Counting Votes - Some Real Data

© Copyright 2006, Paul Kislanko

The poll is a traditional "top 25", with an interactive ballot that requires 25 teams to be ranked and doesn't allow duplicate rankings. (It's the nicest online ranked ballot I've ever used.)

49 teams were ranked in at least one of 52 ballots. There was a remarkable degree of consensus in this ballot set - the 24 teams that were ranked on a majority of ballots can be ordered so that every team is ranked higher than or equal to all teams below it by a majority of voters.

This is not usually possible, but when at least one team pairwise defeats all other teams that team is said to be the Condorcet winner. When there is a subset of teams such that every team within the subset has a pairwise advantage over all teams outside the subset, it is called a Smith Set. In this example, the first 21 teams listed form a Smith set, and Ohio State is the Condorcet Winner.

When we count the votes using the standard "25 points for 1^st, 24 for 2^nd, etc." approach we're using a modified form of the Borda Count, and I showed last time that this can result in a team having a higher ranking even if a majority of voters had it ranked lower than another team that got a worse rank. We can see some of the same effects in this week's Workman Poll when we count the ballots four different ways.

Now, remember these are all results from exactly the same ballots. Does this group of fans collectively think Florida is third or do they think the Gators are sixth? Oregon is the eighth-best team in the country, or they're 12^th. There's a famous (to the Nobel Prize committee, anyway) theorem to the effect that there's no perfect way to decide, but we can categorize the methods by their attributes.

Cond

The Condorcet order is such that every higher-listed team has more ballots where it is listed higher than every lower-listed team. As I noted above this is not always possible, because there may be "cycles" where more voters rank team A higher than team B and team B higher than team C, but more voters rank team C higher than team A. In this example there was also a tie - 46 voters ranked Florida St, Missouri or both, and 23 ballots had FSU higher than Mizzou, with 23 having Mizzou higher than FSU. A typical way to break the tie is to use the stronger defeat over the next non-tied team. In this case FSU's 23-21 margin over Rutgers trumps Missouri's 21-20 margin.

Maj

is my implementation of what is known as Bucklin counting. A team is assigned the highest rank for which a majority of the voters rank the team. If every voter ranks every team (and there are an odd number of voters) this is equivalent to the median rank. There are often ties for this rank, so to order the teams I use the Count of votes for that rank or higher, and if that is also a tie, the Real Borda count for the tied teams.

RealB

is based upon the original definition of the Borda Count. For each ballot, points are assigned to the team at each rank based upon how many teams it is ranked higher than. Since there are 119 teams in Division 1-A, a first place vote is worth 118 points, a second place vote is worth 117, etc. So one vote for 20^th is worth 99 points, and six votes for 25th is worth 564.

25Pts

This is the traditional system that assigns 25 points for 1^st and 1 point for 25^th on each ballot. If one voter ranks team A 20^th and six voters rank team B 25^th the ranking "says" those teams are equally-regarded by the group, which is not likely what all but one of the group would say.

When the AP and USA Today (nee UPI) polls' only purpose was to fill newspaper space on a non-gameday and give fans something to talk about, it really didn't matter how they came up with the answer to "who's number 1?" But now that the "polls" are being used to decide which teams get a $14-17 million payday, it is worth scrutinizing the methods by which they make such decisions. As a minimum they should publish the number of votes for each rank for all teams receiving votes. As I've noted before in order to achieve "transparency in the BCS process" we don't need to see the individual ballots, nor do we need to know who voted how - we just need the counts of votes for each rank.. From that we can calculate each of the orders except the Condorcet order, which usually doesn't exist.

Ballot Summary