Don’t forget to join our ESPN bracket challenge group before Thursday, March 21st!

Use the widget below to explore Tim Chartier’s lectures on March Mathness and to find more advice on how to fill out your brackets this year.

## How to Improve Your Bracket in 2013

Ralph Abbey was a member of the PUP March Mathness ESPN group and completed his bracket in the 91.8 percentile which is a fantastic achievement. However, we’re already looking forward to 2013, so in this post, he shares a few tips for improving your bracket next tournament.

I am a PhD graduate student in mathematics at North Carolina State University. My adviser is Dr. Carl Meyer, coauthor of Who’s #1? While sports ranking isn’t my PhD topic, I do find it very interesting, and it is actually quite a good topic of conversation, even among non-math people.

It was less than 24 hours before the first games and I still hadn’t made a bracket—-to be honest, I had completely forgotten. Somehow the thought came to mind at the last minute, and I realized that I didn’t have enough time to research all the teams in depth to create my own bracket. To put off the stress (and the blame if I got things wrong) I turned to a few ranking algorithms I knew for help. The games data was found on Massey’s website: masseyratings.com

I formed a matrix in which entry i,j was the total number of points team i scored against team j over the entire season. Division 2 teams were included as well. Using this data matrix, I used both the offense defense model, and a pagerank model to rank the teams. I made 2 brackets, compiled by having the higher ranked team always beat the lower ranked team.

Additionally I formed a few other brackets: a “no-upsets” bracket that used the NCAA committee’s rankings. I also created 2 “upset” brackets. The first was a bracket in which if two teams seeded between 4 and 13 (inclusive) played, the worst ranked always won. If one or neither of the teams were in the range, then the better ranked team won. The other upset bracket was formed the same way, except by decreasing the range from 6 to 11.

In the end the winner for me was the offense defense model. It scored 1310 points on the ESPN challenge, placing above 91.8% of all ESPN brackets–absolute rank of 529,254. While it messed up on a lot of the first round upsets, the offense defense model was able to predict 3 of the 4 final four teams, 1 of the 2 championship teams, and it did predict Kentucky to win the whole tournament.

The plans for next year are uncertain, but one thing I do want to try is rank aggregation to see if it can combine the best of multiple models!

## Using Ranking Schemes to Fill in Brackets

 James Keener, Professor of Mathematics and the University of Utah, explains his ranking method.

## More from our leaderboard, students describe their March Mathness brackets

Out of 6.5 million entries, the participants in the March Mathness group of the ESPN Tournament Challenge are doing very well. One third of our group is in the top 20%. Following are summaries from some of those in our group. They describe how they designed their brackets and how they are embracing the excitement of the tournament. The methods mentioned are described in the recently published Who’s #1? By Amy Langville and Carl Meyer.

## Bryan Kelley

Bryan Kelley is a sophomore Math major at Davidson College. He is from Rockland, Massachusetts. In the  following, he describes how he selected his bracket using the PageRank method from Who’s #1? by Amy Langville and Carl Meyer. You can find more on PageRank in Google’s PageRank and Beyond, also by Langville and Meyer.

The experience of filling out a March Madness bracket is completely new to me this year. I have always resisted it in the past because as a die hard competitor and a passionate sports fan, I cannot enjoy the tournament if I have picked a team that I think will win over a team that I want to win. It is a conundrum I’m sure many other Americans also find themselves in this time of year.

This year, however, I put my internal conflict aside, and in the name of math filled out my first March Madness bracket. To create the bracket, Tim Chartier and I referenced Amy Langville and Carl Meyer’s new book, Who’s #1?. Using that book, I coded an algorithm in MatLab that solves for the stationary eigenvector of a stochastic matrix and  used that vector to rank the teams. For the matrix entries, Tim and I decided to use the point differential in teams’ wins/losses.

Considering many of the experts had Kentucky winning this year (which is not surprising considering the season the Wildcats have had) Tim and I did not expect to see the algorithm give us Michigan State as this year’s champion. However, that is what it gave us, and to avoid my internal conflict, that is how I filled out the bracket. In fact, I filled out every spot in the bracket precisely how the algorithm dictated me to fill out the bracket.

