An Objective Approach to Resume-based Ranking

© Copyright 2008, Paul Kislanko

There are two main approaches humans use when forming a top 25 ballot. Some use a "power ranking" based upon their opinion about which pairs of teams might win a hypothetical matchup. Others prefer a "resume-based" approach that uses only the results of games actually played as evidence for a particular ranking.

The computer algorithm analogs of these are the "predictive" and "retrodictive" rankings. The latter employ some form of results versus schedule strength to compare all teams against all other teams to derive a linear scale. The catch is that there are any number of valid SOS definitions, and it's impossible to tell in general which is appropriate for a human to use.

We can define an unambiguous objective resume-based ranking algorithm by borrowing a mathematical object from election-method theory. We begin by finding the set of all teams such that no team outside of the set has a win over any team that is a member. After eight weeks, there are 15 such teams, and we agree they should be ranked 1st through 15th.

The Undefeated teams, U = { Texas, Alabama, Oklahoma St, Ball State, Boise St, Penn State, Texas Tech, Tulsa, Utah }
plus teams who've only lost to teams in set U, V = { Oklahoma, Georgia, Missouri }
plus teams who've only lost to teams in UV, W = { TCU, Cincinnati }
plus teams who've only lost to teams in UVW, X = { BYU }
each have an A→B→...→t path (where '→' means "beat") to all of the other 105 teams, none of which have such a chain to any of these 15.

Now we order these teams by how well they've done against each other. Texas gets credit for { Oklahoma, { TCU, { BYU }, Cincinnati }, Missouri } - 5 of the other 14 teams in this elite grouping by resume.

Alabama gets credit for one - { Georgia }, as does Oklahoma State { Missouri } and Oklahoma gets credit for all three teams in WX = { TCU, { BYU }, Cincinnati } while TCU gets credit for { BYU }.

So one could make a reasonable argument that by the purest of resume approaches the evidence so far is for a list that begins
1. Texas
2. Oklahoma
3. Oklahoma St
4. Alabama
5. TCU
6. Penn State (#17)
6. Missouri (#17)
6. Boise St (#24)
6. Utah (#26)
6. Georgia (#31)
6. Ball State (#46)
6. Texas Tech (#48)
6. BYU (#53)
6. Tulsa (#56)
6. Cincinnati (#80)
The ten teams tied for 6th have no wins against any of the other teams that have no losses to any of the other 105 teams. Essentially any order of these could be justified, but to be consistent we'll use the best win against teams not in this set as ordered by Second Order Winning Percentage.

The 16-25 positions have to be decided based upon how the 105 other teams have done against each other, which even in the purest of resume systems requires some subjective judgement.

For instance, we could go by the teams { T } with the fewest t→...→T chains pointing to them (remember, each of the 15 listed above do, so we subtracted those out) which would give us
16. USC (3)
17. Ohio State (6)
18. Illinois (7)
19. Troy (9)
20. Minnesota (10)
21. Boston College (12)
22. Georgia Tech (13)
22. Kansas (13)
22. Oregon (13)
25. LSU (14)

But does anyone think Troy is better than Oregon State (15), Kentucky (16), Florida(!) or Northwestern (17 each)? Actually, your subjectivity may have caught you if you said "no" - Troy hasn't lost to any team not on this list ahead of them, so as far as we know from the results on the field, they might be. All of the teams I mentioned have lost to teams with worse resumes than they have.