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15. A Progressive System
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From the standpoint of contestants, a rating system
balances gain against risk. A
progressive system, where rating adjustments (when they occur) are
always on the upside, is the choice of those who want to avoid risk.
Virtually any rating system can be made progressive, but the
Berkin System is especially convenient in this regard. We recall the basic performance formula as [15.1] R = S(RcS) / L , L > 0 . Let Lo be the total number of losses which a player incurs against a given sequence of No opposition ratings. Then for the event ratings in this sequence [15.2] DR = S(RcS) / Lo , Lo > 0 , which is always nonnegative. Lo varies with the strength of the rated player against the given sequence of opponents. We may estimate its value as the total of estimated losses in each event. Expressing this estimate in terms of the sampling constant No, [15.3] DR = S(RcS) / [No (1 – Pe)] , where [15.4] Pe = Ro / (Ro + ERc) . Formula [15.3] by itself is the basic tool of an effective progressive system. As we have seen, the sampling constant No essentially provides a means of rating attenuation (see Attenuation). There is consequently no need for a separate performance rating formula. The Progressive System, thus defined, compares well with the Elo System, and even with the Dual Ratio System examined in the last section, under the simulation previously described (see A Test). Figure 5 presents a comparison of standard deviations of percentage expectancies calculated by [15.4] from the percentage expectancies of the predefined crosstable with s = .5. Figure 7 is the same comparison with s = .3. Results are corroborated by Spearman's rank-difference correlation, presented in Figure 6 and Figure 8. A drawback of any progressive system is that the growth of ratings is virtually unconstrained, and ratings may consequently become huge over time. A reasonable workaround is to publish ratings in logarithmic form.
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