This turned out to be a good thing because in my heart  I thought Missouri was going to win the tournament this year, but instead, as we saw on the first weekend, they were taken down by the formidable 15 seed Norfolk St. In addition, as a college student, I don’t have time to follow every Division 1 Men’s basketball team in the country so there were also many picks that I had no idea about. Thus, it was nice to have an algorithm to tell me what to do. By the end of the first night of the tournament, I was in the 98.6 percentile, only missing the VCU upset, and by the end of the first round, I fell a little to the 90.1 percentile. The first time I logged on and saw that I was in the top 98.6% of all brackets was a nice jolt. I probably checked my ranking ten more times that night to make sure there was not a mistake.

What has been most exciting to me this during this experience was finally finding a way to combine my academic passions and my love for sports. I think using math to predict a typically unpredictable tournament is about as cool as it gets because it is exciting to math people and non-math people alike. Even at Davidson alone, it has created a lot of conversation between my math friends and my non-math friends because everyone wants to be able to predict who is going to win. The one downfall of this experience is that I have had the pleasure of watching my Davidson Wildcats all season long and was so excited to see them make it back to the tournament this year that it killed me to pick Louisville over them…and then have it come true.

The first night of March Madness 2012 was pretty standard, and thus my bracket did pretty well. And then came the second day. With two 15 over 2 seed upsets this tournament, my bracket was brought back down to life a little bit. However, I suspect most of the rest of the country also did not see that happening so I expect to move on from those losses and perform well in the Sweet 16. As long as Michigan St. keeps winning, I’ll be happy.

I should also mention that in addition to competing in the PUP group, I have several other brackets  in a different group that is part of ongoing research. In that group, I developed several variations of the PageRank algorithm, and many of those are also doing well. Tim and I are anxious to see how those results turn out in addition to the standard PageRank algorithm.

## David Heilbron

David Heilbron is a junior Math major at Davidson College .  In selecting his bracket he has used the probability of given seeds to advance and then picked a random number to decide if they do. He is doing very well. Here he describes how he filled in his bracket.

So here are the nuts and bolts of my bracket. Basically I went through each bracket from 1985-2011 and saw how many of each seed number made it to each specific round (Sweet Sixteen, Elite Eight, etc). So say over that time period, eight #5 seeds have made it to the Elite Eight. I took this number and then divided it by the total possible #5 seeds that could have been in the Elite Eight over all those years. There were 4 each year,  times 27 years, or 108. I did this for each seed and each round to get all these probabilities of a certain seed making it to each round.

From there, it is a simulation that runs each head-to-head match-up moving through the games. To find the winner of a certain game, we take each seed number’s probability and scale it to make a standard probability that the teams can be judged from. Finally, we get a random number which, if less than or equal to the probability, one team goes on, and if not then the other team goes on.

Long story short, we find each seeds chance of getting to a certain round and then use those in the head to head match-ups to print out the winners of each game and then just put them into a bracket.

In terms of which is more exciting, I really do love the basketball. I grew up watching a lot of ACC basketball so watching the games is so much fun. I actually have been finding the math more interesting though. It is cool to see how everyone thinks up different weightings and strategies.

The Murray State loss may come back to haunt my bracket as the tournament continues and Syracuse really has not looked that good, but I have a good feeling going forward. One game that concerns me though is Louisville vs. Michigan State. I’d love to see Louisville win and keep my bracket going but I do have to be a homer for Davidson and keep licking my wounds after the first round loss.

## How are we doing? Checking in with our March Mathness teams

Out of 6.5 million entries, the participants in the March Mathness group of the ESPN Tournament Challenge are doing very well. One third of our group is in the top 20%. Following are summaries from some of those in our group. They describe how they designed their brackets and how they are embracing the excitement of the tournament. The methods mentioned are described in the recently published Who’s #1? By Amy Langville and Carl Meyer.

