Every March, the country goes mad for March Madness. On Selection Sunday, the initial match-ups in the Division I NCAA Men’s basketball tournament are announced, and with that the madness begins. During the next few days, millions of people fill out brackets requiring predictions for proposed winners in 63 games. This is no easy task. In 2014, Warren Buffet insured a billion-dollar prize for anyone who could perfectly predict the tournament. There were millions of entries. A small number of brackets submitted to the billion-dollar contest perfectly predicted the initial round of 32 games, but no one was perfect after the following round of 16 games, and there has never been a perfect bracket submitted to ESPN, CBS or Yahoo Sports.

While producing a perfect bracket is clearly possible, understanding its improbability requires careful counting. Each year, there are 263 (or 9,233,372,036,854,775,808) which is nine quintillion unique bracket possibilities; in other words, the odds of a perfect bracket are one in nine quintillion. Suppose you decided your goal is to produce every bracket in a given year. If you could produce one billion unique brackets per second and never repeat a bracket you’ve produced, it would take close to 300 years to create nine quintillion brackets. As such, our best attack is improving the odds of winning, which is connected to producing a perfect bracket.

The odds of perfectly predicting the tournament are only 1 in 9 x 10^18 if you assume you are 50% likely to predict the outcome of every game in the tournament. This isn’t true as some game outcomes are much more probable than others. For instance, a #1 seed has always beaten a #16 seed. Still, it’s quite likely that a #16 seed will eventually win.

Historically, people have predicted outcomes in March Madness with 70% accuracy. It’s the 30% of incorrect predictions that creates the “madness” that people want to tame. Assuming a 70% chance of predicting the outcome of any given game in the tournament, the odds of a perfect bracket drop to about one in 5.7 billion. If you improve your predictability to 71%, the odds drop to about one in about 2.3 billon. Things are still not probable but the odds have improved significantly, perhaps enough to win a pool.

At Davidson College, I lead a sports analytics group and teach analytics methods in my applied math courses for both math majors and non-majors. Along with my collaborator, Amy Langville PhD at the College of Charleston, we adapted techniques used by the Bowl Championship Series rankings. These ranking methods use linear algebra, which is used to solve a system of equations like x + y = 5 and 2x - y = 7, requiring x to equal 4 and y to equal 1. In our methods, the ratings of the teams are our unknowns. We also decided who’s #1, #100 and even #350 as we rank all Division 1 men’s basketball teams. So, the linear systems for March Madness involve about 350 unknowns in 350 equations. In the example above, we had two equations with two unknowns: x and y. Our methods produce a ranking of all 350 teams. To form a bracket, you can predict the winner to be the team with the better ranking. So, a team ranked 5th would be predicted to beat a team ranked 11th. In the past years, such methods have resulted in brackets that have outscored 97% and even 99% of the millions of brackets submitted to the ESPN Challenge.

We won’t go into the details of how to form the equations.

It is important to understand the impact of using linear algebra. First, consider winning percentage, where a higher winning percentage implies a stronger team. This isn’t always true. A team from a weak conference may have a strong record but then lose by a large margin to a team like Kentucky, Notre Dame or Duke. Linear algebra enables strength of schedule to be included into the ranking. As such, beating a strong team improves your ranking more than beating a weak team. With linear algebra-based methods, a team’s record is based on the number of wins and losses and the quality of those teams.

Using the BCS linear algebra-based methods, as originally derived, leads to the same bracket for everyone. The key to producing your own mathematically generated bracket is determining the importance of a game. One popular method is breaking a season into four parts. The choice is then how much to count a game in each quarter of the season. Team records do not distinguish a “good win” from a “bad loss”— a win is a win. In our methods, games can vary in their importance. For instance, let’s count (weigh) the games in the first quarter of the season as half a win and loss for the respective teams. In the second quarter of the season, the games will count as 0.75 a game and so on. Is the last part of the season leading into the tournament the most instrumental in predicting a team’s success? If so, maybe you could weight it as 1.25 game, or even 2.

The higher the weight, the more your final ranking will reward teams with momentum in that last quarter of the season.

Now, keep in mind that this step, as well, integrates the strength of schedule. If the final quarter of the season has the highest weight, then a team improves its ranking by beating high quality teams in the final quarter of the season. In a sense, this type of model is developed under an assumption that having momentum in the last quarter of the season is predictive of playing well in the Tournament.

Now, let’s look more closely at the decisions necessary in using, in particular, the online codes. First, we adapted two BCS methods. The Colley method considers only wins and losses and integrates Bayesian probabilities into the calculations. The other method, called the Massey method, integrates scores. We have found that when there is a lot of parity in the teams the Massey method can struggle as scores can create noise rather than differentiation. For instance, a 20-point blow-out might end up as an eight-point win when second teamers finish the game and a two-point game gets stretched to

12 when the leading team hits free throws down the stretch.

Up until now, we have discussed using temporal data to vary the importance of games. One might have games in the final quarter count as 1.4 wins and 1.4 losses. One may also want to weight wins at home with a different weight than wins on the road, weighting the ability to win on the road higher (see “Picking winners,” below).

What parameters should you weight, possibly integrating statistics not listed here? In part, that’s your choice. You may want to use other methods such as regression and machine learning to determine which factors lead to your weightings for games. In this case, finding your ideal weight can lead to big improvements in your bracket performance. Keep in mind, however, that the tournament is inherently maddening and while you might improve your odds of winning, the search for perfection can be quite elusive.