Metric Madness David Clark W hy leave all the March Madness moneymaking to the television networks? ESPN.com and Yahoo! Sports each offers a $10,000 prize for the winner of its bracket competition. Just fill out an online bracket before the tournament starts, and your score is tabulated autopoint for each correct first-round pick, matically: points for each correct second-round pick, up to points for choosing the tournament champion (i.e., the winner of round six). For example, the winner of ESPN s bracket competition last year scored 163 out of a possible 192 points and got 10,000 bucks for it! But there s a catch. What if two or more bracket pickers finish with the same winning score? (Two brackets We can quickly see that the rings in the Manhattan metric are not round; they are diamond rings! can have the same score even if they are distinct: For example, although there is only one bracket that has a perfect score of 192, there are 16 different ways to score 191.) Instead of splitting the prize money among multiple winners, ESPN.com and Yahoo! Sports have devised a tiebreak system. Each bracket picker, at the time his or her bracket is submitted, must also guess the final score of the championship game. Whoever guesses closest to the actual score wins the tiebreak, and thus the $10,000 prize. Here s the rub: What does closer mean here? We all understand that 5 is closer to 7 than 12 is, but a final score is a pair of numbers how do you compare them? To illustrate the difficulty, let s pretend that two bracket pickers, Euclid s Duke Kids and Bourbaki s Bracki, have tied for the winning point total and are headed for a tiebreak. Here are their final score guesses, in addition to the actual final score of the (pretend) championship game: Winning Team Losing Team Bourbaki s Bracki 86 52 Euclid s Duke Kids 74 62 Score 85 72 Whose guess is closer? Bourbaki s Bracki has a very accurate guess of the winning team s score (only off by one), but a comparatively worse guess of the losing score (off by 20). One way to quantify this is to simply add the two errors together: Thus, Bourbaki s Bracki has while Euclid s Duke Kids a total error of By this analysis, each is off by a total of guess is equally good, so our tie remains unbroken. But is this the best way to compare the two guesses? What we re really after is a notion of distance between two scores; whichever guess has a smaller distance to the actual score is the better guess. Mathematically we are speaking of a metric, a function d that outputs a nonnegative real number (the distance) for any input pair of points in a set. (Such a function is meant to generalize our usual notion of spatial distance and so must satisfy a few conditions; for more details, look up metric space in any analysis or topology textbook, in MathWorld, or on Wikipedia.) In the case of our tiebreak system, we are looking for a metric on the set of final scores, 20 April 2012 : : Math Horizons : : www.maa.org/mathhorizons mh-20-24-clark-r3-ad.indd 2
where the winner s score is always written first in the pair. In trying to find such a metric, we can draw some inspiration from the fact that the set S is naturally a subset of and any final score can be easily plotted as a point in the plane. Helpfully, the plane has several well-established metrics, two of which we will consider more closely. One is called the Manhattan metric, also known as the taxicab metric, defined by This is the distance a taxicab would have to travel to get between two points in a city with only east-west and north-south running streets. For example, using the Manhattan metric, the distance between the points and is 8 Figure 2. Standard metric: ring 7 of radius 5 centered at (4, 5). 6 5 4 3 2 1 r 5 1 2 3 4 5 6 7 Of course, this is just the standard distance formula! Using this metric, we see that It s helpful to see a picture of all points in the plane that are a distance of 3 from in the Manhattan metric: See figure 1. Figure 1. Manhattan metric: ring of radius 3 centered at (4, 5). 8 7 6 In this case, the ring of radius centered at is a circle. See figure 2. Each of these metrics on can be applied to our set of final scores. In fact, our first observation, that Bourbaki s Bracki and Euclid s Duke Kids were each off by 21 points, was really an application of the Manhattan metric: 5 4 3 2 r 3 However, our more intuitive idea of geometry says that distances in the plane should be determined by the Euclidean metric. See figure 3. 1 r 17 1 2 3 4 5 6 7 r 21 In fact, let s give a name to this set of points: the ring of radius 3 centered at We can quickly see that the rings in the Manhattan metric are not round; they are diamond rings! Another common metric on the plane is called the Euclidean metric, usually denoted with a plain d. 60 60 Figure 3. Two rings centered at the actual score (85, 72), one in the Manhattan metric and one in the Euclidean metric. www.maa.org/mathhorizons : : Math Horizons : : April 2012 21 mh-20-24-clark-r3-ad.indd 3
With this choice of a metric on S, Bourbaki s Bracki has the lesser distance to the actual score, winning the tiebreak. So, is the Euclidean metric the best one to use for final scores? Let s give it another test in the following scenario: Winning Team Losing Team Bourbaki s Bracki 72 62 Euclid s Duke Kids 72 58 Score 70 60 A note for linear algebra enthusiasts: which contains the set S of final scores, can be parameterized equally well by the two coordinate systems and the former has basis while the latter has basis A consequence is that any score can be written uniquely in either coordinate system. For example, going backwards, we see that if we must have that How does this new metric compare with the Euclidean metric d? It depends on the choice of weights and First, with a bit of (rather painful) algebra we can rewrite in terms of w s and l s: Again, we compute: Both metrics give a tie. But are these two guesses really equally good? Here s where it starts to get interesting. Says Bourbaki s Bracki, Our guess got the point differential (10) perfectly! Clearly it is the better guess! But our guess got the combined points (130) exactly! replies Euclid s Duke Kids. Show us the money! These are both arguments that a basketball fan might make, and they represent ideas used in the arena of sports gambling. Indeed, the two most common ways to gamble on a basketball game are betting the spread (trying to guess the point differential) and betting the over-under (trying to guess the combined points). Let