More reports from our student teams: http://blog.press.princeton.edu/2012/03/23/more-from-our-leaderboard-students-describe-their-march-mathness-brackets/

## Calley Anderson

Calley Anderson is a sophomore English Major with a Film and Media Studies Concentration at Davidson College. She is from Memphis, Tennessee. She is in the 86.1 percentile after the round of 32.

To me, it’s actually pretty shocking that I’m doing so well. I’ve done brackets several times before, but I guess the application of linear algebra gave me an extra kick. That, and the fact that this time around, it was for a class, my decisions were based on the mathematical rankings more so than my personal and emotional thoughts of teams. I used the Colley method (given to our class by Dr. Chartier) and separated the season into 4 parts. If my memory serves me correctly, I weighted Part I as 1/4, Part II as 1/2, Part III as 1, and Part IV as 2. From there, after I put all the teams in the brackets by their mathematical ranking, I used a small amount of personal intuition and changed a few (most notably having Memphis beat St. Louis because it’s my hometown team).

I never thought that my bracket would actually get this far, especially after all of the upsets that occurred in the Round of 32. After taking 2 major hits due to these upsets, I thought that my bracket had reached the end. Being a sports fan in general, I wanted my bracket to have real potential this time around. Most of my previous brackets had Memphis returning to the Championship or going rather far regardless of their season. Everyone else seemed to just fall into a random place, with exceptions for teams that I liked that year. This year, I didn’t let my school or home team influence my decision as much.

Math, however, is far from my favorite. We never really seem to get along. This bracket would be the first case in which I have applauded any type of math as being useful. That’s one of the great things about Dr. Chartier; he takes regular, terrible math and makes its useful and interesting. For this brief moment, I get to be proud that something involving math did me some good. More importantly, this is math that I actually cared about and strived for success with. Math, sometimes, can be awful. But other times, with the right application, it can be fun!

All in all, this has been one of the most memorable experiences in terms of March Madness that I’ve ever had. The intensity that I felt with each game, rather than just a select few, was new but exciting for me. I even went to the lengths to install the Bracket Bound iPhone app so that I wouldn’t miss any game or change in my bracket standings! I feel rather optimistic that I can hold onto my top spot in our class. If I can make it through a round of the most unpredictable upsets, then I can make it to the finish. Even if I don’t, I can still be proud of my short reign of success. I’ve got math on my side and, sometimes, it’s pretty hard to beat that.

## Jonah Galeota-Sprung

Jonah Galeota-Sprung is a junior Math Major at Davidson College. He is from South Orange, New Jersey, and he enjoys birdwatching and pickle making. He is in the 79.0 percentile after the round of 32.

Which method did you chose and why?

I ended up using the Colley ranking method with a cotangent weighting function. The choice of Colley was pretty arbitrary, but I chose the cotangent weighting function because I figured I needed a pretty bizarre bracket if I was to have any chance of doing well, given the unpredictable nature of the tournament. We’ll see how far that idea gets me.

Who do you predict will be in the final 4?

Mr. Colley and the cotangent function predict Kentucky, Florida State, Michigan State, and UNC in the Final Four, but they don’t speak for me personally–I’m seeing a Davidson/’Zags final clearly written in my tea leaves.

Things looked good for about half a day. I was on top of the pool, beating the president and my math professors and about 99 percent of the country, too. Dreams of cash prizes and maybe the Fields Medal for cotangentially managing to predict the VCU and Colorado upsets filled my head. I could practically taste the gold on my tongue (the first thing I do when I receive medals is lick them, just to be sure).

Before long though, it all fell apart. I’ve been told things have a tendency to do that. Pesky NC State kept winning, and peskier Missouri had been knocked out in the first round. The un-predictions piled up, and over the course of a weekend half of my Elite Eight was out of the tournament and my national champion had lost to a six seed.  I was able to take some consolation in the fact that Duke was among the casualties, but that did little to assuage the pain I felt when I looked at my bracket shot through with red holes.

There’s always next year.

## Barbara Sitton

Barbara Sitton is a junior at Davidson College. She plays on the Davidson Division I women’s basketball team and is a huge basketball fan. She is in the 86.1 percentile after the round of 32.

