorrect. In this paper I will show that the correct strategy in this situation is to immediately attempt the two-point conversion.

When their team is down by 14 points late in the game, NCAA and NFL coaches must base their strategy on the assumption that they will score two touchdowns while holding their opponents scoreless. When they score the touchdowns they have three choices of strategy that they can use:

Go for two first: Under this strategy the team attempts a two-point conversion when they score the first touchdown. If it succeeds then they go for the extra point after the second touchdown in an attempt to win the game. If the first one fails, which happens on average about 57% of the time, then they attempt another two-point conversion after the second field goal in an attempt to go into overtime. Although this strategy is apparently never used in professional or college games, and isn’t intuitively very good, I will show that it is clearly the best approach to take, based on well known probabilities for extra point and two-point conversion success rates.

Go for one both times: The commonly used strategy is to attempt the extra point after both of the touchdowns, playing to go into overtime. If the first extra point misses, which happens on average about 6% of the time, then the backup plan is to go for the two-point conversion after the second touchdown. Although this strategy is almost uniformly used, I will show that it is very inferior to the “go for two first” strategy.

Go for one then two: Under this strategy the team attempts the extra point after the first touchdown and then a two-point conversion after the second. This strategy is sometimes used when the coaching staff believes that their team is unlikely to win in the overtime and so they should go for the win now. As an example of this (Mallory & Nehlan, 2004) discuss, without criticism, a game where Bowling Green used this strategy to beat Northwestern in 2001. However I will show that it can never be as good as the “go for two first” strategy, and so it should never be used.

A fourth possible strategy, “go for two both times,” makes no sense logically or mathematically, unless the team’s extra point special team is so terrible that its chance for success is less than the chance for making a two-point conversion, and so I will ignore it here.

In summary, I will show that the “go for two first” strategy is considerably better than the commonly used “go for one both times” strategy, and that the “go for one then two” strategy should never be used.

Assumptions

The NCAA and the NFL have similar statistics for the success rate of two-point conversions and extra points. In the NFL the figures are 43% for the two-point conversion and 94% for the extra point, while in the NCAA the figures are 43.5% and 93.8% (Mallory & Nehlan, 2004). I’ll use the 43% and 94% figures for most examples in this paper, and will also develop the general formulas to show when the “go for two first” strategy is best. I’ll assume initially that the two sides have equal chances of winning the overtime, and will then extend the analysis to consider the more general case of what to do if, for example, the coaching staff believes that they have a higher or lower chance of winning in the overtime.

Mathematical Justification for the “Go for Two First” Strategy

I’ll assume a minimal level of probability knowledge for the rest of this paper. In particular I will assume that if there are two independent events like attempting a two-point conversion after one touchdown and then an extra point after another touchdown then the probability of both succeeding is the product of their probabilities. E.g., if the two-point conversion has a 43% chance of succeeding, and the extra point has 94%, then these will be assigned probabilities of 0.43 and 0.94, respectively, and the probability of both succeeding is 0.43 × 0.94, which is 0.4042, which using percentages is 40.42%.

The Average Case

Initially I’ll just look at the average case where the percentages for the two-point conversion and the extra point are 43% and 94%, respectively, (and so the percentages for failing on the two-point conversion and missing the extra point are 57% and 6%, respectively), and where the teams are equally likely to win if the game goes into overtime. Then I’ll generalize the mathematics to other percentages.

Under the “go for two first” strategy the team will score three additional points (i.e. in addition to the twelve points for the two touchdowns) if they get the two-point conversion and the subsequent extra point, two additional points when either they miss the first two-point attempt and hit the second or when they make the two-point attempt but miss the extra point, or no additional points if they miss both two point attempts. One additional point cannot occur under this strategy. The probability of three additional points, which will win the game, is 0.43 × 0.94 (making the two-point and then the conversion), which is 0.4042. The probability of two additional points (making the two-point and missing the extra point or missing the first two-point but getting the second) is 0.43 × 0.06 + 0.57 × 0.43, which is 0.0258 + 0.2451, which is 0.2709, which will send the game into overtime. The probability of zero additional points, which will lead to a loss, comes from missing both two-point attempts, which is 0.57 X 0.57, which is 0.3249. So there is a 0.4042 chance of winning outright plus a 50% chance of winning the overtime, which adds half of .2709, for a total winning percentage of 0.540.

