How to use the colley matrix ranking calculator
- Enter one game per line as team, score, team, score, for example Hawks 24 Lions 17. Team names are one word.
- Games can be in any order, and each team can appear in any number of games.
- Tied games are left out, because the method counts only wins and losses.
- Read the ranking with each team's rating and record, the top team and the average rating.
Formula
Colley matrix C: the diagonal entry for a team is 2 plus the games it played; the entry for two teams is minus the number of games between them
Right side b for each team: 1 + (wins - losses) / 2
Solve C r = b for the ratings r
The ratings always average 0.5, and a team that wins everything is the highest but never reaches 1
The Colley method starts each team at an equal rating and then adjusts it for its results, with credit for beating teams that are themselves strong.
Two teams, then four teams in six games
With only one game, A beating B, the matrix is [[3, -1], [-1, 3]] and the right side is [1.5, 0.5], so the ratings are 0.625 for A and 0.375 for B, and they average 0.5.
In the four-team example, Hawks won all three games and rate 0.75. Wolves (2 and 1) rate 0.58333, Lions (1 and 2) rate 0.41667 and Bears (0 and 3) rate 0.25. Even though the Bears are the last, the rating differences are smaller than the win counts alone, because every team has played some of the same opponents.
Why the schedule matters
A team that has beaten strong opponents deserves more credit than one that has beaten weak opponents, even with the same record. The Colley method builds this into the equations by solving for all the teams at once, so a win over a high-rated team lifts a rating more than a win over a low-rated team. It uses only who won, so margin of victory cannot be run up, which is why it has been used as one input to ranking systems.
For ratings that update after each game, see the Elo rating calculator. To build a league table with points and goal difference, use the league table generator.
Limits of the method
The results are only as good as the games supplied, and teams that are not linked by any chain of games cannot be compared. The ratings are not probabilities of winning. The method does not account for home advantage or the date of each game. Small numbers of games leave many teams close together. Use the ranking as one view, along with others. To set up a bracket, see the tournament bracket generator.
Assumptions and limits
- Only wins and losses count. Scores decide who won but not by how much.
- Tied games are dropped. Names are single words and are case sensitive.
- Teams that are not connected by any games cannot be compared meaningfully.
- The page allows up to 40 teams.
Frequently asked questions
What is the Colley method?
It is a ranking method that solves a system of linear equations built from wins, losses and the games between teams.
Why do ratings average 0.5?
The equations are set up so the total of the ratings equals the number of teams divided by 2, and so the average is 0.5.
Does the score matter?
Only to decide the winner. A win by 1 counts the same as a win by 30.
What about tied games?
They are left out of the calculation, because the method has no place for a tie.