I’ve only had a little experience with brackets. Before, I chose teams from instinct, it was just for fun. But this time, I used the Colley ranking system to rank teams and predict the outcomes of the games. For the men’s tournament, I predicted Kentucky, Michigan State, UNC, and Ohio State to be in the final four. For the women’s tournament, I predicted Baylor, Stanford, Maryland, and UConn to be in the final four.

I am truly impressed with the way my bracket has held up, although there has been a lot of madness in the NCAA tournament already! I actually have 2 brackets in the group, which I will distinguish as bracket #1 and bracket #2. Bracket #1 has been the most successful so far, and it is in the 86.1 percentile. Some of the biggest upsets have been all of the 12-15 seeded teams who have beaten the 2-5 seeded teams: VCU beating Wichita State, Lehigh beating Duke, USR beating Temple, and Ohio beating Michigan. I’m pretty excited to see what will happen during the Sweet 16!

## Paul Britton

Paul Britton is a senior Math Major and Philosophy Minor at Davidson College. He is a campus tour guide and a member of the Sigma Phi Epsilon fraternity. He is from Castle Rock, Colorado and is ranked in the 86.1 percentile after the round of 32.

My family always sponsored a bracket pool and I started participating when I was 8 or 9 years old. I have done at least one bracket, and often multiple brackets, every year since then.

I submitted 2 mathematically based brackets into the PUP pool this year. The first bracket used the Massey rating method with a piecewise game weighting where I divided the season into thirds and weighted the last third of games at .75, the first third at .5, and the middle third at .25. The first third corresponds to the non-conference schedule, which is an indicator of a team’s talent compared to the rest of the country, and is also indicative of games they will face in the tournament itself. The most recent third is the second half of the conference game slate, which is a strong indicator of a team’s recent performance, and thus was weighted more than the other two segments

The second method was an (imperfect) attempt to weight the teams based on their performance according to Dean Oliver’s “4 Factors of Winning,” which you can read about here: http://www.basketball-reference.com/about/factors.html. Essentially, I gathered statistics on Field Goal percentage and Turnover percentage on offense and defense, and additionally on Rebound rate and Free Throw percentage, then weighted these factors according to Oliver’s specifications. I would have liked to adjust for strength of schedule, but couldn’t figure out an effective way to do so, so I left the initial rankings as they were.

## Who’s #1 on the March Mathness Leaderboard?

Greg Newman is currently at the top of the March Mathness ESPN Tournament Challenge leaderboard with two separate brackets and is ranked 131,052 out of 6.5 million brackets. That means that only 2% are doing better than he is. Greg is a senior Political Science major at Davidson College who spends a lot of time working on Computer Science/Mathematics. Last summer he interned at ESPN and he continues to work for them while in school. We asked him to describe how he picked his brackets this year.

## Greg Newman on his “picks” for March Mathness

I think I would fall into a category of liking both math and sports but, since I have been obsessed with sports since I was three and work at ESPN, I think I’m more of a sports fan. I have two brackets that are doing very well. I joined your group with the second one and think I will explain them separately.

### My “Picks as an analyst” Bracket:

As the name suggests I made picks as an analyst with little math. I did use “simple math” (by simple I mean something that I could explain to someone who never took Calculus but has basic knowledge of probability theory).

The major advantage to this method is the ability to look at match-ups. I would compare the traditional basketball statistics (points, rebounds, strength of schedule) and would do this for each match-up. I also used my own head when picking (this was both an advantage and a disadvantage). Having seen many of these teams play I had an idea of teams that would do well or would not. However, I had not seen all these teams play and did not have full data information.

This was the bracket that took the most time for me to complete since I was using math and my own opinion while making it. For an example of how watching a team can deceive you, I would talk about Missouri. I had seen them play a few times and they always looked very good. In this bracket I had them getting into the elite eight. As most will know, they lost in the first round to Norfolk State (in a huge upset). I had not watched any Norfolk State basketball and did not know how balanced their offense was (in the game against Missouri they had four players score in the double figures).

### My “Harvard” Bracket:

At this point I feel like I should note that I do not attend Harvard nor does it have anything to do with Harvard doing well in the tournament (it had them losing in the first round, which they did).

