The Math Behind March Madness

On March 14, 2017 I gave a talk about bracketology, March Madness, and the College Football Playoff in the Discovery Building on the University of Wisconsin-Madison campus. The talk was recorded and can be viewed here or here:

My slides from the talk are here:

a few last minute thoughts on filling out a perfect bracket

Yesterday, I was on Madison’s CBS affiliate WISC-TV to talk about March Madness and filling out a bracket.

I was also on NBC15 in Madison to talk about the probability of filling out a perfect bracket.

Here is one data point from 2015:

Out of more than 11.57 million brackets entered in ESPN’s Tournament Challenge, one bracket emerged from the round of 64 of the NCAA tournament with a perfect 32-0 record. This is the first time there was a perfect first round in ESPN’s Tournament Challenge since at least 2010 (we’re still trying to find out exactly the last time it happened, but it’s been a number of years).

A little research uncovers 1 perfect bracket after the first 32 games in 54.85 million brackets at ESPN.

Last year, 25,704 of 13 million brackets (0.2%) remained perfect after the first 16 games and none were perfect after the first 32 games. None of those correctly picked the next 16 games.

The probability of a perfect bracket depends on the upsets in any given year. We can see that by observing the number of brackets that correctly selected all Final Four teams:

  • 1140 of 13 million brackets correctly picked all Final Four teams in 2016
  • 182,709 of 11.57 million brackets correctly picked all Final Four teams in 2015. This year stands out because the Final four was composed of #1 Wisconsin, #1 Kentucky, #1 Duke and #7 Michigan State.
  • 612 of 11 million brackets correctly picked all Final Four teams in 2014
  • 47 of 8.15 million brackets correctly picked all Final Four teams in 2013
  • 23,304 of 6.45 million brackets correctly picked all Final Four teams in 2012
  • 2 of 5.9 million brackets correctly picked all Final Four teams in 2011

Of course, these brackets missed several games along the way so none are perfect, but from these data points we can see that it’s inherently more difficult to pick a correct bracket when there are more upsets in any give year or unlikely teams in the Final Four.


Yes, Wisconsin should have been seeded higher

This year I have fielded a lot of questions about Wisconsin’s seed. As the second place finisher in the B1G and runner up in the B1G conference championship, it’s hard to accept an 8 seed. It’s even harder given that lower ranked teams are seeded higher. Maryland finished 3rd in the B1G and received a 6 seed, Minnesota was 4th in the B1G and received a 5 seed, Northwestern finished 6th in the B1G and received an 8 seed, and Michigan received a 7 seed after winning the conference tournament. Wisconsin and Northwestern really don’t deserve the same seeds.

I can’t mathematically prove that Wisconsin was robbed, but I’ll try to step through the process to shed light on what happened.

First, I’ll say a few things about how seeding works.

The first step is selecting the field, and this is started well in advance of Selection Sunday. There are a number of teams who automatically make the tournament, and the committee first chooses who received at large bids. It’s safe to assume that Wisconsin was a lock to make the tournament all along.

The selection committee looks at several ratings/rankings including LRMC (Wisconsin is 22), the Pomeroy ratings (Wisconsin is 23), the Sagarin ratings (Wisconsin is 17), ESPN’s BPI (Wisconsin is 21). Wisconsin is in the top 23 in all of these aside from RPI (where Wisconsin is 36), but RPI isn’t a very good tool and is not used to seed the teams. The committee then ranks the teams 1st to 68th (the “S curve”) to get a larger sense of what the seeds should be and to identify if the seeds in each region are balanced when the seeding is done.

Teams that are consistently ranked in the top 25 are often seeded 6th or so, which is significantly higher than the 8 seed Wisconsin was assigned.

There is not a lot of competition for the 1 seeds because so few teams can make a claim for the one seed. The ends of the distribution are easy but the middle is tougher because there is less difference between the 20th ranked team and the 40th ranked team than between the top ranked team and the 10th ranked team. Therefore, a team in the middle could reasonably be assigned to either a 5 seed or an 8 seed or anywhere in between.

Assigning seeds is not as simple as knowing that there are four of each seed and picking one for each team. There is a lot more to it than that. Scheduling the tournament is hard because there are a lot of constraints. Teams from the same conference cannot meet in the round of 64 or 32. Therefore, a team’s seed might need to be slightly malleable to make it all work while obeying these constraints. Additionally, as mentioned earlier, the seeds need to be assigned so that the strength of each of the regions is roughly balanced. For example, the same region should not contain the best best 1 seed, the best 2 seed, and the best 3 seed.

There is more. Distance is taken into account when assigning the seeds. The committee assigns game locations at the same time it assigns seeds. That makes for a lot of interconnected decisions. All seeds are assigned to one of 8 locations called “pods” where the tournament games are played. For example, the 1, 8, 9, and 16 seeds must play in the same location. The goal is to minimize travel for most of the teams, especially the highest seeded teams who often basically get “home” games where their fans do not have long to travel. There are eight pods, and two of the 4-team groupings are assigned to each pod.

