3. Interval Rating Systems

The typical rating system in chess employs an interval scale. The general formula is

[3.1]          R  =  ER_c + K(P - P_c) ,

where ER_c is the mean rating of opponents. P and P_c are the percentage scores of player and opposition, which reduces to 

[3.2]          P - (1 - P)  =  2P - 1 .

K is an arbitrary constant (-50 in the Ingo System).  The effect of [3.1] is to generate rating differences in proportion to relative performance, which is more easily seen by writing the formula as

[3.1a]          R - ER_c  =  K(P - P_c) .

The latter may be viewed as an arithmetic mean of game instances of the form

[3.3]          R - R_c    K(S - S_c)  ,

where the relative score, S - S_c, has three possible values:  1, 0, and -1.  This equation is generally an approximation for game instances, and the question naturally arises as to how good the approximation is.

Proof of the efficiency of linear rating systems relies on the principle established by Gauss as the first step in his method of least squares:  for a given set of values, the sum of squared deviations from a variable is an absolute minimum where the variable is the arithmetic mean of the set of values.  This can also be demonstrated without much difficulty for the mathematically trained as an exercise in differential calculus.  Formula [3.1] can be represented as the mean

[3.1b]          R  =  E [R_c + K(S - S_c)] ,   

and it follows directly from Gauss's principle that

                   [R  -  R_c  -  K(S - S_c)] ²      

is an absolute minimum over real values of R.  The approximation represented by [3.3]  is optimal with respect to overall results, despite the multitude of inconsistencies that may arise from actual play.