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# Better Know a Computer: Colley Matrix

As the beauty pageant in the human polls begins, Texas stands to gain much, but the computers are proving a wet blanket for our title hopes.  Let's take a look at how another Computer ranking system works.

A couple weeks ago I brought you a breakdown of the Billingsley Report the only computer with Texas in the top 10.  Today I want to talk about the Colley Matrix, because it's incredibly simple.  So simple that Wes Colley, who has a PhD in Math from Princeton, can explain it in 23 math filled pages.  The size of the explanation, however, belies its simplicity.  The Colley method factors in two factors:  winning percentage and strength of schedule.  Colley calculates the latter with what he calls the "iterative method."

Here's how it works

First the formula

Don't worry if you don't know what that Greek letter Sigma means

Colley takes every IA team and first ranks them solely by winning percentage.  The winning percentage is calculated in the first half of the equation, so in the first "rankings" the second half of the equation resolves itself to 1/2, which is the average rating of every team.

In what's called the first iteration, Colley plugs in the rankings of all of each team's IA opponents (Colley doesn't consider IAA teams in his rankings).  The r in the second half of the equation is where the quantified ranking is plugged in and they are all added up (that's what the sigma means).  With this done, the teams are rearranged and the process begins again.

This is repeated until changes in the ratings are negligible.  Thence are the final ratings (Colley took 1 page for this explanation then spent the other 22 talking about why it works and how he figured it out).

There are some obvious weaknesses of this scheme.  When calculating SOS purely by wins and losses.  Hawaii, BYU, and Oregon all get the same initial value as they are all 6-2.  This evens out a little bit with the iterations, but it is still an undue bump for the opponents of BYU and Hawaii.

This week's Colley rankings

1.  Michigan
2.  Ohio State
3.  California
4.  Florida
5.  Notre Dame
6.  USC
7.  Rutgers
8.  Louisville
9.  Auburn
10. Tennessee
11. Boise State
12. West Virginia
13. Texas
14. Texas A&M
15. Missouri
16. Oklahoma
Complete rankings can be found on Colley's website.  Also on the site is a fun feature that let's you subtract and add games and see how it would change the rankings.

I ran some hypotheticals that have been tossed around the site at times.

First I asked what would happen if Texas had not played Ohio State

Without adding another opponent for Texas we jump from 13th to 8th, still behind Michigan, OSU, Cal, Florida, ND, USC, and Rutgers.
When I added in Alabama (a beatable team from a respectable conference), we moved up to 4th, behind only Michigan, OSU, and Cal.
When I added in Central Florida (an opponent from next year), we got to 10th, two spots lower than if we had played no other game at all

Second I asked what difference it would make if Oklahoma had beaten Oregon

The answer is that Oklahoma moves from 23rd to 14th and we move past WVA and Boise State to 11th.

If I can figure out any of the other computer rankings then more segments will be forthcoming.

--AR--