The “go for one both times” strategy requires hitting both extra points, which has a probability of 0.94 × 0.94, which is 0.8836, or missing the first one and then attempting the two-point conversion which has a probability of 0.06 × 0.43 for an additional 0.0258, and then assuming a 50% chance of winning the overtime gives this strategy a winning probability of half of 0.9094, for a winning percentage of 0.455.

The “go for one then two” strategy is the worst. It succeeds and wins the game when both succeed with probability 0.43 × 0.94, which is 0.4042, and ties and goes into overtime if the extra point is missed but the two-point conversion succeeds, which will add half of 0.06 × 0.43, for another 0.0129 and for a total winning probability of 0.417.

In summary, the chances of winning under the three strategies, assuming an even chance in an overtime, 43% for two-point conversions, and 94% for extra points (the NFL average statistics), and assuming that you get two touchdowns without the opponents scoring, are shown in the table below:

Clearly the proposed strategy, even though it is not commonly used, is by far the best strategy to take in this average case.

The General Case

Let’s assume that for your team you estimate that you have a probability x of making a two-point conversion, y of making an extra point, and in this game you believe that you have a probability of z of winning if the game goes into overtime. In the average case example above x = 0.43, y = 0.94, and z = 0.5. The probabilities now become

Interpreting these equations, the “go for two first” strategy wins if either the two-point conversion and subsequent extra point both succeed (probability xy), or the two-point conversion succeeds and the extra point fails but you win in overtime (x(1-y)z), or the first two-point conversion fails, the second one succeeds, and you win the overtime

((1-x)xz). The “go for one both times” strategy wins if either both extra points succeed and you win the overtime (yyz) or the first one misses but the backup two-point conversion succeeds, and again you win in overtime ((1-y)xz). Finally the “go for one then two” strategy wins if the extra point and the subsequent two-point conversion both succeed (yx) or the extra point fails and the subsequent two-point conversion succeeds and you win in overtime ((1-y)xz).

At this point we can completely reject any further consideration of the “go for one then two” strategy because the “go for two first” strategy always has (1-x)xz better probability and this quantity can’t be negative. (In most practical cases it will be about a 0.12 higher probability, or a 12% higher percentage.) The problem with the “go for one then two” strategy is that if the two-point conversion fails then the game is lost, while with the “go for two first” strategy if the two-point conversion fails then the team can attempt a second two-point conversion, going for the tie, after the second touchdown.

Assuming 50% Overtimes

In most games the probability of winning in overtime is dominated by the coin flip and luck, and so it is very close to 50% when evaluated in advance of that flip. Even if a fairer system were developed that took away the 60% advantage of the coin flip, like the pizza splitting system (Smith, n.d.), the success rate in overtime is likely to be very close to 50% for most teams over a reasonable number of overtimes. Given the assumption that z = 0.5, the probability table for the two reasonable strategies becomes:

Given a particular value for y, your probability that you’ll make an extra point, we can compare these two formulas to determine how confident you need to be in your two-point conversion before you decide to use the “go for two first” strategy. As we’ll see, if you believe that you will make at least 38.2% of your two-point conversions (which is significantly below the national average) then you should always use the “go for two first” strategy, even if you know that you’ll never miss an extra point. As you reduce your confidence in your extra point special team, your required confidence in your two-point conversion team can drop even further and still mandate the use of the “go for two first” strategy. So, for example, if you have a poor kicking team and only expect to make 90% of your extra point kicks then you should use the “go for two first” strategy if you expect to make at least 32.8% of your two-point conversions.

