4. Interval Ratings

A statistical theory of ratings begins with the discovery that a typical rating system, such as the Ingo, is tacitly treating of differences in percentage score.  Consider a sequence of games between two players, A and B.  If A's percentage score in this sequence is P, then B's score is clearly 1 – P, and the difference in percentage score P(A) – P(B) is

                         P – (1 – P)  =  2P – 1  =  2(P - .50) .

This last expression recalls the basic Ingo formula [1.1] and suggests the source of  the Ingo System's power.  We can generalize this discovery by a principle of interval systems which states that differences in rating reflect differences in percentage score.  A rating formula that captures this principle is

[4.1]          R  =  ER_c + K(P - P_c) .

The symbol E (expected) is used here to designate an arithmetic mean, so that  ER_c represents the mean rating of opponents.  K is an arbitrary constant (-50 in the Ingo System).  P and P_c are the percentage scores of player and opposition.  The difference may also be written as 

[4.2]          2P - 1  =  (W - L) / N

for points won and lost out of N games.  A term of convenience for this difference in percentage score is relative performance.  

The effect of [4.1] when applied to a competing field is to generate rating differences in proportion to relative performance, which is more easily seen by writing the formula as

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

The latter may be viewed as an equation of means over game instances,

[4.4]           E[R - R_c]  =  E[K(S - S_c)],

where the relative score, S - S_c, in chess evaluates to 1, -1, or 0 (for a win, loss, or draw respectively).  For individual games, rating difference may be thought of as predicting relative score as an approximation, and the question naturally arises as to how good this 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 real values, the sum of squared deviations from a real variable is an absolute minimum where the variable is the arithmetic mean of the set of values, which can be demonstrated by the mathematically trained as an exercise in differential calculus. 

If the rating of formula [4.1] is represented as the mean

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

it follows directly from Gauss's principle that

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

is an absolute minimum over real values of R when [4.1] holds true.  We have only to regroup terms as

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

to show that difference in rating predicts relative score.  This approximation is optimal for ratings calculated by the general linear formula, regardless of the consistency of data on which the ratings are based.