## Hatch Mathematical College Football Rankings - Version 2.3

(System Last Updated June 23, 2017)

*Only three factors are germane to ranking teams:*

(1) Who did you play?(1) Who did you play?

*(2) Where did you play?*

(3) What was the score?(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)**: 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.

- The Initial Ranking is used only in the first iteration of each season's results. For subsequent iterations, the Output Ranking of the previous iteration will be used as the Initial Ranking.
- As a result, the Initial Ranking's effect will be filtered out as each successive iteration is conducted. Repeated testing demonstrates that the effects of the Initial Ranking is filtered out of the system usually by the third week and generates virtually zero statistically significant effect over the Final Ranking.

**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:

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

__<__23,

IPR[A] = PT[B] x 1.3 x (1 + (.008 x M))

IPR[B] = (RF[A] + 20) x -1 x (1 + (.035 x M))

Where M > 23,

**IPR[A] =**PT[B] x 1.192

Where M > 34,

**IPR[B] = -2.225**

Where a bowl game after 1950,

IPR[A] x 1.275

Where a tie,

IPR[A] = .3 x (PT[B] x 1.3) + (-1 x (20 + RF[B]))

IPR[B] = .3 x (PT[A] x 1.3) + (-1 x (20 + RF[A]))

**Part Two. Adjustments for Location**Where A is at home or on a neutral site,

**PR[A] = IPR[A] x 1.00**

**PR[B] = IPR[B] x 1.00**

Where A is on the road,

**PR[A] = IPR[A] x 1.05**

**PR[B] = IPR[B] x 1.05**

Where the game is a Bowl Game (regardless of site) beginning with 1933,

**PR[A] = IPR[A] x 1.15**

**PR[B] = IPR[B] x 1.15**

## 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) ÷ 325**

## 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.
- Only teams considered part of Division 1-A/FBS, or the equivalent for the year in question will be ranked. Exceptions: (1) Teams participating in the Rocky Mountain Conference from 1912-1937 will not be ranked, and (2) Any team in Division 1 that played more than 3 games in the 2020 season will be ranked.
- 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.
**War Years Rules**.*(Used in seasons 1917-1918, 1942-1945, 2020.)*In seasons with Military All-Star teams, those team will not be ranked and will be assigned a rank value of 15 for all calculations involving their games. At the end of the season, an analysis will be conducted to determine the average number of games played by all teams that would ordinarily be ranked. Teams that played more than five games less than the average number of games will be represented by the Division 1-FCS Placeholder for that season only. The next season, such teams' initial rankings will be their final ranking from the last season in which they were ranked.- 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 A’s final rating when it begins in 3rd and 4th position and doing the same for team B. Whichever team’s average rating is higher will be ranked 3rd and the other 4th.) 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 this rule. Ratings for all other teams will reflect the results of the resolution of the loop involving the highest ranked teams.
- For the first week of a season, or during any week thereafter in which ranked teams have played only one opponent, the system will be iterated one time. These rankings will be published as "unofficial," but are listed on the website as evidence that the system is being applied as written in these instructions. After all teams have played, the system will be iterated until it becomes a Stable Ranking Set, and those ranings should be considered official.

## 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.