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## What is the Colley Rating?

Have you ever wondered how the Colley Ratings were calculated?  It's time to brush off those statistics books that are gathering dust on your shelf!

The Colley matrix rating is a rating method developed by Wes Colley and is one of the computer rankings used in determining who plays in the College Football BCS (Bowl Championship Series) games at the end of the season.  I want to thank Wes Colley for granting me use of his copyright formulas in order to make these calculations.  The Colley ranking is the only BCS computer ranking formula that has been published for the public to view.  More detailed information can be found at the Colley rankings site.  Detailed explanations on calculating the rankings can be found here The Colley Matrix Method, 2002 Edition (PDF).

The method for calculating the ratings is based on statistical principles that those without a college degree in math may struggle with.  For those without a background in linear algebra, integral calculus and probability analysis, I will try to give a more basic understanding.  Essentially, every team starts off with a .500 record, thus there is no bias for stronger conferences or historical tradition.  The model also makes minimal assumptions and uses no ad hoc adjustments, reducing the risk of creating a bias based on giving more weight to certain characteristics (points scored, margin of victory, home field advantage, etc.).

The motivation for the model is winning percentage, which is corrected for a team's strength of schedule.  Only games against opponents within the same classificaiton are included in the formula.

Every team starts out with a record of 1-1 and thus, a 0.500 rating.  To explain why this is, I'll use an example.  Let's take two teams that play their first game against each other.  If Team A wins, their 1-0 record would give a rating of 1.000.  Team B, with their 0-1 record gets a rating of 0.000.  If you were to compare the 2 teams after the game, Team A's rating is infinitely higher than Team B's.  Does that make sense?  Maybe, but let's look at what happens when you give everyone a 1-1 record (0.500)

After the first game, Team A's rating is like a record of 2-1 (2/3).  Team B's is like a record of 1-2 (1/3).  It's more realistic at this point to believe that one team deserves twice the rating of the other rather than saying that one team is infinitely better.

So, as a starting point before we make any adjustements for the strength of opponents, we have formula 1.  Using r = rating, w = wins, l = losses, and gp = games played, a team's rating begins as r = (1 + w) / (2 + gp).

Formula 2 shows us a way to calculate a Team A's wins in a way you may not have considered.  Any team's wins can be caluclated by (w - l) / 2 + (gp / 2).  As an example, if a team is 6-4, their wins (6) can be calculated by taking (6-4)/2 + 10/2 = 1+5 = 6.

The term (gp / 2) in formula 2 is the same as (gp * 0.5).  0.5 happens to be the average rating that everyone starts out with in this rating system.  Instead of using 0.5, what if you used your opponents' rating?  You would then create a calculation for wins that is adjusted for strength of schedule.  Instead of getting credit for 1 win by defeating a really good team, you can receive, say, 1.2 wins (or 0.8 wins for beating a weak team).  Losing to a good team may only cost you 0.8 of a loss (or 1.2 losses if you lose to a weak team).

To see this with a real example, let's use the example where two teams play their first game.  From formula 1,

r(winner) = (1 + 1) / (2 + 1) = 2/3 = 0.6667

r(loser) = (1 + 0) / (2 + 1) = 1/3 = 0.3333

The first adjustment to be made is to calculate the adjusted wins for each team.  From formula 2,

wins(winner) = (w - l) / 2 + r(loser) = (1 - 0) / 2 + 1/3 = 1/2 + 1/3 = 5/6

wins(loser) = (w - l) / 2 + r(winner) = (0 - 1) / 2 + 2/3 = -1/2 + 2/3 = 1/6

Thus, the winning team gets credit for only 5/6 of a win and the losing team gets credit for 1/6 of a win, due to the strength of schedule.  Placing these win calculations into formula 1 give us

r(winner) = (1 + 5/6) / (2 + 1) = 11/18 = 0.6111 (slightly lower than the initial 0.6667)

r(loser) = (1 + 1/6) / (2 + 1) = 7/18 = 0.3889 (slightly higher than the initial 0.3333)

We can again calculate the wins with these new ratings.

wins(winner) = (1 - 0) / 2 + 7/18 = 8/9

wins(loser) = (0 - 1) / 2 + 11/18 = 1/9

The new rating after these new win calculations give us

r(winner) = (1 + 8/9) / (2 + 1) = 17/27 = 0.6296

r(loser) = (0 + 1/9) / (2 + 1) = 10/27 = 0.3704

You will note that the ratings will continue to correct up then down then up again after each iteration of this process.  The number will eventually converge into a geometric series.  In this case, you will note that the losing team started with 1/2 then went down to 1/3 then went up to 7/18, then went down to 10/27.  This series can be expressed as 1/2 (1 - 1/3 + 1/9 - 1/27 + ...) and the part in parenthesis is equivalent to the geometric series 1/(1+1/3).  Thus, the true rating for the loser is 1/2 * 1/(1+1/3) = 1/2 * 3/4 = 3/8.  The winning team's rating becomes 5/8.  Note how the average of the two ratings turns out to be 0.5.  This will hold true when calculating ratings for all teams.

The last step is to make an equation for the rating for the teams.  Combining formulas 1 and 2 from above gives us

r(winner) = (1 + 1/2 + r(loser)) / (2 + 1) and r(loser) = (1 - 1/2 + r(winner)) / (2 + 1)

Rearranging these formulas gives

3 * r(winner) - r(loser) = 3/2

-r(winner) + 3 * r(loser) = 1/2

This is a simple two-variable linear system that is solved when r(winner) = 5/8 and r(loser) = 3/8 as calculated above.

When this system is expanded to all teams, a linear system is created with the same number of variables as their are equations. This can be solved using linear algebra matrices (refer to Colley's documentation for an explanation if you're curious on the details).  The result of the calculations is a rating for each team involved where each win or loss is evaluated based on the ratings and results of every team you play and your opponents' interactions with every team they have played.

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