lsumatt computer poll released

A few years ago I created my own computer poll and post it on the rant. I haven't gotten around to yet this year, but since the season is almost done. I am also able to re-create the Colley matrix poll and often like to look at a "future poll" if certain games played out. If I find time, I will post that later this week.

Here is the lsumatt poll

1 {'LSU';} 10-0 1.19
2 {'OSU';} 10-1 1.075
3 {'Alabama';} 9-1 1.072
4 {'Arkansas';} 9-1 1.031
5 {'Kansas St';} 8-2 0.995
6 {'Houston';} 10-0 0.989
7 {'Stanford';} 10-1 0.962
8 {'Oklahoma';} 8-2 0.961
9 {'Boise St';} 9-1 0.946
10 {'Va Tech';} 9-1 0.939


THE BASICS
The basic principle is that when two teams play, there is always a chance either team will win. The computer poll determines each teams ranking in such a way that the number of games they actually won is equal to the number of games they were expected to win based on their rating and the rating of the teams they played. Each team is rated on a scale from 1.0 (very good) to 0.0 (very bad). If two teams play each other that have the same ranking, there is a 50% probability each team will win. If the best team (r =1.0) plays the worst team (r=0.0), then there is a 99.9% chance that the better team will win. Moreover, when a really good team plays a bad team, it makes little difference if they are the 120th best team or the 90th best team. This is taken into account, as the probability a team with r=0.1 beats r=1.0 isn’t much better than a team with r=0.

THE UGLY MATH
The plot below shows the probability of a victory versus the difference in rating of team “i” and team “j”



The above curve can be described by a “sigmoid” equation:
(1)

Where ri and rj are the ratings of the two teams. Now the trick is to determine how many games a team was supposed to win. Suppose a team (LSU) had a rating r=0.9 and they played 3 games; Florida with r=0.95 (43% chance of winning), Auburn with r= 0.8 (64% chance of winning), and Miss State with r=0.45 (93% chance of winning), then they would be expected to have 2 wins (0.43+0.64+0.93=2) and be 2-1 overall. So in summary, the number of expected wins for team “i” is just the sum of all the game probabilities, which will always be less than the number of games they have played (N).

(2)

One catch is that no team is ever expected to be undefeated, because there is never a 100% chance of wining any one game. So even if you are a “perfect” team and played 10 games against awful teams, you would be expected to have 9.9 wins, not 10. To avoid this problem, I assume every team starts out 1-1; beating an imaginary terrible team (r=0) and losing to an imaginary “perfect” team (r=1.0).

In the above example with LSU we knew the teams rating and calculated their expected wins. But we actually know their wins and want to know their rating. So we can write Equation 2 for every team in college football (120) and substitute the number of games they actually won for “Wins”. If LSU were 2-1 (3-2 when you include that 1-1 start) in the above example, their equation would look like:



But we don’t know the rating of team i (LSU) or the ratings of the teams it played (UF, AU, or MSU). In fact there are 120 teams ratings we don’t know, but we do have 120 very complicated equations. Using them we can calculate the ratings of each team in college football and sort them from highest to lowest.

SOME DETAILS
A few things:
1. There is no home/away component yet. But that is easy, I think maybe I will just say that the home team is 10% more likely to win. So if there is a 50% chance of winning on a neutral field, its 55% at home.
2. I have debated how to handle 1AA schools. First, I ignored those games altogether but that wasn’t good (what if a 1A team lost?) Then I thought to just set the rating of all 1AA teams to 0. That wasn’t fair either as many 1AA teams are better than 1A teams. Finally I decided to rank all the 1AA teams independently (from 1.0 to 0.0) and then re-scale it by subtracting by 0.75. So the “best” 1AA team has a rating of 0.25 and the worst of -0.75.

3. The “6” used in the sigmoid equation is arbitrary; I chose it so that the probabilities match what I thought they should be for a #1 team playing a #10 team, etc. If I had time, I think it would be neat to get a bunch of historical data and choose a value that matches the curve.

4. It is possible for a team to be rated slightly higher than 1.0 or lower than 0.0. But that is no big deal.