Mate in 2,  J. Buchwald

4. Ratio Rating Systems

Ratio scales for physical measurements generally convey more information than interval scales, but whether this holds for ratings is an open question.  Elo was impressed enough with the potential of ratio scales to develop a ratio version of his system, eventually implemented by the USCF.  Using logarithms, he created a near clone of his interval system, one that presumably derives percentage expectancies from rating differences in a more fundamental way.  The new system nevertheless treats of rating differences, which ostensibly makes it an interval system. 

The basic ratio formula is

[4.1]        R   =   ER_c (P / P_c) ,            P_c > 0.

Like its interval counterpart, the ratio formula can be written as an arithmetic mean of game instances:

[4.1a]        R   =   E [R_c (P / P_c)] .

Relative performance here must be taken to be a constant over the set of instances.  The minimizing influence of averaging nevertheless applies, and

               [R - R_c (P / P_c)] ²

is a minimum for the calculated value of R.

If the performance ratio is expressed as a logarithm, one arrives at a basic formula of the Elo System:

[4.2]         R   =   ER_c + 400 log₁₀ (P / P_c) ,           P_c > 0.

This is equivalent to taking the logarithm of the performance ratio to a very small base (1.005773).  In the Elo System ratings are implicitly viewed as logarithmic, and ER_c is the logarithm of a  geometric average.  Regarding the second term on the right as a constant over game instances, 

               [R - R_c  - 400 log₁₀(P / P_c)] ² 

is a minimum over real values of R.  The efficiency of the Elo System is thus seen to be a consequence of its use of arithmetic averaging, as is true of rating systems in general.