To justify these numbers I just need to ensure that xy + ½x(1-y) + ½(1-x)x is greater than ½y 2 + ½(1-y)x, cancel out the common term, and solve the equation for x. This gives

xy + ½(1-x)x > ½y 2 which is –x 2 + x(2y+1) – y 2 > 0, which can be solved with the standard quadratic equation formula to get the equation that one should use the “go for two first” strategy whenever:

This looks complicated, but it is easy to apply. E.g., if y = 1.0, and so you believe that your team will never miss an extra point, then substitution shows that if your team can make at least 38.2% of their two-point attempts then the “go for two first strategy” is best. Break points for different expected field goal percentages are shown below:

So, for example, if your kicking team only succeeds 90% of the time with extra points, then if you estimate that you will make a two-point conversion at least one third of the time then you should adopt the “go for two first” strategy. A normal kicking team in either the NCAA or NFL should use the “go for two now” strategy if they can expect to make at least 35% of their two-point conversion attempts.

Assuming Other Overtime Percentages

While most teams will, when they are being honest with themselves, decide that their chances of winning if the game goes into overtime are close to 50%, there might be times when they are more or less confident than that. For example most analysts believe that a team with a much stronger field goal team has an advantage in overtime under either NFL or NCAA rules. In this section I’ll look at how this changes the odds. For any particular expectation of winning or losing in the overtime one can substitute the value in for z in the general equations and solve them using the quadratic equation as I did for the z = 0.5 situation, above. In general this gives the break point on whether or not to use the “go for two first” at

I’ll rebuild the table that I had above for two situations; when the coaching staff aren’t very confident going into overtime, and estimate their chances at 45%, and when they are confident and estimate their chances at 55%.

These figures show that even if you are fairly confident that you will win in overtime (55% confident) then you should still use the “go for two first” strategy unless you think that your chances of making a two-point conversion are way below the 43% average, and that if you believe that your chances are not good in an overtime (45%) then you should use the “go for two first” strategy unless your two-point conversion team is really awful.

They also give rise to one final question: How confident do you need to be in your ability to win in overtime before you reject the “go for two first” strategy and use the “go for one both times” strategy? Some math will provide that information. We know that we should use the “go for one both times” strategy when:

This surprisingly simple condition says that you should only use the “go for one both times” strategy whenever . Assuming the standard NFL values for x and y, 0.43 and 0.96, respectively, then this is , which is 0.633 or 63.3%. So the traditional “go for one both times” strategy should only be used if you believe that your team is nearly twice as likely to win as the opponents in overtime, which seems a wildly optimistic assumption after tying in regular time.

Discussion

I have shown that under nearly all circumstances the “go for two first” strategy is significantly better than the “go for one both times” strategy when trailing by two touchdowns late in the game, and than also the “go for one then two” strategy should never be used.

The only times when the “go for one both times” strategy should be used is when either the coaches believe that they are nearly twice as likely to win as the opponents are (which seems overly optimistic after a tied game unless there are external factors like late injuries to some of the opponent’s important players) or when they believe that their team is far below average at making two-point conversions.

Since the correct strategy never appears to be used, an interesting question is why coaches have always got it so wrong. They have probably been led astray by the expected value of going for two point conversions vs. extra points. The expected value is the expected long term return from taking a particular action. In the case of a two-point conversion it is, for a typical team, (2 points)x0.43, which is 0.86 points each time that you try it. For an extra point it is (1 point)x0.94, which is 0.94 points. So for most of the game kicking extra points after touchdowns is slightly better than going for two-point conversions. When trailing at the end of the game the expected value of the points is no longer relevant, since all that matters is whether you are more or less likely to win. Looking at it differently, if the coaches use the “go for two first” strategy then, as we saw earlier, there is a 0.4042 of winning outright, a 0.3249 of losing outright, and the rest of the time (0.2709) you’ll go into overtime. So you are more likely to win than lose. Using the “go for one both times” strategy there is no chance of an outright win, a 0.9094 of going into overtime, and the rest of the time (0.0906) you will lose outright. So with this strategy you are more likely to lose than win. One reason expected values don’t help here is that if you lose outright with the “go for two first” strategy it will be by two points, but with the “go for one both times” strategy it will sometimes be by only one point, but a loss is a loss, so this isn’t relevant.

In this paper I haven’t discussed how to handle other situations like trailing by seven points (attempt the extra point) or by 21 points (go for two first). I also haven’t discussed high school football because two-point conversion attempt and extra point percentages vary so spectacularly across high school teams. However once high school coaches have some estimates for their team’s percentages in these two areas they can use the formulas in this paper to determine their best approach. It appears that for all practical cases the “go for two first” strategy will also be best for them.

