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15. A Progressive System
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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. [15.1] R = S[RcS] / No 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 = RcS / No , 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 No, and [15.3] DR1 + DR2 + . . . + DRN = S[RcS] / No . 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] Dloga 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] Dloga R @ DR / (R . ln a) . Substituting [15.2] into this formula, [15.6] Dloga R @ RcS / (RNo . 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 . aRc-R / (No . ln a) , which gives a manageable progressive formula for logarithmic ratings to base a. |