15. 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 the problem of accuracy.  A workable system would require careful management over long periods of time since a rating would progress over the entire career of a chess player.  Virtually any rating system can be made progressive simply by ignoring negative results, but the Berkin System seems especially suitable for this adaptation.  The formula

[15.1]        R  =  S[R_cS] / N_o

is equivalent to the basic Berkin formula [14.2] when L reaches the value of the arbitrary constant.  Thereafter the change formula for game-by-game results is simply

[15.2]       DR  =  R_cS / N_o ,

which is always nonnegative.  Note that [14.2]  results in rating decreases for losses.  This can be avoided by applying [15.2] 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

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

Formula [15.2] thus turns out to be an all-purpose progressive formula of remarkable simplicity.  It is simulated in Table 10, which is discussed in Rating by Single Games.

A drawback of progressive systems is that the growth of ratings is unconstrained, and ratings may consequently become huge over time.  A rating by [15.2] increases exponentially, and a mere thousand wins or so against equal-rated opponents would bring it to the point of overflow as a double precision variable in ordinary computer languages.  A logarithmic version of the system is possible, however, using an approximation from calculus.  The limit

[15.4]        Dlog_a R / DR  =  1 / (R ^. ln a) 

as DR goes to zero is a consequence of the derivative of a logarithmic function..  For small values of  DR then

[15.5]       Dlog_a R   @  DR / (R ^. ln a)  .

Substituting [15.2] into this formula,

[15.6]      Dlog_a R   @  R_cS / (RN_o ^.  ln a) .

Even now calculations might require the use of unwieldy values.  If the rating variables are assumed to be logarithmic,

[15.7      DR   @  S ^. a^Rc-R / (N_o ^.  ln a) ,

which gives a manageable progressive formula for logarithmic ratings to base a.