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Playing with the Percentages When Trailing by Two TouchdownsJ. Denbigh Starkey Computer Science Department Montana State University Bozeman, MT 59717
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 twopoint 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 twopoint 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 twopoint 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 twopoint 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 twopoint 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 twopoint 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 twopoint 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 twopoint 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 twopoint 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 twopoint conversions and extra points. In the NFL the figures are 43% for the twopoint 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 twopoint 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 twopoint 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 twopoint conversion and the extra point are 43% and 94%, respectively, (and so the percentages for failing on the twopoint 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 twopoint conversion and the subsequent extra point, two additional points when either they miss the first twopoint attempt and hit the second or when they make the twopoint 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 twopoint and then the conversion), which is 0.4042. The probability of two additional points (making the twopoint and missing the extra point or missing the first twopoint 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 twopoint attempts, which is 0.57 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 twopoint 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 twopoint 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 twopoint 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 twopoint 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 twopoint conversion and subsequent extra point both succeed (probability xy), or the twopoint conversion succeeds and the extra point fails but you win in overtime (x(1y)z), or the first twopoint conversion fails, the second one succeeds, and you win the overtime ((1x)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 twopoint conversion succeeds, and again you win in overtime ((1y)xz). Finally the “go for one then two” strategy wins if the extra point and the subsequent twopoint conversion both succeed (yx) or the extra point fails and the subsequent twopoint conversion succeeds and you win in overtime ((1y)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 (1x)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 twopoint conversion fails then the game is lost, while with the “go for two first” strategy if the twopoint conversion fails then the team can attempt a second twopoint 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 twopoint 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 twopoint 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 twopoint 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 twopoint conversions. To justify these numbers I just need to ensure that xy + ½x(1y) + ½(1x)x is greater than ½y 2 + ½(1y)x, cancel out the common term, and solve the equation for x. This gives xy + ½(1x)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: x > 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 twopoint 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 twopoint 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 twopoint 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 x > 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 twopoint 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 twopoint 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: x < Rearranging gives: Squaring gives: Getting rid of the three common terms and then the common 4z factor gives: and finally . 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 twopoint 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 twopoint conversion it is, for a typical team, (2 points) 0.43, which is 0.86 points each time that you try it. For an extra point it is (1 point) 0.94, which is 0.94 points. So for most of the game kicking extra points after touchdowns is slightly better than going for twopoint 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 twopoint 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 Mallory, W. & Nehlan, D. (eds.) (2004). Complete Guide to Special Teams, American Football Coaches Association, ISBN 0736052917. 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.

