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14. 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 questions of
accuracy. [14.1] R = [S(RcS)] / No 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 = (RcS) / No , 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 No, and [14.3] DR1 + DR2 + . . . + DRN = S[RcS] / No . 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) ≈ RcS / (RNo) . If the rating variables are assumed to be logarithmic, R may be substituted for (ln R), eR for R, and eRc for Rc, giving [14.7] DR ≈ S . eRc-R / No . Ratings may be initialized to zero. The change in rating for a result between two players new to the system would be S / No. A newcomer defeating a player rated lnNo would gain one point. |