References

1. Mallory, W. & Nehlan, D. (eds.) (2004). Complete Guide to Special Teams, American Football Coaches Association, ISBN 0736052917.
2. Smith, M. (n.d.) Splitting the Overtime Pizza, Football Outsiders Web Page, Retrieved September 19, 2005, from http://www.footballoutsiders.com/ramblings_print.php?p=87&cat=1.

Intercollegiate Athletic Corporate Sponsorships and the First Amendment:

In response to the recent escalation of head coaches’ salaries in intercollegiate athletics, and the constant pressure for athletic directors to adhere to shrinking budgets, many athletic directors are turning to corporate sponsorships to increase revenue. Athletic departments can raise money from corporate sponsors in a variety of methods ranging from selling advertising on the outfield fence of the baseball stadium to allowing corporate sponsors to set up an information booth and distribute advertisements on the concourse in the basketball arena. While an athletic director may be eager to welcome with open arms corporate sponsors with open checkbooks, an athletic director may at times be placed in a situation where he or she does not want to allow a corporate sponsorship to a company or organization that could portray a negative image on the athletic department or the university. For example, an athletic director may decide to decline an offer from Hooters, or a local “gentlemen’s club” to become a corporate sponsor for fear that it will negatively impact the family atmosphere and image the athletic department would like to portray. However, an Athletic Director at a state institution must be aware that providing sponsorship opportunities to certain companies while refusing sponsorship opportunities to others could implicate the right to free speech under the First Amendment. This article will attempt to provide an Athletic Director with information regarding lawful distinctions between corporate sponsors in order to avoid infringing on the First Amendment rights of the members of a corporation who are seeking to become a corporate sponsor for a state institution’s athletic department.

From the outset, it must be noted that this article only applies to state institutions because the First Amendment is not implicated with regard to private institutions. Thus, if a private religious university such as Brigham Young University refuses to allow Budweiser or Coors to become corporate sponsors (since alcohol consumption is contrary to the religious beliefs supported by B.Y.U.), it can lawfully decline such a corporate sponsor without any possible First Amendment implications.

The first step in analyzing a freedom of speech problem such as whether the athletics department can constitutionally deny a corporate sponsor an opportunity to engage in athletic department promotions is to determine the type of forum in which the intended speech is to take place. See Cornelius v. NAACP Legal Defense & Educ. Fund, 473 U.S. 788, 800 (1985). The 9 th Circuit in Diloreto noted that “where the government acts in a proprietary capacity to raise money or to facilitate the conduct of its internal business, the Supreme Court generally has found a nonpublic forum, subject only to the requirements of reasonableness and viewpoint neutrality.” Diloreto v. Downey Unified School Dist. Bd. of Educ , 196 F.3d 958 (9 th Circuit, 1999), citing Lehman v. City of Shaker Heights, 418 U.S. 298, 303-304 (1974). The Diloreto court held that where a high school offered advertising opportunities to businesses allowing the business to post an advertisement on a sign at the high school baseball field, and where the intent of the school in opening the baseball field to advertising was to raise funds, not to create a forum for unlimited public expression, the forum was a nonpublic forum open for a limited purpose. Id. at 966.

When an athletic department offers corporate sponsorships, it is clearly doing so to raise funds for the athletics department, not to create a forum for unlimited public expression. Thus, when an athletic department solicits and receives a corporate sponsorship such as for the signage at the athletic field or arena, the forum that has been created will likely be found to be a nonpublic forum open for a limited purpose.
In a nonpublic forum open for a limited purpose, restrictions on access “can be based on subject matter…so long as the distinctions drawn are reasonable in light of the purpose served by the forum” and as long as the restrictions discriminate on the basis of content rather than viewpoint. See Id.; Rosenberger, 515 U.S. at 829 (1995); Lamb’s Chapel, 508 U.S. at 392-93 (1993); Cornelius at 806. Thus, before denying an opportunity to a corporate sponsor, an athletic director must ensure that the he or she is making a reasonable distinction based on the purpose of the sponsorship opportunities it is offering, as well as ensuring that he or she is not making a distinction between two corporate sponsors on the basis of a corporation’s viewpoint.

