Hatch Mathematical College Football Rankings - Version 3.7.2
(System Last Updated June 17, 2024)

Only three factors are germane to ranking teams:
(1) Who did you play? 
(2) Where did you play?
(3) What was the score?

1.  Definitions

Rank (RK): The rank of a given team is its position in the order of teams' Performance Rating scores, sorted descending. (E.g., a team that has the highest Rating score’s Rank is 1.) The team ranked 1st is considered to be the "highest" ranked team.

Rank Factor (RF): The rank of a team divided by the total number of teams ranked (including the Division 1-FCS Placeholder), then multiplied by 150.

Initial Ranking (IR): All teams' initial ranking is 1. After the initial iteration (and all subsequent iterations), the next iteration begins with the final output rankings from the previous iteration.

Output Ranking: The ranking of all teams' Output Ratings for any given iteration in which all available game scores in a current season have been evaluated based on the Performance Rating Formula and produced Output Rating scores for all teams.

Stable Ranking Set: The ranking set is Stable when one full iteration produces an Output Ranking that is identical to its Initial Ranking.

Final Ranking: The Output Ranking of the last iteration in a Stable Ranking Set.

Rating Points (PT): Each Team is assigned a “Points Value” based on its Rank in each iteration.  The Points for a given team is the result of the following formula: 151 - RK = PT.

2.  Calculation of the Performance Rating

Each game generates a Performance Rating for both participating teams. The average of a team's Performance Ratings across the course of the season constitutes that team's Output Rating for the current iteration.

​The calculation of the Performance Rating for each game is as follows:

Part One. Margin of Victory
For a game in which Team A defeats Team B by a margin of M:

• Where M ≤ 24:
  • IPR[A]=(PT[B]×1.15)×(1+(0.01×M))
  • IPR[B]=−1.1×((RF[A]+40)×1.15)×(1+(0.02×M))
• Where M ≤ 34:
  • IPR[A]=(PT[B]×1.15)×1.25
  • IPR[B]=−1.1×((RF[A]+40)×1.15)×(1+(0.02×M))
• Where M > 34:
  • IPR[A]=(PT[B]×1.15)×1.25
  • IPR[B]=−1.1×((RF[A]+40)×1.15)×1.70

Part Two. Preliminary Modifications

• Piecewise Function (pf):
  • If M <=25: 1 + 1 / (1 + exp(-0.3 * (x - 12)))
  • Otherwise: 2
• IPR[A] = IPR[A] * (exp(-0.01 * RF[B]) + 1)
• IPR[B] = IPR[B] * pf(M)
• IPR[B] = IPR[B] * (1 + (1 - exp(-0.01 * RF[A])))

Part Three. Site Based Modifications:

• Where A is on a neutral site:
  • ​PR[A] = IPR[A] * 1.2
  • PR[B] = IPR[B] * 1.2
• ​Where A is on the road:
  • ​PR[A] = IPR[A] * 1.3
  • PR[B] = IPR[B] * 1.3
• ​Where a bowl game:
  • ​PR[A] = IPR[A] * 1.45
  • PR[B] = IPR[B] * 1.45

3.  Synthesizing Game Performance Ratings Into an Output Rating

                        OUTPUT RATING = ((PR[Game 1] + PR[Game 2] + PR[...] + PR[Final Game] + (130-IR)) ÷ (Games Played) + 150) ÷ 475

4.  Ranking Teams Based on Output Rating & Miscellaneous Rules

1. Rating scores will be rounded to one ten-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-hundred thousandth of a point, they will be treated as tied.
2. Teams will be included if they were part of the University Division after 1950 until 1972, or Division I from 1973 until 1977, or Division I-A, or I-FBS, since 1978. Prior to 1950, teams were included based on a centrality analysis, if they played a number of games within 1 standard deviation of the mean number of games played by a team against other ranked teams in the given season.
3. All teams outside of those ranked will be represented by the Division 1-FCS Placeholder. The Division 1-FCS 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. 
4. The system will be iterated until it becomes a Stable Ranking Set. In the event that the results of a given season would create an irresolvable infinite loop for two or more different teams, all teams will be ranked based on an average of their ratings in the 99th and 100th iterations. 

5.  Determination of the National Champion

The team with the highest Final Ranking after all scheduled games within one season have been played will be declared the National Champion.