5. Ratio Ratings

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

                     =   ER_c (P / 1 - P) ,        P < 1 ,

                     =   ER_c (W / L) ,            L > 0 ,

with the same variables as the interval formula.  Here relative performance cannot be defined for individual games because of the possibility of division by zero, and it must therefore be treated as a constant.  Moving R to the right and relative performance to the left and reversing the equation,

[5.2]          ER_c / R  =  L / W

which may also be written

 [5.3]          E (R_c / R)  =  L / W.

We now have an equation of means to which Gauss' principle can be applied.

                 S (R_c / R  -  L / W) ²

is a minimum over real values of R for the calculated value.  Rating ratios in individual games may thus be said to predict overall relative performance.  The reciprocals in this result can be eliminated by using a harmonic mean in [5.1], but this would be creating practical problems to avoid a theoretical quirk.

Elo regarded ratio systems as an important advance over interval systems, perhaps because a ratio scale for physical measurement has distinct advantages.  The two rating systems are distinguished primarily by the fact that aggregate scores can be related either as ratios or differences.  When logarithms are employed to deal with the large values of a ratio system, the distinction becomes somewhat blurred, and it is not altogether clear that a ratio system is to be preferred.

Elo's focus in his ratio system was on established ratings.  Performance ratings were left to be estimated by linear processes, but a performance rating can be derived from his formula (46),

[5.4]         P(D)  =  1 / (1 + 10^-D/2C) .

Solving for D gives

[5.5]         D(P)  =  2C ^. log₁₀(P / P_c) ,

where C is the class interval of 200 rating points.  A performance formula would therefore be

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

The conditions on this formula are no doubt reason enough to avoid it.  The formula follows directly from [5.1] by the application of logarithms, although its use of arithmetic averaging implies a geometric average in the original formula.  There is no special notation for ratings as logarithms. The base of logarithms b may be calculated from the coefficient of relative performance,

[5.7]         log_b R  =  400 log₁₀ R ,

and by the rules of logarithms

[5.8]         log_b R  =  log₁₀ R / log₁₀ b .

Setting equals to equal,

[5.9]         log₁₀ R / log₁₀ b  =  400 log₁₀ R .

Consequently,

[5.10]         log₁₀ b  =  1/400

and

[5.11]         b  =  10^1/400

or about 1.00577.  Aside from inaccuracies arising from averaging, Elo's ratio system appears to be a logarithmic version of the system described here, and this confluence of ideas is telling.  It suggests that reliance on a probability distribution based on the logistic function is unnecessary.  A performance formula derived from [5.4], as noted, is not actually used in the system.  Established ratings do the main work, and their formulas make assumptions about percentage expectancy that are characteristic of the Elo System.  We shall postpone definite conclusions till the discussion of established ratings.  It is perhaps worth noting that taking the logarithm of [5.1] does not produce a linear formula equivalent to [4.1] since relative performance in the result would be the nonlinear

                  log_b(P) - log_b(P_c) .

Another ratio system, one with unique properties, is the Berkin System, which will be discussed under that heading.