Making a Reasonable Distinction

In Lehman, the U.S. Supreme Court addressed the issue of whether it was reasonable for a city transit system to decide which type of advertising may be displayed in its buses; an issue that arose when a political candidate was not allowed to advertise on the city buses. Lehman at 298. The Lehman court held that the city’s decision to exclude political advertising from bus signs was reasonable given the city’s desire to generate revenue and the potential for “ lurking doubts about favoritism, and sticky administrative problems [that] might arise in parceling out limited space to eager politicians.” Id. at 304.
An athletic director’s reason to decline offers from corporate sponsors such as Hooters or a local “gentlemen’s club” may be to keep a family oriented atmosphere at its games, and maintain an environment that is suitable for children. It is likely that making a distinction between a corporate sponsor such as Hooters and a corporate sponsor such as Ford or Coke based on the atmosphere that the corporate sponsor may create would be seen as a reasonable distinction.

Viewpoint Versus Content Discrimination

However, even assuming that an athletics department’s reason to deny a sponsorship opportunity to a certain corporation is reasonable, it may still violate the First Amendment if doing so discriminates on the basis of viewpoint, rather than content. Diloreto at 969, citing Cornelius at 811.

The U.S. Supreme Court noted that the distinction between viewpoint discrimination and content discrimination is not a precise one. Rosenberger at 831. The Diloreto court attempted to explain the distinction when it stated, “ Permissible content-based restrictions exclude speech based on topic, such as politics or religion, regardless of the particular stand the speaker takes on the topic.” Diloreto at 969, citing Children of the Rosary, 154 F.3d at 981. The Diloreto court further stated, “In contrast, impermissible viewpoint discrimination is a form of content discrimination in which the government targets not subject matter, but particular views taken by speakers on a subject.” Id. citingRosenberger at 829.
In Diloreto, a businessman was not allowed to buy advertising space at the high school baseball field because his proposed sign listed the text of the Ten Commandments, which was against the school’s policy of not allowing religious advertising. Id. at 962. The school district refused to post the sign “based on (1) concern about running afoul of the Establishment Clause; and (2) disruption, controversy and expensive litigation that might arise from community members seeking to remove the sign or from religious or political statements that others might wish to post.” Id. at 963. The Diloreto court stated, “We conclude that the District’s decision not to post Mr. DiLoreto’s sign was pursuant to a permissible, content-based limitation on the forum, and not viewpoint discrimination.”
If an athletic director allows corporate sponsors that promote a family environment to engage in sponsorships, while he or she refuses to allow corporate sponsors that promote an adult oriented environment to engage in the same sponsorships, the athletic director would likely be seen to be engaging in permissible content discrimination rather than viewpoint discrimination. By way of example, the athletics department would not be allowed to refuse to allow Hooters to be a corporate sponsor, while at the same time allow another adult oriented business such as a local “gentlemen’s club” to be a corporate sponsor because doing so would be deciphering between two different businesses that are adult oriented, which would likely be seen as viewpoint discrimination because the decision to exclude Hooters instead of the local “gentlemen’s club” would appear to be based on the particular views or specific stance Hooters takes. Similarly, an athletic director could not allow certain religious organizations such as the local Baptist Church to become a corporate sponsor while refusing to allow other organizations such as an atheist group to become a corporate sponsor because doing so would likely be seen as impermissible viewpoint discrimination.

However, the Diloreto case offers support for an athletic director to refuse to allow a religious organization or an adult oriented business to become corporate sponsors so long as the athletic director refuses to allow all religious organizations and adult oriented businesses to become corporate sponsors because doing so would likely be viewed as permissible content discrimination rather than impermissible viewpoint discrimination.

Conclusion

Therefore, since an athletic department, through its corporate sponsorships has likely created a nonpublic forum open for a limited purpose, if the distinction the athletic director is making between different corporations is a reasonable distinction that does not amount to viewpoint discrimination, it is likely that the athletic director’s actions of refusing to allow such a corporation to be one of its corporate sponsors would be found to be constitutional.