This was very mathy. I had looked into many different methods including Colley, Massey, LRMC, Pythagorean ratings, Power rankings, S-Curve rankings, ELO and a bunch of other “saber metric” like ratings and rankings. The reason I called it “Harvard” was that it is based off of the Harvard College Sports Analysis Collective blog. Specifically, I looked at their “Survival bracket.” I really liked how they used numbers/analytics to try to make “intangibles” tangible. An example would be tournament experience, which experts agree is important. You can look at past tournament minutes play and say it correlates (and I would say correlates pretty well) with experience. It also had a ranking for consistency, which is hard to measure but incredibly important.

The idea of survival is also very important and is crucial to the success of this bracket. Even with the crazy upsets (that this bracket did not get) all of the elite eight teams are still playing. This means that my PPR (Points Possibly Remaining) is very high at 1280 and means that this bracket could possibly end up doing even better (since many people have lower PPRs) at the end of the tournament.

### Tips in selecting a bracket:

Pay less attention to seeds/history.

A committee chooses seeds and we can’t always figure out why a team is given the seed. The art of “bracketology” is something entirely different and hard to understand. At some point a number 1 seed will lose to a 16 seed (even though it hasn’t happened yet). Coming into the tournament 2 seeds were 104-4 against 15 seeds (96%), this year they are 106-6 (95%) but does that mean number 15 seeds have a better shot next year? Also, even if you magically figure out what seeds will advance there are four of each seeds!

Don’t try to be perfect.

A perfect bracket is hard! Of everyone in the ESPN bracket, everybody had at least two wrong after the first round. The current leader had six incorrect picks. Don’t worry about being perfect, chances are nobody else will be either.

Rankings are good but Ratings are better.  Some people just looked at rankings. Many of the teams were rated very closely, so a team ranked five spots ahead of another may not be a sure win.

### What I wish I did:

Game Theory

I really focused on getting the most right, not highest percentile. I wish I had tried to pick the upsets (almost) nobody else picked. This would make me seem very smart, help me in brackets with different formats and possibly given me an advantage. If 95% of the country has team A beating team B but my model was team A only has a 60% chance, it would probably make sense to pick team B (note: I probably would not pick team B to advance further as to minimize the penalty and because neither team seems as good as everyone else thinks).

Tried another style of tournament

I like that each round is worth the same number of points (so each game as you get further in the rounds is worth more points). I wonder how well different models would work in different bracket scoring systems.

Saved/Organized all of my data better

I got a lot of data and all of it around the same time. I’m not sure that I will even be able to find all of it after the tournament is over.

I made ten brackets in total. I would have made so many more with each method individually and different combinations of all of them but scoring them myself was not something I had time for.

## Math Improves March Madness Predictions: Tim Chartier Interviewed on Inside Science

Inside Science Television spoke with Tim Chartier about how math can be used to predict the winners of March Mathness. They also provide additional resources for those who wish to learn more or teach this in their classrooms.

Tim notes in the interview that: “To do well in bracketology, you need to know how teams will do earlier. It’s often those teams that are very difficult to predict and those games that often our methods are picking up.”

He also reveals that, using data available at the time of the interview, his methods currently predict Kentucky will be the winner. Does that match up with your bracket?

You can view the complete article and the video here: http://www.insidescience.org/television/1-2563

## Tim Chartier’s rating method explained

In this Huffington Post article, Tim Chartier explains the method he uses for filling in his bracket.

You still have until noon today to compete with us at PUP! Don’t forget to join our ESPN group.

From Tim’s article:

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.

## Still need bracket advice? Tim Chartier offers some insight in a MAA Distinguished Lecture

The Math Association of America writes up Tim’s Chartier’s distinguished lecture on bracketology.

http://maa.org/news/2012DL-Chartier.html

From their description it sounds like Tim provides absolutely essential background to fill out those brackets:

Improbable as it sounds, Tim Chartier can leverage math to foretell such outcomes, to “predict how 18- to 22-year-old young men will perform in high-pressure situations.” The basis of Chartier’s mathemagic is the Colley Method, named for its developer, Wesley Colley.