s now create a third metric, based on the Euclidean metric but with a gambling twist. To simplify our equations a bit, let s use delta notation; thus, for example, Then we can rewrite the original Euclidean metric as We want a metric that reflects information about spread and combined points This is easily done: Just replace and with and We ll also throw in some weights, and that we can tweak later. Our new metric then looks like Notice that if the formula above collapses back to the usual distance formula, giving that For any other choice of the weights, we get a different metric. In particular, if then our will give more weight to point spread. For example, if and then Here, Bourbaki s Bracki has a guess that is closer to the actual score, thus winning the tiebreaker. On the other hand, if then our will more heavily weight the combined points. Thus, for example, if and then in which case, Euclid s Duke Kids has the better guess. Thus, there are $10,000 riding on our choice of and This is a lot of pressure! In choosing weights, we should ask ourselves the following: In the context of NCAA men s basketball championship games, which is harder to guess: spread or combined points? Perhaps history can be our guide here. Looking at all NCAA championship games going back to 1950, and with the help of a spreadsheet, it s easy to compute that the mean spread is 9.4, with a standard deviation of 6.5. On the other hand, the mean combined score is 143.8, with a standard deviation of 19.3. This says the combined score is historically much more variable and, 22 April 2012 : : Math Horizons : : www.maa.org/mathhorizons mh-20-24-clark-r3-ad.indd 4
thus, harder to forecast in any given year. Let s choose our weights so that How much smaller? One natural option is to set the ratio of the weights equal to the ratio of the corresponding standard deviations: 64 62 r 3 60 This still doesn t pin down our weights completely it only sets up their proportion. To get unique values, we need one more equation: 58 r 2 2 This secretly comes from linear algebra. Remember the two coordinate systems and above? Well, when you compose the change-of-basis transformation between them with a scaling transformation by the weights and you get a new invertible linear transformation. The equation above forces that linear transformation to have determinant one, which is desirable for reasons we ll soon see. Now we just solve the system to get and After rounding off these constants, we finally have our new metric let s call it the gambling metric on the set S of final scores: Applying this formula to our tiebreak, we compute that 56 66 68 70 72 74 Figure 4. Two rings centered at the actual score (70, 60), one in the Euclidean metric and one in the gambling metric. These rings enclose equal areas. where and are constants. This is the equation of an ellipse; see figure 4. Of course, the eccentricity of these gambling rings was determined by our choice of weights and However, assuming we ve chosen them to produce a linear transformation with determinant one, any gambling ring of radius r encloses the same area as a Euclidean ring of radius r. Let s test our gambling metric on one final scenario: Winning Team Losing Team Bourbaki s Bracki 105 85 Euclid s Duke Kids 95 75 Score 95 85 Here we compute that Yet again, we have a secondary tie. Are these guesses equally good? Having already taken spread and combined points into account, we must appeal to a different notion from probability. Remember that the mean score of all NCAA men s basketball championships, since 1950, is about See figure 5 for a plot of this mean along with our guesses. Observe that, while both guesses have equal distance from the actual score, one is much closer to the mean score: Euclid s Duke Kids guess lies inside the ring of radius 30 (in the gambling metric) centered at the mean score, while Bourbaki s Bracki lies outside. One interd g ((105,85),(95,85)) 10.73 So, using our new metric, Euclid s Duke Kids wins. Let s also look at the geometry here. Remember that these two guesses tied using the Euclidean metric: In particular, they both lie on the ring of radius centered at We also know that, in the gambling metric, Euclid s Duke Kids guess lies inside the ring of radius 3, while Bourbaki s Bracki s guess lies outside. But what, exactly, do the rings in the gambling metric look like? A little algebra shows that the ring of radius 3 centered at is the set of points that satisfy d g ((95,75),(95,85)) 10.73. www.maa.org/mathhorizons : : Math Horizons : : April 2012 23 mh-20-24-clark-r3-ad.indd 5
What metric do these online bracket competitions use? It turns out that ESPN.com breaks its ties with the Manhattan metric; Yahoo! Sports compares only the combined score. It is also relevant to ask how often these tiebreak systems are employed. While both companies are tight-lipped about data from their competitions, it is a safe bet that the number of entries in each competition is large. In other words, ties are probably common, and eventually a tie will arise that their basic tiebreak systems will resolve unjustly, or not at all. Although it may be hard to imagine that such a simple game would require the consideration of metrics in it is the opinion of this mathematician that these are precisely the ideas that deserve our attention. And it should be done soon, before the inevitable perfect tie creates some real March madness. 110 r 30 r 10.73 90 70 Mean 70 90 110 Figure 5. Two gambling rings: one centered at the actual score (95, 85), and one centered at the mean score (77, 67). pretation of this fact is that Bourbaki s Bracki made a bolder guess. We d like to put forth the following idea: that a bold guess should be rewarded, since it is less likely to be a good guess by accident. Using this criterion, Bourbaki s Bracki wins the secondary tiebreak and the $10,000. David Clark is an assistant professor of mathematics at Randolph-Macon College. He takes pleasure in topology, geometry, and the arc of a well-shot three-pointer. Email: DavidClark@rmc.edu http://dx.doi.org/10.4169/mathhorizons.19.4.20 GRAND OPENING! MAA Store powered by Amazon > Enhanced search function > Easy navigation > Access to MAA merchandise > Link to your Amazon password & username > All major credit cards accepted > Wish list MAA Store > Special sale section Visit us at: http://maa-store.hostedbywebstore.com/ MATHEMATICAL ASSOCIATION OF AMERICA 24 April 2012 : : Math Horizons : : www.maa.org/mathhorizons mh-20-24-clark-r3-ad.indd 6