charlie burrus's college football rankings
I started playing with computer football rankings quite a few years ago. Here's what I learned: you can make them say almost anything. I started doing it thinking that I was going to find the ultimate correct way to rank the teams, but what I found is that there is almost an infinite number of fair ways to grade the teams, and the way that someone ends up judging which is best is just by looking at it and trying to see if it looks reasonable. So here is the result of what I think is a pretty reasonable algorithm. The rankings look only at the performance of a team; they do not look at potential. Strength of schedule, margin of victory and the date of each game are all taken into account, although there's a cap on the margin to prevent blowouts from skewing the data too much.
I started with a ranking system developed by three mathematicians (Thomas Callaghan, Mason Porter and Peter Mucha). You can read a very entertaining article about their rankings at
How Well Can Monkeys Rank Football Teams?,
but here is the basic idea:
Imagine a whole bunch of voters (sort of like what we have in the AP and USA Today Polls). Count up the total number of games that have been played so far this year, and deal out the voters evenly to each game.
For each game, 75% of the voters go with the winners and the remaining 25% go with the loser.
The voters are now aligned with particular teams. Deal the voters out evenly to the games that team has played. So for each game, some voters are coming from one team, and some are coming from another. The teams are probably not sending the same number of voters, but that's OK.
Go back to step 2, and loop through this process LOTS of times. The number of voters per game that each team has eventually stabilizes. For all you linear algebraists, this is essentially an eigenvalue problem where you find the eigenvector for a big ol' whoppin' matrix that corresponds to an eigenvalue of 1.
Rank the teams based on the number of voters per game. (If you do it per game, it keeps teams from being rewarded for simply playing more games.)
So I tweaked the algorithm a bit, making the following changes:
I look at more teams. Callaghan, Porter and Mucha looked only at FBS teams, and I look at all Division I teams (since there really is a pretty big variance in quality).
Instead of always having 75% of the voters go with the winner I have between 60% and 90% of the voters go with the winner, based on margin of victory and whether the game was a home or away game.
I changed the way that the voters are dealt out to the games: instead of dealing them out evenly to all games played by a team, slightly more voters go to more recent games and slightly fewer to less recent games.
I liked the algorithm so much that I started ranking basketball teams in much the same way.
I am lucky and grateful that Peter R. Wolfe of UCLA compiles all the scores each week and posts both
basketball scores on his website.
I would have a very, very difficult time producing these rankings without access to that data.
By the way, I don't think that the ranking paradox means that there should be a college football playoff. It's pretty much common knowledge that anything can happen in a game, meaning that the
better team doesn't always win, so why would the best team always win a playoff? That's one of the great things about sports and games: there's always an element of chance, and the question of