Random Walker Rankings for NCAA Football
2005 Random Walker Rankings
Starting with 2005, we introduce a new spin for you. The original random walker rankings were designed primarily for ease of explanation: "But my team beat your team" on the scale of an arbitrarily large number of statistical "random walking" voters carrying out this argument ad infinitum. We will continue to call this the "Random Walkers" (RW) system.
In our investigations into the effects of the choice of the bias parameter p, we uncovered that a slight twist on the random walkers could lead to slightly better results. It's technical, and we can discuss it later in these pages (or you can look at our manuscripts); but the better idea is that one gets better behavior ranking round-robin tournaments if we do both the "But my team beat your team" in conjunction with a mirror "But my team lost to your team" argument (imagine trying to argue that you've identified the absolutely worst team in Division I-A football), which we will denote the "Random Walkers with First & Last" (RWFL) system. Kenneth Massey wasn't surprised when I told him this, summing it up perfectly with "That makes sense, since your original system is biased towards beating highly ranked teams; this will account for losing to lower ranked teams" (paraphrasing perhaps, I didn't record his exact words; but that's the essential gist of the idea).
The very nice properties of the RWFL system on round robins (again, we'll have to save the detailed discussion for another time) begin to break down at bias parameters p above a cutoff value that is near 0.75. As such, we will use p=0.75 as our preferred bias parameter weighting, since it gives us the largest direct effects due to wins and losses while still retaining the behavior of interest in the RWFL system. This rationale is philosophically similar to one used in the recent network-based ranking system developed by Juyong Park and Mark Newman.
Since we have a reason to use p=0.75 in the RWFL system, we will use the same p value for the RW system, with the added benefit that this has been the same primary p value we have been using in the RW system these past years. We continue to be grateful to Kenneth Massey for donating his time and energy to maintaining his comparisons page, where we contribute the RW and RWFL rankings at p=0.75. We are also grateful to Peter R. Wolfe for making game results available online, without which we would be lost.
Below we list the Top 25 according to each system. We also post rankings of all 119 Div-IA teams at p=0.75. 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.
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