Description of Maas' Rating System.

Introduction.

Already more than thirty years ago I felt a need for a Rating System that could do better than just counting the number of wins and losses. In my opinion a more sophisticated Rating System was needed, especially for unbalanced competitions, where due to draws and/or a knock-out systems not all participating teams play each other and where not all participating teams even play the same number of matches.

I did develop a Rating System that took the strength of a team's opponents into account and that appeared to have a number of properties in common with the well-known ELO-Rating System.

In 1971 for the first time a computer based Ranking Table derived from Maas' Rating System was published in a Dutch Magazine (Goal) for European Champions Club Competitions. Due to lack of computer facilities and problems in finding correct results I did not further maintain this Rating System.

Since the availability of powerful Personal Computers and Computer Languages and the accessibility to all information about Rating Systems via the Internet I became more interested again, especially after reading about College Football related Rating Systems.

Hereafter I will describe the rational of Maas' Rating System.

Assumptions.

The assumptions of Maas' Rating System are:

1. Winning is the only thing that matters.
   (Margin of victory is not taken into account).
2. Wins against stronger opponents should be "rewarded" more than wins against less stronger Opponents
   (Losses to stronger Opponents should be "punished" not as much as Losses to less stronger Opponents).

Formula.

Each team's Rating equals its number of Wins, minus its number of Losses, plus the Sum of the Ratings of all its Opponents,
all divided by the total number of Matches played.

The above described model can be presented as formula:

     Wi - Li + Sum(Ropp,i)
Ri = ---------------------
              Mi 

Where:  Ri = Rating of Team i
        Wi = Wins of Team i
        Li = Losses of Team i
Sum(Ropp,i) = Sum of Ratings of all Opponents of Team i
        Mi = Matches (total) of Team i

If a team plays an opponent two or more times, then, for the determination of Sum( R_opp,i ) it is counted as that many opponents.

Computation.

Unlike others I do not try to find a satisfactory solution by means of iterative calculations. My solution in fact exists of the following steps:

• I define a set of linear equations that is based on the number of Wins against various Opponents. All those equations together constitute a Matrix.
• Futhermore a solution Vector is created that consists of all the teams' scores, calculated as: Number of Wins minus Number of Losses.
• The Rating scores are then obtained by inverting the Matrix and multiplying that inverted Matrix by the solution Vector.

Details about mathematical, technical and computational issues are presented in a separate PDF file that can be accessed here.

Ranking Table.

Like for many other Rating Systems, the absolute value of a team's Rating Score is rather arbitrary: the only thing that really matters is the Ranking Order.

In order to present Ranking Tables for which the Rating scores fit into an appropriate scale, various types of linear transformation are used, depending on the type of application (USA College Football, UEFA Champions League etc.).

Note that these types of transformation procedures do not change the order in which teams are ranked based on the originally computed Rating Scores.

Discussion.

The Model of Maas' Rating System uses a minimum of input and assumptions and is based on common sense: the "reward" for a Win depends on the difference in strength between the two opponents. Maas' Rating System adjusts for strength of schedule.

I do claim that Maas' Rating System has no subjective inputs (and hence no bias) at all. The only input contains of number of Wins for Team X versus Team Y. All other parameters can be derived from that Number of Wins and all Ratings will be determined by the System itself. Hence the assumption: "Winning is the only thing" is fulfilled.

Furthermore it can easily be derived from the presented Formula that with equal number of Wins (and Losses) a team's Rating depends on the Ratings of that team's Opponents: the higher the um of the Ratings of all Opponents, the higher the team's Rating will be. And this fulfils another assumption: "Wins against stronger opponents should be rewarded more than Wins against less stronger opponents".

Like for most other Rating Systems, the absolute value of the Rating Score is not important. It is the Ranking Order (based on these Rating Scores) that matters.

Based on the examples (see PDF file) and on the comparison with other Rating Systems (see Massey Ratings) I am confident that Maas' Rating System produces effective Ranking Tables.

Taking into account some characteristics of a specific tournament (e.g. low number of matches or Home Field Advantage etc.) adjustment of the initial Ranking System can be justified.

• Maas' Rating System
  is really unbiased: it has no subjective inputs at all. The only input data consist of number of Wins of Team X versus Team Y.
• Maas' Rating System
  is developed by Martien W. Maas (Nijmegen, The Netherlands) and results based on this System have been published already in 1971.
• Maas' Rating System
  is based on only one assumption: that a win over a stronger team should be rewarded more than a win over a less stronger team.
• Maas' Rating System
  computes rating scores based on the solvation of a set of equations (derived from the number of wins against various opponents).
• Maas' Rating System
  produces Ranking Tables that highly correlate with Ranking Tables produced by other well accepted Rating Systems.
• Maas' Rating System
  is especially applicable for other than round-robin competition formats and can be applied for all types of sport events
• Maas' Rating System
  is included in the College Football Ranking Comparisons by Kenneth Massey.