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Wednesday, October 3, 2007

The Value of a Tournament

In the next few paragraphs, I will present the solution to the most pressing problem (in my opinion) facing college football. (How about that for an intro!)

The primary goal of a college football season is to recognize a national champion. Logically, there has been a lot of discussion about how this is best done. The focus of this discussion has been a lack of benchmarks: too few games and too many teams. But a tournament is not a magic solution.

There are two problems with a tournament. First, it places too much emphasis on a few games at the end of the season. Second, because a team's performance may vary from game to game the team that wins the tournament is often not the best team overall.

In a tournament, performance through the season is recognized by allowing the team to participate in the tournament and matching up better teams against worse teams as much as possible. If we believe that a college football season does not offer the necessary match-ups throughout the season, a short tournament that provides those match-ups may be the best available option.

Theoretically, the superior system would give entrance only to those teams that have a legitimate claim at the national championship after their performance during the season. The superior system should also be more likely to crown the superior team with a minimal number of games. A seven game series would be preferable if it didn't take two months to play out.

Including too many teams, therefore, is a double wammy. First, it increases the number of games to be played. With 8 teams there are seven elimination games and the finalists must play three extra games. Because teams vary in their performance from one week to the next, the more teams and the more games a team must win, the less likely it is that the best team will actually win.

This can be easily demonstrated with the equations developed for week 4.

Assume we have 8 teams, A through H, with ratings: A = 35, B = 34; C = 33, etc., such that A would be favored by 1 over B, by 2 over C, and by 3 over D. These are, in this case, objective ratings, such that A has been the best team over the course of the season and is currently the best team in the country. But the probability that A will win the national championship varies with the type of tournament employed.

In a single elimination tournament with 8 teams:
A 24.71%
D 12.33%
E 9.60%
H 4.39%
B 19.93%
C 15.80%
F 7.48%
G 5.76%

In a single elimination tournament with 4 teams:
A 32.19%
B 26.89%
C 22.37%
D 18.55%

In a double elimination tournament with 4 teams:
A 34.36%
B 27.30%
C 21.51%
D 16.83%

In a single championship game with 2 teams:
A 53.06%
B 46.94%

Therefore, in this scenario, the best predictor is, in fact, the single championship game, and logically so. This format, though, is flawed because we cannot always objectively identify the top two teams.

In recent years, three of the four top conferences (PAC 10, Big 12, SEC and Big 10) have produced a title contender. We have also seen challenges made by representatives of the Big East and WAC, and, before long, the ACC will begin to produce contenders again. Therefore, a four or five team tournament (with a play-in game) should be sufficiently inclusive to identify the nation's top team.

Another advantage of a four team tournament is that placement plays almost no role in deciding the winner.

This narrows our options to two potential formats, single or double elimination. The double elimination format is more likely to crown the superior team, but requires almost as many games and more games per team than the 8 team format.

But the double elimination format is only significantly more likely to produce the correct national champion if the best team is significantly better than the rest of the field. For example, if we reproduce the earlier experiment but with larger differences between teams (which is, admittedly, the less likely scenario), such that A=35, B=31, C=27, D=23, etc., we arrive at the following probabilities:

In a single elimination tournament with 8 teams:
A 54.34%
D 4.28%
E 1.31%
H 0.03%
B 27.76%
C 11.79%
F 0.38%
G 0.11%

In a single elimination tournament with 4 teams:
A 53.72%
B 27.58%
C 12.84%
D 5.86%

In a double elimination tournament with 4 teams:
A 61.22%
B 26.03%
C 9.52%
D 3.23%

In a single championship game with 2 teams:
A 62.01%
B 37.99%

In this case, the three or four extra games would definitely favor the better team at all in the end. But, assuming we would want a single championship game (such that the team emerging from the losers bracket would not need to defeat the winner bracket representative twice), the odds of A winning would fall to 56.24% from 61.22%. And, since the best team is still almost twice as likely to win the tournament as any other team using this format, it can still efficiently identify the champion.

A play-in game should increase the probability that the correct team is identified as the national champion (unless the best team is ranked 4th at the time) and could be played before the rest of the bowl season. As such, a system could be devised that the conference champions of the major conferences could be reserved a play-in spot if they do not otherwise qualify by ranking as long as, I believe, the top three teams are not forced to win more than two games to win the tournament.

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