14. A Progressive System

In a progressive rating system players never lose rating points. This would be an obvious boon to organized chess if it were not for questions of accuracy.  A workable progressive system would require careful management at the very least. Virtually any rating system can be made progressive simply by ignoring negative results, but the Berkin System seems especially suitable for this purpose.  The formula

[14.1]        R  =  [S(R_cS)] / N_o

is equivalent to the basic Berkin formula [13.2] when L reaches the value of the arbitrary constant. As we saw in Established Ratings, the sample size in cumulative averaging can be maintained at a constant value, if sufficiently large, with negligible loss of accuracy.  Thereafter the Berkin change formula for game-by-game results is

[14.2]       DR  =  (R_cS) / N_o ,

which is always nonnegative.  Note that the basic performance formula [13.2] produces rating decreases for losses.  This can be avoided by applying [14.2] from the outset of the rating process, which is justified by the fact that after N successive applications, where N may be very large, the number of lost points eventually reaches N_o, and

[14.3]       DR₁ + DR₂ + ^{. . .} + DR_N  =   S[R_cS] / N_o .

 If the inaccuracies involved are deemed acceptable, Formula [14.2] becomes an all-purpose progressive formula.  A drawback of progressive systems is that the growth of ratings is exponential, and ratings may consequently become huge over time. A logarithmic version of the system is possible, using an approximation from calculus.  The limit

[14.4]        lim D(ln R) / DR  =  1 / R

as DR goes to zero defines the derivative of a natural logarithm by the delta method.  For small values of DR this gives the approximation

[14.5]        D(ln R)  ≈  DR / R  .

Substituting by [14.2] into this formula,

[14.6]       D(ln R)  ≈  R_cS / (RN_o) .

 If the rating variables are assumed to be logarithmic, R may be substituted for (ln R), e^R for R, and e^Rc for R_c, giving

[14.7]        DR  ≈  S ^. e^Rc-R / N_o . 

Ratings may be initialized to zero. The change in rating for a result between two players new to the system would be S / N_o.  A newcomer defeating a player rated lnN_o would gain one point.