Math and mathematicians, in other words, work in mysterious ways.

Chartier dispelled the mystery a little, though, by describing how to derive from team statistics—number of wins, total games played—values that, slotted into a massive 350 by 350 matrix, represent a system of equations that can be solved to yield team ratings. To make a bracket based on these ratings, you just assume that the higher rated team always wins.

You can, of course, “usurp the math and make your own decisions,” Chartier said, but consider the Colley Method’s none-too-shabby track record: When Chartier entered a Colley-generated bracket in the 2009 ESPN Challenge, it beat 62 percent of the more than 4 million entrants. In 2010 Colley outperformed everyone from LeBron James and Davidson alum Stephen Curry to Dick Vitale and Barack Obama.

Sheldon Howard Jacobson, Professor of Computer Science at the University of Illinois at Urbana-Champaign has this helpful website to share: http://bracketodds.cs.illinois.edu/BI.html

Remember you only have until Thursday, March 15 to submit to our ESPN site and be sure to check out Who’s #1 for any last-minute tips.

## A Word on Ranking from Team Sport Analyst Anjela Govan

Anjela Govan, an expert on the Markov method of rating and ranking, weighs in on the “science” behind getting a jump on the competition.

The topic of ranking (or the question “Who’s #1?”)   is usually accompanied by the question “Who will win this game?” Granted, ranking does not apply to sports only, but sports is viewed as less academic than most applications. The question of “Who will win?” essentially asks us to somehow identify which team is stronger, better, of higher quality, or is higher ranked. We need to recall that even though we often interchange words “ranking” and “rating,” they do have different meaning. Rating may somehow summarize the quality of a team based on some associated criteria. Ranking is simply an indication of the place in the list that reflects relative importance of teams to each other. The difficulty of this topic is to determine what constitutes quality and importance as far as this particular set of teams is concerned.  Ideally, to determine rating we would need to know the characteristics of the perfect team and measure all the teams against it, thus arriving at an absolute measure. Given that we know how far any given team is from the ideal, we can compute the absolute ratings. However, very few (if any) real world applications allow this ideal method. What we are able to do is measure relative difference in quality between teams, thus arriving at a rating based on these relative measurements. Now we are waxing  philosophical!

Back to game predictions:  This question has two aspects to it, first being “which team in a given competition will  win?” and the second being “by how much?”  The first is easier to answer.   Suppose we pick a method that produces rating scores, a favorite one from the great collection introduced by Dr. Langville and Dr. Meyer.  For a game between teams A and B to determine a winner we simply compare each team’s rating scores, rA and rB , the better rated team wins! Now for the by how much, referred to in chapter 9 of Who’s # 1 as point spread: there are many ways to estimate point spread. One of the simplest approaches is to think of the point spread being proportional to the difference between the ratings of the teams. That is, point spread for a game between team A and team B = α|rA rB |, where α is some constant. In this simplistic point spread approach, the work is concentrated in estimating an appropriate constant α. This constant could be the same for all the games, and could be determined  using the previous  season, that is the point spreads from the previous  season are known and least squares could be a way to approximate α. Another way is to customize α to the pairs of teams. Maybe there is a trend between teams that could be observed across a number of seasons. The described method is simplistic, perhaps it is evident why.  For a more in depth discussion do consider the well laid out chapter on point spreads in Who’s #1?

## Using Darwin to Rank: Bracket Advice from Kathryn Pedings-Behling

Kathryn Pedings-Behling is is a high school math teacher by day and a graduate student by night. She teaches math at the Charleston Charter School for Math and Science where she heads the math department and the AP math program. She graduated from the College of Charleston in 2008 with a B.S. in Mathematics where she met Dr. Amy Langville who became her master’s thesis adviser. Kathryn will graduate with an M.S. in Mathematics from the College of Charleston in May. In her post below, she explains how Darwin comes into play when ranking sports teams.

