The math of March Madness brackets
Let’s say you somehow manage to sleep through all of the National Collegiate Athletic Association’s March Madness and wake up in a stupor after all 64 college basketball teams in the main tournament have faced off and a champion has been crowned. You drop down to find your friends’ boards, in which they predicted which team would win each game, as well as a list of how many points each of your friends ended up scoring. Is it possible to know which team won each match? According to mathematicians, it depends on how many of these friends you have and how each one filled in their boxes.
The March Madness main tournament begins by pitting 64 of college basketball’s highest-level teams in 32 games. The 32 winning teams advance to the next round, during which they are divided into 16 matches. Then come eight games, then four, then two and finally a championship game. Whether in national pools for cash rewards or just for fun with friends, millions of Americans each year fill out templates called “brackets” while trying to predict which team will win each game. For each of the 63 matches, the decision makers have two choices for the winner, resulting in 263 possible configurations of the supports. That’s more than nine quintillion! It should therefore come as no surprise that perfect support has never been verifiably achieved.
Instead of trying to create the perfect medium, Georgia State University mathematician Sam Spiro took a different direction. He asked: Given a set of scores scored based on the number of correctly predicted matches, can I reconstruct how the tournament actually went?
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Suppose someone gets a perfect score in their category. Their predictions were then completely correct and you know precisely what happened during the tournament. But what if you only had a side note from a poor, unfortunate soul who didn’t score any points? (A randomly chosen medium has a 1⁄232or one chance in more than two billion of winning nothing. Quite impressive!). This bracket tells you the winners of all the first round matches (the opposite of what your wonderfully incompetent friend selected), and you know who played in the second round matches, but not much else.
Spiro got the idea to study this while writing a web novel about “a presidential fighting tournament.” Friends began sending him ranges of predictions about which former U.S. presidents they thought would win. “I realized that if I kept telling them how well they scored against each other, I would end up revealing the entire tournament… So I just asked myself, ‘How much information am I giving away?’ At what point, if they colluded, would this determine everything?'” Spiro says. “And then I thought it would be better to present this as a story about March Madness rather than a story about my president’s obscure novel.”
From these silly beginnings came serious new mathematical results. Spiro discovered that in any standard single elimination tournament with n teams, there is a set of n⁄2 supports that can be used to determine the actual results of tournament matches, regardless of how the tournament plays out. This means that for a 64-team tournament like March Madness, you can strategically choose 32 different brackets in advance that, once scored, will reveal all the results to you. This is true for any scoring system that assigns each game a positive number of points for a correct prediction, provided you know the scoring system beforehand.
But what happens if your friends don’t respect the specially chosen brackets? How many different brackets would you need to ensure you can determine the outcome of the tournament, regardless of which brackets you choose? According to Spiro, this number is harder to pin down, but it is “very, very high.” He determined a bracket that depends on the number of teams participating in the tournament. For March Madness, that’s between 8.9 quintillion and nine quintillion, which is pretty close to the total number of possible brackets. That’s about a billion times the number of people on Earth. So you better get to work making some new alien friends — or maybe you’re better off just watching March Madness after all if you want to know how it’s done.
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