It's March. This past Sunday, the first round match-ups in the Division I NCAA Men's basketball tournament were announced and with that, the madness began! Sports channels offer (seemingly, if not literally) nonstop analysis on the match-ups. Over the next few days, millions of people will decide how to fill out a bracket. They use a variety of techniques ranging from one's preference of mascots to personal algorithms based on one's knowledge of the season. Even with over 5 million brackets submitted to ESPN alone last year, there has never been a perfect bracket submitted to ESPN, CBS or Yahoo Sports. Surprising? Keep in mind, there are 9,233,372,036,854,775,808 (which is said 9 quintillion) ways to fill out a 64-team bracket.
Want some help? Turn to algebra. Remember solving equations like x + y = 5 and 2x - y = 7? You can add the equations together and get 3x = 12, implying x = 4, and thus y = 1. The Bowl Championship Series, which ranks college football teams for the bowl games, already uses such equations in one of its ranking methods. What equations? Above, the variables x and y are unknowns (i.e. we don't know their values in advance) since we must determine their value to make both equations true at the same time. Having 2 equations with 2 unknowns could rank 2 teams. The BCS must find the values for about 700 unknowns to make 700 equations true at the same time. Cool application of algebra but not quite (or even close to) a homework problem. The equations are formed using the win-loss record of each team and enable the rankings to take into consideration a team's strength of schedule. Beating strong teams improves one's ranking more than beating weak teams.
We can leverage the techniques of the BCS to create a bracket. Just find the values for 350 unknowns that make 350 equations true at the same time! Ready to run to the water cooler for a variety of ad hoc bracket advice? Not so fast. What if a computer program formed the equations for you and found those 350 values? Further, what if you could make a decision, all your own, that would create your own mathematically-generated BCS-style bracket? Let's do it.
First, we won't go into the details of how to form the equations. A computer will do this for us. To dig deeper into the math, read the new book by Amy Langville and Carl Meyer Who's #1? The Science of Rating and Ranking, published by Princeton University Press. It discusses two BCS ranking methods, the ranking algorithm of Google that was instrumental in their success, and how to rank chess, movies, and more.
The key to our method will be determining the importance of a game. Let's think of breaking a season into 4 parts as seen below.
The choice now is how much to count a game in each part. Generally, a game counts as 1 win and 1 loss for the respective teams. In our method, we alter this to make some games more important than others. For instance, let's count 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. Given these choices, what would you choose for the third and fourth pieces of the season? 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 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. The values of 0.5, 0.75 and such are called weights, as we are weighting the games.
How do you form your bracket? Look at your ranking and for any game the higher ranked team wins. Can this really work? 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. How will yours do? Keep in mind, March is inherent with madness. Your choice in weighting will pick some winners and almost undoubtedly some losers. How many? Depends on your math modeling decision regarding the weights and, of course, how the tournament unfolds. Think about using algebra to alleviate some of the madness of March, fill out your bracket and let the games begin! To rank teams for March Madness, click here.