Description of Maas' Rating System.

* Introduction

* Assumptions

* Formula

* Computation

* Ranking Table

* Characteristics

Already more than forty 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 match data 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.

The assumptions of Maas' Rating System are:

- Winning is the only thing that matters.

(Margin of victory is not taken into account). - 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).

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:

W_{i}- L_{i}+ Sum(R_{opp,i})

R_{i}= ---------------------

M_{i}

Where: R_{i}= Rating of Team i

W_{i}= Wins of Team i

L_{i}= Losses of Team i

Sum(R_{opp,i}) = Sum of Ratings of all Opponents of Team i

M_{i}= 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.

The presented Formula shows that with equal number of Wins, Losses (and Draws) a team's Rating
depends on the Ratings of that team's Opponents:

the higher the Sum of the Ratings of all Opponents,
the higher the team's Rating will be.

This conclusion is perfectly in line with the assumption:

*"Wins against stronger opponents should be rewarded more than Wins against less stronger opponents"*.

Unlike other systems Maas' Rating System does not try to find satisfactory Rating Scores by means of iterative calculations. Maas' Rating System in fact exists of the following steps:

- Each match of the competition is defined as a linear equation.
- For each Team separately all those equations are pooled which yields one linear equation for each Team in competition.
- All linear equations of all Teams together constitute a set of linear equations.
- The Rating scores for all Teams in competition are obtained by solving this set of linear equations.

Details about mathematical, technical and computational issues as well as an example of this procedure are presented in a separate PDF file that can be accessed here.

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* and the relative differences between the Rating Scores.

For publishing purpose all originally calculated Ratings are linearly transformed:

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

Finally we can summarize the characteristics of Maas' Rating Systems as follows:

- Maas' Rating System

is really unbiased: it has no subjective input at all. The only input data consist of the results (Win, Loss or Draw) of all matches between Team X and 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.