HATCH MATHEMATICAL COLLEGE FOOTBALL RANKINGS
Last Updated: 1/6/13
A number of factors have caused me to believe that it is time to tweak the formula that I have used to rank teams for the past 9 years. Although I have previously avoided editorializing on my website, I believe that I should explain my reasons for making these changes. My commentary will be found in red throughout.
1. Initial Ranking (IR): The final results from the previous season's poll. In the event that new teams join Division 1-FBS, new teams will be assigned a preseason ranking of a position immediately lower than the lowest ranked team from the previous season.
a. The Initial Ranking is used only in the first iteration of each season's results. For subsequent iterations, the final ranking of the previous iteration will be used as the Initial Ranking.
b. As a result, the Initial Ranking's effect will be filtered out as each successive iteration is conducted.
I have seen on at least one message board that my system was criticized for having an Initial Ranking. I would answer that (1) all systems have some initial input (even if the initial input is to assign the same value to all teams) and (2) that there is virtually zero impact (probably around 1/1000th of a teams actual Rating) from the Initial Ranking by Week 3, if not sooner.
2. Stable Ranking Set: achieved when one set of final rankings is used as the Initial Rankings for a further iteration which produces the same set of final rankings that are identical to the Initial Rankings.
What this means is that I keep iterating the system until the initial and final rankings are identical. This usually takes several iterations.
3. Rank (RK): the rank of a given team is its position in the order of teams' Rating scores, sorted descending. (E.g., a team that has the highest Rating scoreÕs Rank is 1.)
4. Points (PT): Each Team is assigned a ÒPoints Value.Ó
a. For seasons in which less than 129 teams are ranked, the Points total for a given team is the result of the following formula:
130 - RK = PT
(E.g., a team that has a Rank of 1 has a Points total of 129.) Points should not be confused with Rating score or Rank.
b. For some seasons in which 130 or more teams are ranked, the Points total for a given team is the result of the following formula:
130 – (RK/#) = PT, where # = 130/the number of ranked teams.
I use this same approach in my basketball ratings. Some historical ratings had more than 129 teams eligible for ranking and I didnÕt want to arbitrarily exclude teams. The ever-expanding Division 1-FBS is also likely to wind up with more than 129 teams within the next few years, so this calculation may wind up being used in contemporary ratings before too long.
CALCULATION OF THE PERFORMANCE RATING
1. Part One.
For a game in which Team A defeats Team B by a margin of M,
Where M < 11,
IPRA = PTB x 1.05
IPRB = (RKA +10) x -1.05
Where 10 < M < 20,
IPRA = PTB x 1.1
IPRB = (RKA + 10) x -1.25
Where M > 20,
IPRA = PTB x 1.2
IPRB = (RKA + 10) x -1.6
Where a tie,
IPRA = (.3 x (PTB x 1.05)) + ((RKB +10) x -1.05))
IPRB = (.3 x (PTA x 1.05)) + ((RKA +10) x -1.05))
After an extensive review of the 100 previous seasons I have ranked, I came to two conclusions:
(1) I had not sufficiently punished teams for losing, especially by large margins. Simply because a team shouldnÕt be rewarded for winning by a big margin doesnÕt mean that a team that loses by a big margin shouldnÕt be punished. Losses by less than 21 points were also assigned more weight.
(2) My system failed to adequately deal with tied games. This system was born in the Era of Overtime and I have never been satisfied with the historical rankings that treated ties as generally positive results. This change sees a tie as a mostly negative result. The object of the game is to win. Obviously, ties are still decidedly more favorable than losses.
2. Part Two.
Where A is at home:
PRA = IPRA x 1.00
PRB = IPRB x 0.95
Where the Away Team wins:
PRA = IPRA x 1.05
PRB = IPRB x 1.05
Where the game is at a neutral site:
PRA = IPRA
PRB = IPRB
Where the game is a Bowl Game (regardless of site):
PRA = IPRA x 1.2
PRB = IPRB x 1.2
These values remain unchanged, except for the weighting of Bowl Game results.
3. Part Three.
Ratings are Calculated.
RATING = (PRGame 1 + PRGame 2 + PR... + PRFinal Game + (130-IR)) Ö (Games Played)
a. Rating scores will be rounded to one thousandth of a point. Decimal values beyond this value will not be considered, except in case of a tie. If teams are tied at the level of one-ten thousandth of a point, they will be treated as tied.
b. Only teams considered part of Division 1-A/FBS, or the equivalent for the year in question will be ranked. Exception: Teams participating in the Rocky Mountain Conference from 1912-1937 will not be ranked.