It sounds crazy, doesn’t it? What in the world does Darwin have to do with ranking sports teams? This was the same question I asked myself as I entered an independent study course with Dr. Amy Langville on the topic of evolutionary optimization. Okay, so now you are possibly seeing the connection: Darwin and evolution. But you are still stuck on this ranking thing. How is it related?

The process of evolutionary optimization takes Darwinian ideas of mating, mutating, fitness, and survival of the fittest and puts them in the context of ranking. I must admit that I am a total sucker for topics like this. The hardest part of making math your life is trying to have conversations with non-math folks about what it is that you are passionate about in this field. However, this was something that I knew “normal” people could wrap their head around. Any person who has been through an acceptable level of education knows the basics about evolution. The members of a population continue to mate and have offspring. Every now and then a mutation is thrown in. Only the fittest population members survive to then create more offspring. If you know these things, you can understand my ranking method! It’s that simple!

Before getting to the meat of the ranking method, there are a few important fundamental concepts. The members of your population are each a different possible ranking of your teams that is expressed mathematically as a permutation vector. You can mate these rankings by performing any of the rank aggregation methods found in the book Who’s #1?, such as average rank or Borda count. Mutating uses much simpler, random actions such as switching two teams or swapping the order of a group of teams, and is much more cost effective to perform on a large set of teams.

How do you know if your mating and mutating has produced a good offspring? This leads to a much deeper question: how do you define what it means to have a quality ranking? Well, if I knew the exact to answer that question, I would win the ESPN March Madness Bracketology Competition every single year, significantly boosting my popularity as a high school math teacher. However, we believe that we have a way to measure the relative quality of a ranking.

To understand this measure (which in Darwinian terms is the fitness of our population members), you need a new linear algebra definition called hillside form of a matrix. Hillside form is very easy to understand: the entries of each row are increasing from left to right, the entries of each column are decreasing from top to bottom, and the lower triangular of the matrix is filled with zeros (this last part of the definition is optional based on what type of data matrix you start with). Below you will find an example of a matrix in hillside form:

Hillside form makes sense in the context of ranking sports teams because, in a perfect scenario, you want the first place team to beat the second place team by a little, the third place team by a little more, etc. Unfortunately, the reality is that no season is perfect so all we can do is find the ranking that symmetrically reorders the columns and rows of a data matrix to be as close to hillside form as possible. This is the motivating factor behind our ranking method.

Now you are ready for the steps of the evolutionary optimization ranking method:

1. Collect the data matrix you will use for your ranking – I tended to use game score data to make my head-to-head matrix, but that doesn’t have to be the case.
2. Gather your original population – This can be done randomly or using the rankings of other methods you can learn about in Who’s #1?.
3. Using each of your original population members, symmetrically reorder your data matrix and count the number of violations you have to hillside form. The lower the number, the better the ranking!
4. Now it’s time to make an offspring! Either mate or mutate members of your population.
5. Repeat step three for that offspring.
6. If your offspring has a lower number of violations to hillside form than the weakest member of your population, congratulations – it made the cut! Drop the weakest member from your population. However, if the offspring did not make the cut, it’s history.
7. Repeat steps 4 – 6 until you have a satisfactory solution – we chose to continue gathering offspring until the change in our fitness values got very small. Just realize when you are setting your stopping criteria that the longer you allow your algorithm to run, the better your solution will be since this method is constantly finding a solution that is getting closer and closer to hillside form.

There you have it! It’s really as easy as these seven steps. The thing I really love about this process is that it is so bare bones that you can pick any of those steps and make it as sophisticated as you like! Just remember two things before you start doing that: 1. Evolutionary optimization requires a certain amount of randomization in order to find “good” solutions. 2. The more sophisticated the method; the longer it will take to run.

Before ending this post, I need to share a deep, dark secret with you. I’m not a fan of sports. It’s true. Now, don’t get me wrong, I enjoy attending a live sporting event as much as the next person, but I actively dislike watching sports on television. I just can’t seem to get interested. However, what I am a fan of is ranking methods. So with that being said, don’t ask me if the bracket you’ve created is good because I have no idea, just be ready to talk to me about the math behind your ranking. That is something I can follow any day of the week.