For March Madness, the Mathematics Behind Bracketology

March 11th, 2012 by Erwin Gianchandani Post a comment »

The 2012 NCAA Final Four in New Orleans [image courtesy NCAA].Just in time for the kickoff of March Madness later today, University of Illinois at Urbana-Champaign computer science professor Sheldon Jacobson describes the mathematics behind bracketology — and BracketOdds, a website his research team developed that uses data from 27 past tournaments “to identify a distribution that models the probability of certain seed combinations playing each round of the tournament.”

From the interview, posted on UIUC’s website:

Sheldon Jacobson, UIUCThe tournament is exciting for its upsets and seeming unpredictability. Yet your research has found distinct patterns. How can that help people trying to make sense of it all?

 

Each game in the tournament can be viewed as a random experiment, with a different probability for each game (or each pair of seeds pitted against each other). Our research suggests that in the Elite Eight and beyond, we can model the performance of how far seeds progress. An implication of such a model is that it is less important which teams are playing each other, but rather, which seeds are playing each other.

 

How does the BracketOdds site help aspiring bracketologists?

 

Working with my students, we created the BracketOdds website to assist fans filling in their brackets. The site translates our model into a Web tool for anyone interested in assessing the seed combinations for their brackets in the Elite Eight and beyond. Let’s take the Final Four, for example. The most likely Final Four seed combination is 1, 1, 2, 3. The odds against this occurring are about 16 to 1. It has occurred three times in the past 27 years. The odds against the four No. 1 seeds reaching the Final Four are about 48 to 1, just about three times less likely. This has occurred just once in the past 27 years.

 

The site can compute the odds against seed combinations occurring that have never been observed. For example, that odds against a 1, 1, 2, 4 Final Four is about 26-1, the highest odds Final Four that has yet to occur. The odds against one or more No. 16 seeds reaching the Final Four are about 791 to 1. The odds against all four No. 16 seeds reaching the Final Four are about 100 trillion to 1, which is just over six times the size of the national debt.

 

How should people interpret the odds that the site calculates?

 

The odds provide a measure of likelihood for a certain set of seed combinations to occur in a given round. Relative odds rather than absolute odds are the best way to use the information from the site. To illustrate this point, the odds against a 1 vs. 2 national finals is about 3.7 to 1, while the odds against a 2 vs. 3 national finals is about 13.6 to 1. This means that a 1 vs. 2 national final seed combination is just over three times more likely to occur than a 2 vs. 3 final combination. In other words, comparing the odds of different seed combinations can help people assess the likelihood of their bracket compared to other people’s brackets…

 

Are there any other insights you can share with the millions of people who will be filling out brackets after Selection Sunday?

 

On our [BracketOdds] website, we have a section called “Help With Building Your Bracket” that highlights numerous observations to help people calibrate the right number of upsets in each round. For example, the 12-5 upset in the round of 64 is often discussed, yet the 11-6 upset is just as likely to occur. In all but three of the past 27 tournaments, an average of just over 3 teams seeded No. 7 or lower have reached the Sweet Sixteen. As for upsets, in 18 of the past 27 tournaments, eight or fewer of teams seeded No. 1, 2 or 3 reached the Sweet Sixteen (in other words, four or more did not). On the other hand, for the risk averse, in 11 of the past 27 tournaments, only teams seeded No. 1 or 2 have reached the Final Four; the odds against this occurring are 6-1. The Web tool used to compute many of these odds can be found at here.

To learn more, check out the full interview and try your hand at BracketOdds.

(Contributed by Erwin Gianchandani, CCC Director)

  • mystery_meat

    “The most likely Final Four seed combination is 1, 1, 2, 3. The odds against this occurring are about 16 to 1. It has occurred three times in the past 27 years. The odds against the four No. 1 seeds reaching the Final Four are about 48 to 1, just about three times less likely. This has occurred just once in the past 27 years.”

    This doesn’t help a whole lot when actually picking teams in the tournament. If you are picking seeds, then yes, I would choose the first scenario. There are four teams for each seed so there are twelve possible combinations to get a 1, 1, 2, 3 scenario. While there is only one possible way to get a 1, 1, 1, 1 scenario.

    For example, let’s list possible 1, 1, 2, 3 seed scenarios in the 2012 NCAA Tournament:

    Kentucky, Michigan State, Florida State, and Kansas.
    Kentucky, Michigan State, Ohio State, and Georgetown.
    Duke, Marquette, Syracuse, and North Carolina.

    And the list goes on….there are a total of twelve combinations.

    But there is only one combination in which all one seeds advance: Kentucky, Michigan State, Syracuse, and North Carolina. So yes, the first scenario is more likely when considering which seeds will be in the Final Four, only because there are so many more combination possiblities.

    All in all, when talking about picking actual TEAMS, not seeds, advancing all of the top ranked seeds from each region (the one seeds) is probably the most likely scenario (since they are considered the best teams in their region). So whoever is giving “bracket tips” and telling people to not pick all one seeds in the Final Four is pretty much advising people to not pick the most likely Final Four scenario.

    That is all.