It’s possible to assign a team a lower seed so that they would have substantially less travel. That was not the case for Wisconsin, who was assigned an 8 seed in Buffalo.Initially, I suspected that Wisconsin was assigned an 8 seed in the Milwaukee pod, which would have been essentially a home game. That was not the case. Instead, the Badgers are getting the worst of both worlds: a bad seed and a long distance to travel. I’m puzzled by this.

You cannot change one team’s seed or a pod location without creating ripple effects that affect many other teams so these decisions are not easy to make. You can see the bracket with pod locations here. I’m sure that every year a team or two gets under/over-seeded because of these constraints. It’s easy to point out teams that are badly seeded but it’s much harder to know how to fix the problems. The tournament scheduling part is so difficult that some (like Cole Smith, Barbara Fraticelli and Chase Rainwater!) have published research papers on optimization models to design good tournaments that balances these constraints while getting the seeds right. Despite this challenge, it seems that swapping Wisconsin with another B1G team such as Minnesota would be feasible, not cause other ripple effects, and would better reflect the teams’ rankings. Scheduling might be hard, but a lot is on the line so the committee should get it right most of the time.

I also may be biased, but in sum, I cannot find a part of the seeding process where Wisconsin being assigned an 8 seed makes sense. I’m disappointed. Getting the seeds right matters because, in this case, an 8 seed for Wisconsin means playing overall top seeded Villanova in the round of 32, assuming that Villanova is not the first 1 seed to be upset in the round of 64. That makes for a rough path to the Final Four. In any case, this is all part of the game. I’m looking forward to the tournament and rooting for the Badgers.

I talked to WKOW in Madison about the seed. You can read about it and watch the video here.

I’ll be talking about the math behind #MarchMadness on March 14 on the UW-Madison campus

I will be talking about bracketology, March Madness, and the College Football Playoff on Tuesday, March 14 at 7:30 pm in the H.F. Deluca Forum in the Discovery Building on the University of Wisconsin-Madison camps. More information can be found here. I hope to see you there!


Bracket tips for winning your #MarchMadness pool

Today I have four tips for winning your March Madness bracket pool.

1 Ignore RPI, use math based rankings instead to take strength of schedule into account.

Ken Massey has a rankings clearinghouse here: I’m happy to say that I’m the only women contributor to this list 🙂 My rankings are here.

2 Pay attention to the seeds

The seeds matter because they determine a team’s path to the Final Four. Some seeds generate more upsets than others, such as 7-10 seeds and 5-12 seeds. Historically, 6-11 seeds go the longest before facing a 1 or 2 seed. Teams with an 8 seed face a tough Round-of-64 opponent and have to face a 1 seed next (sorry Badgers).

However, there are plenty of upsets. The Final Four has been composed of all 1 seeds only once. See BracketOdds at Illinois for more information on how the seeds have fared.

Having said that, the committee doesn’t always get it right. There are Some teams like SMU, Wichita State, and Xavier are underseeded and are poised to upset. Also, Villanova is the overall #1 seed and has a 15% chance of winning the entire tournament, which is low, meaning that there isn’t a strong favorite this year.

3 Don’t pick Kansas to win it all

Be strategic. The point is NOT to maximize your points, it’s to get more points than your opponents. I’ve been getting in the habit of picking my Final Four first and filling in the rest later.  You can pick the eventual winner (say, Villanova) and still lose your pool if everyone else picks Villanova. FiveThirtyEight estimates that Villanova has a 15% chance of winning the tournament, meaning that another team is probably going to win.

One way to be strategic is to pick an undervalued top team to win the tournament. For example, last year Kansas was selected as the overall winner in 27% of brackets on ESPN and in 62% of Final Fours) despite having an overall 19% chance of winning (538). On the other hand, UNC was selected as the overall winner in 8% of brackets (with a 15% win probability). Getting UNC right last year helped vault past those who picked Kansas.

4 It’s random

The way brackets are scored means that randomness rules. It’s easy to forget that a good process does not guarantee the best outcome in any give year. A good process yields better outcomes on average but your mileage may vary any given year (at least that what I tell myself when I don’t win my pool!)

Small pools are better if you have a good process. The more people in a pool, the higher chance that someone will accidentally make a good bracket with a bad process. It’s like stormtroopers shooting at a target. They’re terrible, but if they take enough shots they’ll hit the target once.


For more reading:

OR history and traditions

The INFORMS History & Traditions Committee, added six new videos representing interviews of luminaries in the field of Operations Research and Management Science.  All of these interviews were conducted at the recent INFORMS conference in Nashville, TN, in November. Thank you to the committee and Irv Lustig for this great resource. The committee website has more resources.

Algorithms to live by

Algorithms to Live By: the Computer Science of Human Decisions by Brian Christian and Tom Griffiths is a nice book about how to live an efficient life. The authors interviewed me and included some of our conversations in the book. Below I include a snippet from the book summarizing how I once used the Secretary Problem to accept and reject bids when selling my house. It was ultimately successful and also pretty stressful. I blogged about it here. Other excerpts are about my research in emergency response and how I use the critical path method to get my three daughters to school on time every morning.



Algorithms to live by on Punk Rock OR: