Hatch Mathematical College Football Rankings - Version 3.8
(System Last Updated February 9, 2026)
Only three factors are germane to ranking teams:
(1) Who did you play?
(2) Where did you play?
(3) What was the score?
(1) Who did you play?
(2) Where did you play?
(3) What was the score?
1. Definitions
Rank (RK): The rank of a team is its position when all teams are sorted by Output Rating, descending. (The highest Output Rating has RK = 1.)
Rank Factor (RF): A rank-normalized value on a 0–150 scale: RF = RK / (N+1) ×150, where N is the number of rated teams (excluding the Placeholder).
Points Value (PT): PT=151−RF
Initial Ranking (IR): The ranking set used to begin an iteration. In the first pass, teams begin from seeded starting ranks; each subsequent iteration begins from the prior iteration’s Output Ranking.
Output Rating: The season average of a team’s game-level Performance Ratings in a given iteration, transformed to display scale.
Output Ranking: The ranking produced by sorting all teams by Output Rating in a given iteration.
Stable Ranking Set: Achieved when one full iteration reproduces the same ranking order as the ranking used to start that iteration.
Final Ranking: The Output Ranking of the last iteration in a Stable Ranking Set.
Division I-FCS Placeholder: Synthetic fallback opponent used when a game references a non-rated or missing team. It is fixed at last place (RK = N+1), is never output as a ranked team, and is never allowed to move.
Rank Factor (RF): A rank-normalized value on a 0–150 scale: RF = RK / (N+1) ×150, where N is the number of rated teams (excluding the Placeholder).
Points Value (PT): PT=151−RF
Initial Ranking (IR): The ranking set used to begin an iteration. In the first pass, teams begin from seeded starting ranks; each subsequent iteration begins from the prior iteration’s Output Ranking.
Output Rating: The season average of a team’s game-level Performance Ratings in a given iteration, transformed to display scale.
Output Ranking: The ranking produced by sorting all teams by Output Rating in a given iteration.
Stable Ranking Set: Achieved when one full iteration reproduces the same ranking order as the ranking used to start that iteration.
Final Ranking: The Output Ranking of the last iteration in a Stable Ranking Set.
Division I-FCS Placeholder: Synthetic fallback opponent used when a game references a non-rated or missing team. It is fixed at last place (RK = N+1), is never output as a ranked team, and is never allowed to move.
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:
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:
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
- 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.
- 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.
- 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.
- 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.