The general idea is to provide the ranking information in all formats that would be needed to count "votes" in a
team-quality poll were the computer rating systems "voters" in such a poll. There are many ways to count ranked
ballots to create a composite rank, and since there's no best way to do that, I try to provide enough
information to use any way to do so.
I only include computer ratings that rank all teams in the field, so my list will never exactly match Dr. Massey's, which includes human
top 25's and a few computer "top n" where n is less than the number of teams in the field.
| Top 25 | Truncates every computer rating's ranking at 25 and then counts the
ballots the same way media polls do, using a 25-24-23-... point assignment for teams ranked 1-2-3....
I only include this report to demonstrate how much information is left out of the usual media presentation of their poll results.
In addition to the number of "points" I include the number of ballots that listed the team in the top 25 and the number of votes
for each rank for which the team was voted in the top 25.
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| Top 25 Correlations | See below for a general discussion of rank-correlations. The top 25 correlations are a bit different, since not every ranking includes the same teams. The reports are described in Top 25 Correlations.
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| Borda | The usual method of counting top 25 ballots is a variation on
the Borda Count. In its basic form, teams get one point for each team they are ranked better than. In a 130-team field,
a #1 vote is worth 129, #2 worth 128 down to #130 worth 0. When all teams are ranked the order is the same as the average
rank over all ratings.
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| Majority Consensus |
My consensus rank is based upon the Bucklin vote-counting method. For each team find the best rank for which a majority
of the ratings agree the team should be ranked at least that highly. I use a strict majority, namely 50% + one rating.
When there are an odd number of ratings this is the same as the arithmetic median. For an even number of ratings it is
the best rank worse than the median.
Unless otherwise stated this is the ranking used in reports that include a team's rank.
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| Pairwise Matrix | Even when the majority ranks team A better than
team B, it is possible that team B is ranked better on more ballots than team A. In Condorcet voting, the ballots are
translated into pairwise comparisons between alternatives. The method suffers from a lack of transitivity: team A > team B and
team B > team C does not imply team A > team C!
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| Playoff Approval |
Approval voting does not require a ranked list. Instead the voter just lists the alternatives that meet or exceed their minimum criteria. For division 1A football the criterion for selecting the four playoff teams is "one of the best four teams in the field." As I suggested in Committee "consensus" this would be a much better way for the committee to amalgamate its collective thoughts than the "top-25" poll they chose. My report just counts the number of voters that rank the team in its top 4.
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| Correlations |
The basis for measuring how alike two ordinal rankings
are is the distance metric. This is the number of swaps required by a bubble sort to place one of the lists in the
same order as the other. The distance varies from zero (the lists are identical) to the total number of team-pairs (the lists
are reversals of each other; 8,128 for a 128-team field.) For each ranking I report the contribution to the distance function by each team.
The distance is the number of discordant pairs - the number of pairs where the teams' relative orders are reversed in the
two rankings. When the teams are in the same relative order in both lists the pair is said to be concordant. When
the teams have the same rank in either list the pair is ignored.
These can be turned into rank correlation coefficients in several ways. The two I calculate are:
Kendall's tau:
| τ = | #Concordant pairs - #Discordant pairs |
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| ½ × #Ranked × (#Ranked-1) |
Goodman and Kruskal's gamma:
| γ = | #Concordant pairs - #Discordant pairs |
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| #Concordant pairs + #Discordant pairs |
These give -1 ≤ { τ, γ } ≤ 1 with |τ| ≤ |γ|. Both will be -1 if the teams are in exactly reverse order, 0 if the relationship
is perfectly random (whatever that means!) and +1 if the rankings are identical. The τ and γ are the same
if there are no ties (but notice that ties in the Majority Consensus rank are to be expected, in which case τ will be closer to zero
than γ.)
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| Conference Ranks |
There are more ways to aggregate team ranks by conference than ratings, but I have chosen these.
- Rank Distribution by Conference
- includes every rank for every team in the conference. In addition to the average
rank there's a count of team ranks in the ranges, 1-25, 26-50, and so on.
- Team Consensus Ranks by Conference
- lists the consensus ranks of teams by conference.
- Pairwise Comparison of Teamranks by Conference
- compares the consensus rank of each conference team to that of every member of other conferences. The
entries represent the number of times out of a thousand that a team from the row conference would be expected to
have a better rank than that of a team from the column conference.
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| Weighted Violations | Roughly one in five games result in the
worse-ranked team winning no matter which rating produces the ranking. Were it not so sport would not be interesting. One
measure of how well a rating represents results-to-date is the count of Retrodictive Ranking Violations, the number
of games in which after the ranking takes into account team A beat team B it still ranks team B better than team A.
Motivated by Potemkin's idea that instead of just counting the
number of RRVs we shuold take into account the size of the violation (rank difference) and the importance (how
highly the loser is ranked) I came up with a Weighted RRV value that combines the size of the upset (in scores and
rank difference) with importance (loser's rank.) The "importance" component also takes into account that violations later in the
season (when the rating has more input) should count more.
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WRRV = ⌈ (WS - LS)÷S ⌉ × (WR - LR) × ƒ(LR) |
- WS
- Winner's score
- LS
- Loser's score
- ⌈ (WS - LS)÷S ⌉ =
-
the margin of victory in number of scores
(⌈x⌉ is the least integer ≥ x)
S=8 for football
- WR
- Winner's rank
- LR
- Loser's rank
Note that (WR-LR) is positive for all ranking violations by definition.
- ƒ(LR) =
-
Log LRi Mj Mi+j
where i and j are chosen to make ƒ(1) any predetermined value
ƒ(M) is exactly 1.
M is the number of ranked teams (130 in 2018 for Division 1A.)
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See WRRV 2.1 for graphs of ƒ(LR) for different choices of i and j.
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