Last time I "introduced" the idea of using Condorcet's election method to order teams by how many computer rating systems rank a team higher than (some number of) other teams. I'd forgotten that this is exactly the basis of the Dolphin Majority meta-ranking, and I should have referenced that. The raw data that produces it can be presented in the (large, for football) Pairwise Matrix where each team is compared to all of the others.
In the matrix, the number in row i and column j is the number of rankings that have team i ranked higher than team j. The number associated with each team is not exactly the Dolphin/Condorcet ranking - instead it is the Bucklin order I defined in the referenced article. When the Bucklin Rank and disagrees with a pairwise Condorcet comparison I have indicated that by bold numbers in the j, i and i, j positions.
I have made what turns out to be a significant change since the last article. In the previous version I assigned the team's "majority rank" as the best rank for which at least half the ratings agreed when there was an even number of ratings. For several reasons, a strict majority (50 percent plus one) works better for such a small number of "voters."As an aside, a set of teams such that all members pairwise defeat all teams outside of the set is called a "Smith Set" in social choice (election methods) terminology. When there's one team with that characteristic, it is called the Condorcet winner, and in this sample Texas pairwise-beats every other team.
I have reproduced the input copied from Kenneth Massey's source to preserve the integrity of my point-in-time copy of his page and to provide a visual clue as to how the ratings correlate to the Bucklin majority consensus. The input shows the rankings by a rating system that agree with the consensus in color and the ratings are oriented left to right by the number of teams for which the rating agrees with the majority. Ties are broken by the sum of the squares of the ranking difference from the consensus for all teams.
You may notice that I do not include the human polls in this analysis. There are a couple of reasons for this, not least of which is that they do not rank the entire field. In addition, as I have written before the method typically used to conduct the polls can be unduly influenced by "strategic voting" on the part of the poll-ees. If they used a Condorcet or Bucklin majority -method instead of the 25 points for first, 24 for second, etc., or used (number of teams minus 1) for first, (...minus 2) for second, etc. then it would be trivial to incorporate them into a correlation such as these.
Next time I'll describe how we can use Massey's comparison data along with David Wilson's rating descriptions to derive a sound meta-ranking suitable for BCS purposes. Stay tuned!