Sunday, December 7, 2008

Random Walker Rankings through December 6th

We remain grateful to both Kenneth Massey for maintaining his comparisons page and to Peter R. Wolfe for making game results available online.

Below we list rankings according to the Random Walker (RW) system of first-place votes and the Random Walker First-Last (RWFL) system at our selected p=0.75. We use the random walkers to rank all 120 Football Bowl Subdivision teams, plus the extra made-up "team" representing their collective Football Championship Subdivision opponents. The "votes" in the Random Walker (RW) system are all essentially first-place votes, teams with more votes ranked more highly. The Random Walker with First & Last (RWFL) system is a combination of "first-place" and "last-place" votes. We have determined that our bias parameter p=0.75 has nice properties for RWFL (near the minimum crossing probability when compared against round-robin standings), so we use p=0.75 for both RW and RWFL and are happy that this is coincidentally the same primary value we have used in the past for the RW rankings.

The breakdown of the top teams across different p values for both RW and RWFL can be seen in the figures below. Additionally, we plot the fraction of ranking violations---that is, the fraction of game outcomes that have been contrary to the rankings---for each system across the p values. Ranking violations like these are also included on Kenneth Massey's site, and can be used as one measure of the efficacy of a rank ordering; indeed, one could instead choose p to minimize such errors, but we are more comfortable at the selected fixed values for ease of explanation.

[See http://rankings.amath.unc.edu/old/2008.htm for these pre-bowl rankings. Post-bowl rankings are also available.]