The RMCC was, for all intents and purposes, hermetically sealed from playing other teams during this period. Inclusion of this conference skews the results badly because there are no external forces (i.e. inter-conference games) to help properly ascertain the strength of RMCC teams. This is an editorial decision, but obviously itÕs an exception dealing with teams that played 75 years ago or more. I donÕt think this decision is substantively problematic.
c. All teams outside of those ranked will be represented by the Division 1-AA Placeholder. The Division 1-AA Placeholder will always be ranked below all rated teams in all rankings and will not be moved from the last place position for any reason.
In the past, I had treated the Division 1-AA Placeholder as a regular team that could move up or down in the system. That solution may have worked in, for example, 1993 where only 47 games were played against 1-AA opposition (about 6 games to every 13 teams), or as late as 2005 with only 53 games (a similar ratio as to 1993). In 2011, In 2011, 97 games were played against 1-AA teams (approaching 3 games to every 4 teams). In 2012 that number was 108 (in comparison to 124 teams).
I donÕt know about you, but I am sick of seeing allegedly strong teams play 1-AA opponents. My system should not reward Rutgers in 2011 for playing North Carolina Central (2-9 in the MEAC, with a loss to Savannah State(!)) more than if Rutgers had played Tulane or Akron. The idea of abstracting 1-AA teams is a necessary fiction (or else IÕd quickly find myself having to rate every team), but I will not allow teams to profit by playing 1-AA teams when there are 1-A teams available.
In my view, the trend towards playing increasing numbers of smaller division opponents will undermine various rating systemsÕ (including the future Playoff Selection CommitteeÕs) abilities to properly value the relative strengths of different leagues and will, resultantly, skew their final results.
d. In the event that the results of a given season would create an irresolvable infinite loop for two or more different teams, teams will be rated based on the average of the results produced. (e.g., when team A is ranked 3rd and team B is ranked 4th in the initial ratings it produces a final rating having team B in 3rd and team A in 4th. Further iterations keep all other ranked teams in identical positions except for A and B that alternatively switch positions without any possibility of a conclusion. The dilemma is resolved by averaging team AÕs final rating when it begins in 3rd and 4th position and doing the same for team B. Whichever teamÕs average rating is higher will be ranked 3rd and the other 4th.)
1. In the event that multiple infinite loops occur within the same season, the final rankings for all teams not caught such a loop will be determined following all averaging under Rule 2.3.d. Ratings for all other teams will reflect the results of the 2.3.d resolution of the loop involving the highest ranked teams.
I donÕt think I wrote this very well. If I have two loops, each time I iterate the system the looping teams will be in different positions which impacts many other teamÕs ratings. Where this happens (I think it only happened 2 times in the historical results), every one elseÕs rank will be determined by the ratings when the highest-ranked loop is resolved under 2.3.d. (In my example of A and B above, if A finished 3rd and B finished 4th, every other team not involved in a loopÕs ratings will be calculated as if A finished 3rd and B finished 4th).
e. For the first week of a season, the system will be iterated a maximum of five times. Thereafter, the system will be iterated until it becomes a Stable Ranking Set as defined in Rule 1.2, and subject to the procedures in Rule 2.d. and 2.d.1.
DETERMINATION OF THE NATIONAL CHAMPION
Teams will be ranked according to their IR. Scores from each game will be assigned Ratings as discussed in 2.3. The final rankings of each team based on their overall Rating will then become the IR for a next iteration. This process will continue until all rankings have become a Stable Ranking Set, and any teams that are caught in an irresolvable infinite loop have their ratings determined by Rule 2.3.d as described above. The team with the highest Rating after the bowl games will be declared the national champion.
It is my view that ultimately, only three factors are germane to ranking teams. (1) Who did you play? (2) Where did you play? (3) How did you do? (Margin of Victory or Defeat – capped at 21 points, to limit the effects of running up the score.)
The touchstone of this system is that the quality of your opponent is the most important of the 3 factors. Although these changes have no impact on (2) and increase the effect of (3), the overriding consideration in this system rightly remains strength of schedule.
Having spent the holidays reworking the past 100 seasons, the effects that I have noticed are that teams are punished more for losing games (which was intentional), allowing for some teams with better records to be pushed up past where they used to be. To be honest, I am not sure if this will increase or decrease accuracy in predicting future games. But this system is not a predictive indicator. It is an indicator of past results. If past results now put more of a premium on winning, then that canÕt be a bad thing.
Thanks for visiting the site and please feel free to send me your hate mail.